Concepts

The canonical list of every concept the curriculum names. Each maps to at most one unit. Browse 523 concepts; 438 have shipped units.

• spin-geometry.clifford.clifford-algebraopen unit 03.09.02 →

  Clifford algebra

  Graded algebra structure; relation to exterior algebra and Grassmann algebra; real vs complex Clifford algebras.

  requires: linalg.bilinear-form, linalg.vector-space, algebra.tensor-algebra, algebra.quotient-algebra

• spin-geometry.clifford-chessboardopen unit 03.09.11 →

  Clifford algebra classification — the 8×8 chessboard

  ABS classification of ${\mathrm{Cl}}_{p,q}$ as matrix algebras over $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$. Bridging identity ${\mathrm{Cl}}_{r+1,s+1}\cong {\mathrm{Cl}}_{r,s}\otimes {\mathrm{Cl}}_{1,1}$. Real eight-fold periodicity ${\mathrm{Cl}}_{p,q+8}\cong {\mathrm{Cl}}_{p,q}\otimes M_{16}(\mathbb{R})$; complex two-fold periodicity. Module quotient ${\hat{\mathfrak{M}}}_n\cong K{O}^{-n}(\mathrm{pt})$.

  requires: spin-geometry.clifford.clifford-algebra, algebra.tensor-algebra, algebra.quotient-algebra

• spin-geometry.kr-theoryopen unit 03.09.12 →

  KR-theory and the $(1,1)$-periodicity theorem

  Atiyah's bigraded $K{R}^{p,q}$ unifying $K$, $KO$, and $KSp$. The $(1,1)$-periodicity $K{R}^{p,q+1}\cong K{R}^{p+1,q}$ is the $K$-theoretic incarnation of the Clifford bridging identity. Bott periodicity in eight steps via the volume element of ${\mathrm{Cl}}_{0,8}$.

  requires: spin-geometry.clifford-chessboard, k-theory.bott-periodicity, topology.classifying-space

• spin-geometry.trialityopen unit 03.09.13 →

  Triality on $\mathrm{Spin}(8)$ and exceptional Lie group constructions

  Triality as the outer automorphism of $\mathrm{Spin}(8)$ permuting the vector representation $V$ and the two half-spinor representations ${\mathfrak{S}}^{\pm }$. Spinor squaring builds the octonions and constructs $G_2\subset \mathrm{Spin}(7)\subset \mathrm{Spin}(8)$, $F_4$, $E_6,E_7,E_8$ via the Freudenthal magic square.

  requires: spin-geometry.spin-group, spin-geometry.clifford-chessboard, lie.classification.cartan-weyl

• spin-geometry.dirac.dirac-operatoropen unit 03.09.08 →

  Dirac operator on a spin manifold

  First-order elliptic operator on the spinor bundle, $D=c\circ {\nabla }^S$. Principal symbol is Clifford multiplication, hence elliptic. Role in Lichnerowicz formula, positive scalar curvature, and Atiyah-Singer index theorem.

  requires: spin-geometry.clifford.clifford-algebra, spin-geometry.structure.spin-structure, spin-geometry.spinor-bundle, diffgeo.connection.connection, diffgeo.elliptic-operators, functional-analysis.fredholm.operators

• foundations.real-numbersopen unit 00.01.01 →

  Real numbers, integers, rationals

  Number-system hierarchy $\mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}$. Field axioms (closure, associativity, commutativity, distributivity, identities, inverses) and order axioms for $\mathbb{Q}$ and $\mathbb{R}$. Completeness axiom (least upper bound) and Archimedean property. Two parallel constructions of $\mathbb{R}$ from $\mathbb{Q}$: Dedekind cuts and Cauchy-sequence completion modulo null sequences. Uniqueness of complete ordered fields up to unique order-isomorphism. Irrationality of $\sqrt{2}$ (classical contradiction proof). Density of $\mathbb{Q}$ in $\mathbb{R}$ via Archimedean property. Cardinality jump $|\mathbb{Z}|=|\mathbb{Q}|={\aleph }_0$ versus $|\mathbb{R}|=2^{{\aleph }_0}$ via Cantor's diagonal argument. Foundational floor for the algebra strand (linalg.field), the analysis strand (metric-space completion), and the complex-analysis strand.

  requires: none

• foundations.absolute-valueopen unit 00.01.02 →

  Absolute value and the triangle inequality

  Definition $|x|=x$ if $x\ge 0$ else $-x$, the four signature properties (positivity, multiplicativity, triangle inequality, reverse triangle), and the case-on-sign proof of the triangle inequality. The triangle inequality is the load-bearing axiom — the absolute-value function on $\mathbb{R}$ is the prototype of a norm, and the metric $d(x,y)=|x-y|$ is the prototype of a metric. Master-tier scope adds the ${\ell }^p$ norms via Minkowski's inequality, the discrete metric as a non-norm, the $p$-adic absolute value with its ultrametric strengthening, and Ostrowski's theorem classifying all non-trivial absolute values on $\mathbb{Q}$ as either Euclidean or $p$-adic. Foundational floor for sequence convergence (analysis.sequence-limit), metric-space axioms (topology.metric-space), normed vector spaces (functional-analysis.banach-spaces), and number-theoretic completions (number-theory.p-adic-numbers).

  requires: foundations.real-numbers

• foundations.polynomialsopen unit 00.01.03 →

  Polynomials and rational expressions

  A polynomial over a field $k$ is a finite formal sum $p(x)={\sum }_{i=0}^n{a}_i{x}^i$ with $a_i\in k$ and $a_n\ne 0$ (the leading coefficient); $\mathrm{deg}p=n$. The set $k[x]$ is a commutative ring under term-by-term addition and convolution multiplication. For $k$ a field, $k[x]$ is a Euclidean domain via the polynomial division algorithm: for any $p,q\in k[x]$ with $q\ne 0$ there exist unique $a,r\in k[x]$ with $p=aq+r$ and $\mathrm{deg}r<\mathrm{deg}q$. Factor theorem: $r$ is a root of $p$ iff $(x-r)\mid p(x)$. A degree-$n$ polynomial has at most $n$ roots in any field. Rational expression: ratio $p/q$ with $q\ne 0$, organised into the field of fractions $k(x)$ when one quotients by the equivalence $p_1/q_1\sim p_2/q_2⟺p_1{q}_2=p_2{q}_1$. Master-tier scope: $k[x]$ Euclidean $\Rightarrow$ PID $\Rightarrow$ UFD, with the chain breaking for $k[x_1,\dots ,x_n]$ ($n\ge 2$) which is UFD (Gauss) but not PID; fundamental theorem of algebra (every non-constant $p\in \mathbb{C}[x]$ has a root, Gauss 1799 thesis); real-coefficient polynomials factor into real linear and irreducible real quadratic factors via conjugate-pair roots; algebraic closure $\bar{k}$ and the algebraic numbers $\bar{\mathbb{Q}}\subset \mathbb{C}$; partial-fraction decomposition over $k(x)$ as the dual to division. Originators: Diophantus Arithmetica (~250 CE); al-Khwārizmī al-jabr (~825 CE); Cardano-Tartaglia 1545 Ars Magna (cubic formula); Ferrari 1545 (quartic formula); Abel 1824 / Ruffini 1799 (no general quintic formula in radicals); Galois 1832 (the symmetry theory behind solvability); Gauss 1799 (fundamental theorem of algebra). Foundational floor for the algebra strand (linalg.field, polynomial-ring algebra), complex analysis (complex-analysis.fundamental-theorem-of-algebra), and integration of rational functions (partial fractions).

  requires: foundations.real-numbers

• foundations.linear-equations-lineopen unit 00.03.01 →

  Linear equations and the line

  A linear equation in $n$ variables over a field $k$ is an equation $a_1{x}_1+\cdots +a_n{x}_n=b$ with coefficients $a_i,b\in k$ and the $a_i$ not all zero; the solution set in $k^n$ is a hyperplane (codimension-$1$ affine subspace). For $n=2$ over $\mathbb{R}$, the solution set of $ax+by=c$ is a line in ${\mathbb{R}}^2$, equivalently the affine $1$-flat $\{p_0+tv:t\in \mathbb{R}\}$ where $p_0$ is any solution and $v$ is a direction vector orthogonal to the normal $(a,b)$. Slope-intercept form $y=mx+b^{\prime }$ exists for non-vertical lines, with $m$ the slope and $b^{\prime }$ the y-intercept; two non-vertical lines are parallel iff their slopes agree and perpendicular iff $m_1{m}_2=-1$. Classification of two-line intersections: the system $\{a_{1}x+b_{1}y=c_1,a_{2}x+b_{2}y=c_2\}$ has a unique solution iff the determinant $D=a_1{b}_2-a_2{b}_1\ne 0$ (Cramer's rule: $x=(c_1{b}_2-c_2{b}_1)/D$, $y=(a_1{c}_2-a_2{c}_1)/D$); if $D=0$, the lines are parallel (no solution) or coincident (every common point), distinguished by whether the triples $(a_i,b_i,c_i)$ are proportional. Master-tier scope: linear-versus-affine distinction (linear subspace through origin, affine subspace = translate); hyperplanes and flats in ${\mathbb{R}}^n$; Frobenius / Kronecker-Capelli theorem ($Ax=b$ consistent iff $\mathrm{rank}(A)=\mathrm{rank}([A\mid b])$); Cramer's rule for $n\times n$ systems via the adjugate-inverse formula; projective line ${\mathbb{P}}^1(\mathbb{R})=\mathbb{R}\cup \{\infty \}$ with $\mathrm{PGL}(2,\mathbb{R})$ acting by Möbius transformations, sharply $3$-transitive on triples of distinct points; linear codes over ${\mathbb{F}}_q$; convex polytopes as bounded intersections of half-spaces (Minkowski-Weyl duality). Originators: Euclid Elements Book I (~300 BCE) for the geometric line; Descartes 1637 La Géométrie and Fermat ~1636 letters (algebra-geometry correspondence); Cramer 1750 (determinant rule); Cauchy 1812 (systematic determinant theory); Klein 1872 Erlangen Programm (projective hierarchy). Foundational floor for the determinant in linear-algebra.determinant, linear maps and rank-nullity in linear-algebra.linear-transformation-rank-nullity, multivariable Jacobian determinants in multivariable-calculus.jacobian, Möbius transformations and the Riemann sphere in complex-analysis.mobius-transformations, and the affine-versus-linear distinction throughout differential geometry.

  requires: foundations.real-numbers

• foundations.quadratic-formulaopen unit 00.03.02 →

  Quadratic equations and the quadratic formula

  A quadratic equation over a field $k$ of characteristic not two is $a{x}^2+bx+c=0$ with $a,b,c\in k$ and $a\ne 0$. Completing the square rewrites the equation as $(x+b/(2a){)}^2=(b^2-4ac)/(4a^2)$, and taking square roots gives the quadratic formula $x=(-b\pm \sqrt{\Delta })/(2a)$, where $\Delta =b^2-4ac$ is the discriminant. Over the real numbers the discriminant classifies the solution count: $\Delta >0$ gives two distinct real roots, $\Delta =0$ a single repeated real root (the vertex of the parabola $y=a{x}^2+bx+c$ touches the $x$-axis), and $\Delta <0$ no real roots (two complex-conjugate solutions $x=(-b\pm i\sqrt{-\Delta })/(2a)$). The Vieta formulas express the elementary symmetric functions of the two roots in terms of the coefficients: $x_1+x_2=-b/a$ and $x_1{x}_2=c/a$. Master-tier scope: the discriminant of a degree-$n$ polynomial ${\Delta }_n=a_n^{2n-2}{\prod }_{i<j}(r_i-r_j{)}^2$ as a symmetric function of the roots, expressible in the coefficients by Vieta; Cardano's 1545 cubic and Ferrari's 1540 quartic formulas; Abel-Ruffini 1824 and Galois 1832 (no general formula in radicals for the quintic and beyond, originating Galois theory); the binary quadratic form $Q(x,y)=a{x}^2+bxy+c{y}^2$ and the conic-section discriminant $b^2-4ac$ classifying hyperbola / parabola / ellipse; the general quadratic form on ${\mathbb{R}}^n$ as $Q(v)=v^{\top }Av$, Sylvester's law of inertia on the signature $(p,q)$ under change of basis; Gauss's 1801 quadratic reciprocity for Legendre symbols; the fundamental theorem of algebra (Gauss 1799) extending complete factorisation from quadratics to all polynomials over $\mathbb{C}$. Originators: Babylonian scribes ~2000 BCE (geometric completion-of-the-square on specific quadratics); al-Khwārizmī ~825 CE Kitāb al-jabr wa-l-muqābala (systematic Arabic method, with the word algebra deriving from al-jabr); Cardano 1545 Ars Magna (cubic formula and modern algebraic form); Viète 1591 In artem analyticen isagoge (letter notation for coefficients, sum-and-product of roots); Gauss 1799 doctoral thesis (fundamental theorem of algebra); Gauss 1801 Disquisitiones (quadratic reciprocity). Foundational floor for conic-section classification (foundations.conic-sections), bilinear and quadratic forms (linalg.bilinear-quadratic-form), the algebra-strand discriminant of a polynomial, the Galois-theoretic solvability story (algebra.galois-theory), and the algebraic-closure picture (complex-analysis.fundamental-theorem-of-algebra).

  requires: foundations.polynomials

• foundations.inequalitiesopen unit 00.04.01 →

  Inequalities (linear and quadratic)

  An inequality in one real variable replaces the equals sign in an equation by one of $<,\le ,>,\ge$. The solution set is a region of the number line rather than a finite set of points. The basic manipulation rules match those for equations except that multiplying or dividing both sides by a negative number reverses the direction of the inequality, since the order on $\mathbb{R}$ is incompatible with multiplication by negatives in the opposite sense from addition. A linear inequality $ax+b\le c$ has solution set $\{x:x\le (c-b)/a\}$ when $a>0$ (and $\{x:x\ge (c-b)/a\}$ when $a<0$), a half-line in either case. A quadratic inequality $a{x}^2+bx+c\le 0$ is solved by sign analysis: factor as $a(x-r_1)(x-r_2)$ when the discriminant $\Delta =b^2-4ac$ is non-negative, then read the sign of the product in each of the three regions cut off by the roots. The solution set is a closed interval when $a>0$ and $\Delta >0$, the complement of an open interval when $a<0$ and $\Delta >0$, a single point when $\Delta =0$, and either the entire line or the empty set when $\Delta <0$. Named load-bearing inequalities at this tier: the triangle inequality $|x+y|\le |x|+|y|$ on $\mathbb{R}$, the arithmetic-mean-geometric-mean inequality $(x_1+\cdots +x_n)/n\ge (x_1\cdots x_n{)}^{1/n}$ for non-negative reals with equality iff all $x_i$ coincide, and the Cauchy-Schwarz inequality $|u\cdot v|\le \Vert u\Vert \Vert v\Vert$ for vectors in a real or complex inner-product space with equality iff the vectors are linearly dependent. The standard proof of Cauchy-Schwarz observes that the quadratic $t\mapsto \Vert u-tv{\Vert }^2$ is non-negative everywhere, hence its discriminant in $t$ is non-positive — a direct invocation of the discriminant trichotomy from foundations.quadratic-formula. Master-tier scope: Hölder's inequality $\sum |x_i{y}_i|\le (\sum |x_i{|}^p{)}^{1/p}(\sum |y_i{|}^q{)}^{1/q}$ for conjugate exponents $1/p+1/q=1$ (with Cauchy-Schwarz as the case $p=q=2$); Minkowski's inequality $(\sum |x_i+y_i{|}^p{)}^{1/p}\le (\sum |x_i{|}^p{)}^{1/p}+(\sum |y_i{|}^p{)}^{1/p}$ supplying the triangle inequality on ${\ell }^p$ and $L^p$; Jensen's inequality $\varphi (\mathbb{E}[X])\le \mathbb{E}[\varphi (X)]$ for $\varphi$ convex (generalising AM-GM via $\varphi =-\log$); the isoperimetric inequality on ${\mathbb{R}}^2$ (and Lévy-Gromov on manifolds); polynomial inequalities and the Tarski-Seidenberg theorem (1948) on the decidability of the first-order theory of $\mathbb{R}$ via quantifier elimination over real-closed fields; semi-algebraic sets as a category. Originators: Cauchy 1821 Cours d'analyse (the original finite-sum form of the inequality, in the language of bilinear sums); Schwarz 1885 (the inner-product generalisation used in his minimal-surfaces work); Hölder 1889 (the eponymous inequality, generalising Cauchy-Schwarz to conjugate exponents); Minkowski 1896 (the triangle inequality in ${\ell }^p$); Jensen 1906 (the convex-function inequality and its probabilistic formulation); Tarski 1948 (the decision procedure for the elementary theory of $\mathbb{R}$). Foundational floor for the metric-space triangle inequality (metric-spaces.metric-space), the $L^p$-norm theory (functional-analysis.lp-spaces), inner-product geometry (functional-analysis.hilbert-space — the Cauchy-Schwarz inequality is the reason the angle $\cos \theta =⟨u,v⟩/(\Vert u\Vert \Vert v\Vert )$ takes values in $[-1,1]$), and concentration inequalities in probability (probability.concentration).

  requires: foundations.quadratic-formula

• linalg.fieldopen unit 01.01.01 →

  Field

  Commutative division rings with $0\ne 1$, field homomorphisms, characteristic, prime fields, finite fields in elementary examples, and field extensions as the scalar context for vector spaces and algebras.

  requires: set-theory.function

• linalg.vector-spaceopen unit 01.01.03 →

  Vector space over a field

  The single most-reused concept in the curriculum. Definition over arbitrary field, not just $\mathbb{R}$ or $\mathbb{C}$. Counterexamples: function spaces, polynomial rings as $K$-vector spaces.

  requires: linalg.field, linalg.set-and-function

• linear-algebra.subspace-basis-dimensionopen unit 01.01.04 →

  Subspace, basis, dimension

  The structural triple every linear-algebra computation rests on. Subspace = subset closed under the two operations; basis = linearly independent spanning set; dimension = common cardinality of any basis. Steinitz replacement theorem (1913) is the load-bearing result. Generalises to rank over a PID and to module-theoretic invariants for general modules; in the categorical view, finite-dimensional $K$-vector spaces form a category equivalent to $\mathbb{N}$ with morphisms given by matrices, and $K_0(K{\text{-Vect}}^{\mathrm{fd}})=\mathbb{Z}$.

  requires: linalg.vector-space

• linear-algebra.linear-transformation-rank-nullityopen unit 01.01.05 →

  Linear transformation: kernel, image, rank-nullity

  The fundamental dimension-counting theorem for linear maps. Linear map $T:V\to W$ is the morphism in the category of vector spaces; kernel $\ker T$ and image $\mathrm{im}T$ are its canonical sub-objects. Rank-nullity $\dim V=\dim (\ker T)+\dim (\mathrm{im}T)$ packages dimension as a conserved quantity through any linear map. Categorical sharpening: in $K\text{-Vect}$ every short exact sequence splits, so $V\cong \ker T\oplus \mathrm{im}T$ — the splitting fails for general modules (e.g. $0\to \mathbb{Z}\stackrel{2}{\to }\mathbb{Z}\to \mathbb{Z}/2\to 0$). Functional-analytic generalisation: the index $\mathrm{ind}(T)=\dim \ker T-\dim \mathrm{coker}T$ for Fredholm operators (Atiyah-Singer). Pairs with the first isomorphism theorem $V/\ker T\cong \mathrm{im}T$.

  requires: linear-algebra.subspace-basis-dimension

• linear-algebra.determinantopen unit 01.01.07 →

  Determinant: axiomatic + expansion + properties

  The scalar invariant of a square matrix. Three equivalent definitions: axiomatic (the unique multilinear-alternating-normalised function on rows / columns), Leibniz permutation sum $\det A={\sum }_{\sigma \in S_n}\mathrm{sgn}(\sigma ){\prod }_i{a}_{i,\sigma (i)}$, and recursive Laplace / cofactor expansion. Geometric content: signed volume of the parallelepiped spanned by the columns; $\det A=0$ iff the columns are linearly dependent. Multiplicativity $\det (AB)=\det A\cdot \det B$ from the axiomatic characterisation. Builds toward (i) the change-of-variables Jacobian $|\det J|$ in multivariable calculus, (ii) the characteristic polynomial ${\chi }_A(t)=\det (tI-A)$ for eigenvalues, (iii) the top exterior power ${\Lambda }^{n}A:{\Lambda }^{n}V\to {\Lambda }^{n}V$ acting as multiplication by $\det A$, and (iv) the determinant line bundle $\det V={\Lambda }^{\mathrm{rank}V}V$ in geometry. Originators: Seki Takakazu (1683) and Leibniz (1693); Cauchy (1812) unified the modern theory; Cayley (1858) the matrix notation; Vandermonde (1771) the special case; Artin (1942) the modern axiomatic presentation; Bourbaki the multilinear-algebra framing.

  requires: linear-algebra.linear-transformation-rank-nullity

• linear-algebra.eigenvalue-eigenvectoropen unit 01.01.08 →

  Eigenvalue, eigenvector, characteristic polynomial

  The spectral structure of a linear operator on a finite-dimensional vector space. Eigenvalue equation $Tv=\lambda v$ with $v\ne 0$; eigenspace $E_{\lambda }=\ker (T-\lambda I)$; characteristic polynomial ${\chi }_T(t)=\det (tI-T)$ with eigenvalues as roots; algebraic multiplicity vs geometric multiplicity. Key result: eigenvectors for distinct eigenvalues are linearly independent, so an operator with $n=\dim V$ distinct eigenvalues is diagonalisable. Cayley-Hamilton: every operator satisfies its own characteristic polynomial, ${\chi }_T(T)=0$; the spectral theorem (finite-dim) refines this for self-adjoint operators on an inner-product space, giving an orthonormal eigenbasis with real eigenvalues; Jordan canonical form classifies operators on a finite-dim $\mathbb{C}$-vector space up to similarity by Jordan-block data, with the Segre characteristic recording block sizes. Generalises to Banach-space spectral theory (point / continuous / residual spectrum), the resolvent $(T-\lambda I{)}^{-1}$ as an analytic function off the spectrum, spectral measures for self-adjoint operators on Hilbert space (Stone-von Neumann), and Frobenius eigenvalues for $\ell$-adic Galois representations. Originators: Cauchy (1829) for symmetric matrices; Cayley (1858) and Frobenius (1878) for Cayley-Hamilton; Jordan (1870) for Jordan canonical form; Hilbert (1904-) for the infinite-dim spectral theorem; von Neumann (1929) for self-adjoint spectral measures.

  requires: linear-algebra.determinant

• linear-algebra.jordan-canonical-formopen unit 01.01.11 →

  Jordan canonical form and minimal polynomial

  The complete similarity classification of linear operators on a finite-dimensional $\mathbb{C}$-vector space. Jordan block $J_k(\lambda )$: $k\times k$ matrix with $\lambda$ on the diagonal and $1$s on the superdiagonal. Minimal polynomial $m_T(t)$: monic generator of the ideal $\{p\in K[t]:p(T)=0\}$; $m_T\mid {\chi }_T$ by Cayley-Hamilton, and $T$ is diagonalisable iff $m_T$ is square-free. Primary decomposition: $V={⨁}_i{G}_{{\lambda }_i}$ with $G_{{\lambda }_i}=\ker (T-{\lambda }_{i}I{)}^{m_i}$ the generalised eigenspaces. Existence + uniqueness theorem: over an algebraically closed field every operator is similar to a direct sum of Jordan blocks, unique up to block reordering. Segre characteristic: multiset of block sizes at each eigenvalue, encoding the partition of $\dim G_{\lambda }$ by Jordan-chain lengths. Builds toward (i) rational canonical form over arbitrary fields via companion matrices of invariant factors, (ii) Smith normal form for matrices over a PID, (iii) the structure theorem for finitely generated modules over a PID via $V\cong {⨁}_{i}K[t]/(p_i^{e_i})$, (iv) holomorphic functional calculus $f(J_k(\lambda ))={\sum }_{j<k}\frac{f^{(j)}(\lambda )}{j!}{N}^j$ used to compute $e^{tA}$, $\sin A$, $\cos A$, (v) GIT quotient ${\mathrm{GL}}_n//{\mathrm{GL}}_n$ parametrising conjugacy classes. Originators: Weierstrass (1858, elementary divisor theory); Smith (1861, PID version); Jordan (1870, Traité des substitutions); Frobenius (1879, rational canonical form); Lang (1965, modern module-theoretic packaging in Algebra Ch. III).

  requires: linear-algebra.eigenvalue-eigenvector

• linear-algebra.singular-value-decompositionopen unit 01.01.12 →

  Singular value decomposition (finite-dim)

  The factorisation $A=U\Sigma V^{\ast }$ for an arbitrary $m\times n$ matrix $A$ over $\mathbb{R}$ or $\mathbb{C}$: $U$ unitary $m\times m$, $V$ unitary $n\times n$, $\Sigma$ an $m\times n$ diagonal matrix with non-negative entries ${\sigma }_1\ge {\sigma }_2\ge \cdots \ge {\sigma }_r>0$ called the singular values of $A$, with $r=\mathrm{rank}A$. The singular values are the non-negative square roots of the eigenvalues of $A^{\ast }A$ (equivalently of $A{A}^{\ast }$); the right singular vectors (columns of $V$) are an orthonormal eigenbasis of $A^{\ast }A$, the left singular vectors (columns of $U$) are an orthonormal eigenbasis of $A{A}^{\ast }$, and the two bases are paired by $A{v}_i={\sigma }_i{u}_i$ for ${\sigma }_i>0$. Existence proof: spectral theorem on $A^{\ast }A$, then define $u_i=A{v}_i/{\sigma }_i$ and verify orthonormality. Uniqueness: the singular values are uniquely determined; $U$ and $V$ are unique up to a unitary block on each constant-singular-value subspace (and a phase rotation on simple singular values in the complex case). Companion structures: (i) Moore-Penrose pseudoinverse $A^{+}=V{\Sigma }^{+}{U}^{\ast }$ with ${\Sigma }^{+}$ the transpose of $\Sigma$ with non-zero entries reciprocated; gives the minimum-norm least-squares solution to $Ax=b$. (ii) Polar decomposition $A=QP$ with $Q=U{V}^{\ast }$ unitary and $P=V\Sigma V^{\ast }$ Hermitian positive semidefinite — the matrix analogue of the polar form $z=r{e}^{i\theta }$. (iii) Eckart-Young theorem: for any unitarily invariant norm, the best rank-$k$ approximation to $A$ is $A_k={\sum }_{i=1}^k{\sigma }_i{u}_i{v}_i^{\ast }$; in the operator norm $\Vert A-A_k\Vert ={\sigma }_{k+1}$, in the Frobenius norm $\Vert A-A_k{\Vert }_F=({\sigma }_{k+1}^2+\cdots +{\sigma }_r^2{)}^{1/2}$. (iv) Schmidt decomposition for compact operators on Hilbert space: $A={\sum }_n{\sigma }_n{u}_n{v}_n^{\ast }$ with ${\sigma }_n\to 0$, foundation for trace-class and Hilbert-Schmidt operators. (v) GL action: the bi-unitary action of $U(m)\times U(n)$ on $m\times n$ complex matrices has orbits parametrised by the tuple $({\sigma }_1,\dots ,{\sigma }_r)$ of singular values — SVD is the orbit-decomposition. Applications: principal component analysis (right singular vectors = principal directions, ${\sigma }_i^2/N$ = variances); least-squares regression via the pseudoinverse; low-rank approximation in image compression, latent semantic indexing, recommender systems; condition number $\kappa (A)={\sigma }_1/{\sigma }_r$ for numerical sensitivity; numerical rank via small-singular-value thresholding. Originators: Beltrami (1873) and Jordan (1874), independently for square matrices; Sylvester (1889) for the rectangular case; Schmidt (1907) for the integral-operator / Hilbert-space generalisation; Weyl (1912) for the modern unified treatment; Eckart-Young (1936) for the low-rank approximation theorem; Mirsky (1960) extended Eckart-Young to all unitarily invariant norms; Golub-Kahan (1965) gave the numerical SVD algorithm.

  requires: linear-algebra.eigenvalue-eigenvector

• set-theory.functionopen unit 00.02.05 →

  Function

  Functions as total single-valued relations, graphs, image, composition, identity maps, inverse criterion for bijections, and categorical monomorphism / epimorphism behavior in Set. Foundational prerequisite for vector spaces, groups, maps, and actions.

  requires: none

• linalg.bilinear-formopen unit 01.01.15 →

  Bilinear form / quadratic form

  Symmetric / antisymmetric / hermitian sub-cases. Polarization identity. Gram matrix. Signature for real symmetric forms (Sylvester's law of inertia).

  requires: linalg.vector-space, linalg.dual-space

• algebra.tensor-productopen unit 03.01.01 →

  Tensor product of vector spaces

  Universal bilinear map $V\times W\to V{\otimes }_{K}W$, pure tensors, basis $e_i\otimes f_j$, functoriality, uniqueness by representing property, and the bridge to tensor algebra, tensor powers, vector-bundle operations, and Clifford algebra.

  requires: linalg.field, linalg.vector-space

• algebra.associative-algebraopen unit 03.01.02 →

  Associative algebra

  Unital associative $K$-algebras, bilinear multiplication, central scalar action, algebra homomorphisms, kernels as two-sided ideals, commutator Lie algebra, matrix and polynomial examples, and the ambient structure for tensor and quotient algebras.

  requires: linalg.field, linalg.vector-space

• algebra.idealopen unit 03.01.03 →

  Ideal in an algebra

  Left, right, and two-sided ideals in associative algebras; kernels of algebra homomorphisms; intersections and preimages; two-sided ideals as the data needed for quotient algebra multiplication; polynomial examples and relation to Clifford algebra.

  requires: algebra.associative-algebra

• algebra.tensor-algebraopen unit 03.01.04 →

  Tensor algebra of a vector space

  $T(V)=⨁V^{\otimes n}$. Universal property among unital associative algebras under $V$. Distinguish from symmetric / exterior / Clifford algebras as quotients of $T(V)$.

  requires: linalg.vector-space, linalg.tensor-product

• algebra.quotient-algebraopen unit 03.01.05 →

  Quotient algebra by a two-sided ideal

  Universal property: factoring through the quotient is equivalent to killing the ideal. Foundational for: Clifford algebras (quotient of $T(V)$), exterior algebras, polynomial rings modulo relations.

  requires: algebra.associative-algebra, algebra.two-sided-ideal

• algebra.groupopen unit 01.02.01 →

  Group

  Algebraic groups in the elementary sense: identity, inverse, associativity, homomorphisms, subgroups, kernels, quotient-ready normal subgroups, and the foundation for group actions and Lie groups.

  requires: set-theory.set-and-function

• algebra.group-actionopen unit 03.03.02 →

  Group action

  Left and right group actions, orbit-stabilizer theorem, torsors, equivariance, homogeneous spaces, and the principal-bundle fiber action.

  requires: algebra.group, set-theory.set-and-function

• lie-groups.orthogonal-groupopen unit 03.03.03 →

  Orthogonal group

  Orthogonal group of a real inner-product space, matrix equation $A^{T}A=I$, determinant components, special orthogonal group, Lie algebra of skew-symmetric matrices, and role in frame bundles and spin geometry.

  requires: algebra.group, linalg.bilinear-form, lie-groups.lie-group

• spin-geometry.structure.spin-structureopen unit 03.09.04 →

  Spin structure on an oriented Riemannian manifold

  Lift of the orthonormal frame bundle through Spin(n) → SO(n). Existence obstructed by w_2; classification (when nonempty) by H^1(M; Z/2). Pin± analogues for non-orientable case.

  requires: spin-geometry.clifford.clifford-algebra, spin-geometry.spin-group, bundle.principal-bundle, bundle.frame-bundle.orthonormal, topology.cover.double-cover, char-classes.stiefel-whitney

• topology.cover.double-coveropen unit 03.05.05 →

  Double cover

  Two-sheeted covering maps, fibers with two points, deck involutions, relation to principal $\mathbb{Z}/2$-bundles, and the role of $\mathrm{Spin}(n)\to \mathrm{SO}(n)$ as the double cover behind spin structures.

  requires: topology.topological-space, topology.covering-space

• topology.continuous-mapopen unit 02.01.02 →

  Continuous map

  Continuity between topological spaces, preimage characterization, composition, homeomorphisms, metric-space epsilon-delta comparison, and use in pullbacks and classifying maps.

  requires: topology.topological-space

• bundle.frame-bundle.orthonormalopen unit 03.05.03 →

  Orthogonal frame bundle

  Orthonormal frames of a Riemannian vector bundle, principal $\mathrm{O}(n)$-bundle structure, oriented $\mathrm{SO}(n)$ reduction, tautological frame action, and use as the bundle lifted by a spin structure.

  requires: bundle.principal-bundle, bundle.vector-bundle, diffgeo.smooth-manifold, lie-groups.orthogonal-group

• diffgeo.stokes-theoremopen unit 03.04.05 →

  Stokes' theorem

  ${\int }_{M}d\omega ={\int }_{\partial M}\omega$ on oriented compact manifolds with boundary. Unifies fundamental theorem of calculus, Green's, classical Stokes, divergence theorem. Extends to chains (de Rham theorem) and supports Poincaré duality.

  requires: diffgeo.differential-forms, topology.integration-on-manifolds, diffgeo.exterior-derivative, diffgeo.smooth-manifold

• diffgeo.exterior-derivativeopen unit 03.04.04 →

  Exterior derivative

  Unique linear operator $d:{\Omega }^k\to {\Omega }^{k+1}$ characterised by action on functions, linearity, graded Leibniz, $d^2=0$. Local formula. Naturality (commutes with pullback). Cartan magic formula. Poincaré lemma. Maurer-Cartan equation on Lie groups. Covariant exterior derivative on bundle-valued forms.

  requires: diffgeo.differential-forms, diffgeo.smooth-manifold

• topology.homotopyopen unit 03.12.01 →

  Homotopy and homotopy group

  Homotopy as continuous deformation; homotopy equivalence; fundamental group ${\pi }_1$ via loop concatenation; higher homotopy groups ${\pi }_n$ for $n\ge 2$ (abelian by Eckmann-Hilton). Functoriality, homotopy invariance, Seifert-Van Kampen, Hurewicz, universal cover, long exact sequence of a fibration. Eilenberg-MacLane spaces.

  requires: topology.topological-space, topology.continuous-map

• topology.fundamental-groupopen unit 03.12.00 →

  Fundamental group

  ${\pi }_1(X,x_0)$ as homotopy classes of based loops $\gamma :[0,1]\to X$, $\gamma (0)=\gamma (1)=x_0$, under concatenation. Identity = constant loop; inverse = reversed loop. Functoriality $f_{\ast }:{\pi }_1(X,x_0)\to {\pi }_1(Y,y_0)$, $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$. Homotopy invariance: $f\simeq g$ implies $f_{\ast }=g_{\ast }$. Basepoint independence up to inner automorphism via path conjugation ${\beta }_{\eta }$. Standard examples: ${\pi }_1({\mathbb{R}}^n)=0$, ${\pi }_1(S^1)=\mathbb{Z}$ (winding number via universal cover), ${\pi }_1(S^n)=0$ for $n\ge 2$, ${\pi }_1({\mathbb{T}}^n)={\mathbb{Z}}^n$, ${\pi }_1({\mathbb{RP}}^n)=\mathbb{Z}/2$, ${\pi }_1({\Sigma }_g)$ surface group, ${\pi }_1$(figure-eight) $=F_2$. Loop space $\Omega (X,x_0)$ with compact-open topology; ${\pi }_0(\Omega X)={\pi }_1(X)$ and ${\pi }_n(X)={\pi }_{n-1}(\Omega X)$. Galois correspondence between subgroups and connected covers (topology.covering-space). ${\pi }_1(X,x_0)={\mathrm{Aut}}_{{\pi }_1(X)}(x_0)$ in the fundamental groupoid (topology.fundamental-groupoid). Originator Poincaré 1895.

  requires: topology.topological-space, topology.continuous-map, topology.homotopy

• topology.fundamental-groupoidopen unit 03.12.08 →

  Fundamental groupoid

  Small category ${\pi }_1(X)$ on points of $X$ with morphisms = path-homotopy classes; partial composition (concatenation when endpoints match); inverses by reversed paths; identity = constant path. Functor ${\pi }_1:\mathbf{Top}\to \mathbf{Groupoid}$. Equivalent to one-object $\mathrm{B}{\pi }_1(X,x_0)$ when $X$ is path-connected. Brown's groupoid Seifert-van Kampen: ${\pi }_1(X,A)$ as a pushout, no path-connectedness hypothesis on $U\cap V$. Galois correspondence as equivalence $\mathbf{Cov}(X)\simeq {\mathbf{Set}}^{{\pi }_1(X)}$.

  requires: topology.homotopy, topology.topological-space, topology.continuous-map

• topology.singular-homologyopen unit 03.12.11 →

  Singular homology

  Singular $n$-simplex $\sigma :{\Delta }^n\to X$. Free abelian chain group $C_n(X)$. Boundary $\partial =\sum (-1{)}^i{d}^i$, ${\partial }^2=0$. Homology $H_n(X)=\ker {\partial }_n/\mathrm{im}{\partial }_{n+1}$. Functoriality, homotopy invariance via prism/chain-homotopy. Reduced homology, augmentation. Coefficients $H_n(X;G)$. Mayer-Vietoris. Long exact sequence of a pair. Computations $H_n(\mathrm{pt})$, $H_n(S^k)$. Originator: Eilenberg 1944 Singular homology theory (Ann. Math. 45); precursors Vietoris 1927, Lefschetz 1933.

  requires: topology.topological-space, topology.continuous-map, topology.homotopy

• topology.simplicial-homologyopen unit 03.12.12 →

  Simplicial and $\Delta$-complex homology

  $\Delta$-complex / semi-simplicial structure on $X$ via characteristic maps with face conditions. Simplicial chain complex $C_n^{\Delta }(X)$ — finite-dim when $X$ is finite. Simplicial homology $H_n^{\Delta }(X)$. Comparison theorem (Hatcher 2.27): $H_n^{\Delta }(X)\cong H_n(X)$ via the natural chain map. Standard computations: $S^n$, ${\mathbb{RP}}^n$, ${\mathbb{CP}}^n$, genus-$g$ surfaces, lens spaces. Originator: Poincaré 1895 (simplicial chain complex); Eilenberg-Zilber 1950 / Eilenberg-Steenrod 1952 modern $\Delta$-framing.

  requires: topology.topological-space, topology.continuous-map, topology.homotopy

• topology.cellular-homologyopen unit 03.12.13 →

  Cellular homology and cellular approximation

  For a CW complex $X$, $C_n^{\mathrm{cell}}(X)=H_n(X^{(n)},X^{(n-1)})$. Cellular boundary via long-exact-sequence connecting map; explicit degree formula $d_n(e_{\alpha }^n)=\sum \mathrm{deg}({\varphi }_{\alpha \beta })e_{\beta }^{n-1}$. Cellular = singular: $H_n^{\mathrm{cell}}(X)\cong H_n(X)$ (Hatcher 2.35). Computational power: ${\mathbb{CP}}^n$ has zero boundaries (no consecutive cells); ${\mathbb{RP}}^n$ has $\partial =1+(-1{)}^k$. Cellular approximation theorem: every continuous CW-map is homotopic to a cellular map (Hatcher 4.8). Originator: J.H.C. Whitehead 1949 Combinatorial homotopy II.

  requires: topology.cw-complex, topology.singular-homology

• topology.excisionopen unit 03.12.14 →

  Excision theorem

  For $Z\subseteq A\subseteq X$ with $\overline{Z}\subseteq \mathrm{int}(A)$, the inclusion induces $H_n(X\setminus Z,A\setminus Z)\cong H_n(X,A)$. Equivalent open-cover form: $X=\mathrm{int}(A)\cup \mathrm{int}(B)\Rightarrow H_n(B,A\cap B)\cong H_n(X,A)$. Proof via barycentric subdivision + Lebesgue-number argument. Consequences: Mayer-Vietoris derivation; $H_n(D^k,S^{k-1})\cong {\tilde{H}}_{n-1}(S^{k-1})$; cellular boundary formula. Originator: Eilenberg-Steenrod 1952 (axiomatic elevation); precursor in Eilenberg 1944.

  requires: topology.singular-homology, topology.cellular-homology

• topology.eilenberg-steenrodopen unit 03.12.15 →

  Eilenberg-Steenrod axioms

  Axioms for ordinary homology: homotopy, long exact sequence of a pair, excision, additivity, naturality, dimension ($h_n(\mathrm{pt})=0$ for $n\ne 0$), and weak-equivalence. Uniqueness theorem: any two theories satisfying dimension and agreeing on $\mathrm{pt}$ are naturally isomorphic on CW pairs. Generalised cohomology theories drop dimension axiom (K-theory, cobordism, stable homotopy). Brown representability via spectra. Singular = simplicial = cellular = Čech all satisfy axioms; uniqueness explains agreement.

  requires: topology.singular-homology, topology.excision

• topology.poincare-dualityopen unit 03.12.16 →

  Poincaré duality

  For closed oriented $n$-manifold $M$, cap with $[M]$ gives $H^k(M;\mathbb{Z})\cong H_{n-k}(M;\mathbb{Z})$. $\mathbb{Z}/2$-coefficients version: works without orientation. Lefschetz duality for manifolds with boundary. de Rham version: integration pairing on closed oriented smooth manifolds. Consequences: vanishing $\chi$ for odd-dim closed manifolds, signature theorem, Hirzebruch L-genus. Originator: Poincaré Analysis Situs 5th supplement; modern proof Lefschetz 1930 + Eilenberg-Steenrod 1952; de Rham 1931 (smooth version).

  requires: topology.singular-homology, topology.cellular-homology, topology.excision, topology.eilenberg-steenrod

• topology.cap-productopen unit 03.12.17 →

  Cap product

  Front-face/back-face split: $\varphi ⌢\sigma =\varphi (\sigma {|}_{[v_0,\dots ,v_k]})\cdot \sigma {|}_{[v_k,\dots ,v_n]}$. Bilinear $⌢:C^k\otimes C_n\to C_{n-k}$, descends to homology via Leibniz rule $\partial (\varphi ⌢\sigma )=(-1{)}^k(\delta \varphi ⌢\sigma -\varphi ⌢\partial \sigma )$. Makes $H_{\ast }(X)$ a graded $H^{\ast }(X)$-module. Naturality (projection formula): $f_{\ast }(f^{\ast }\varphi ⌢\alpha )=\varphi ⌢f_{\ast }\alpha$. Cap-cup compatibility: $\varphi ⌢(\psi ⌢\alpha )=(\psi ⌣\varphi )⌢\alpha$. Cap with fundamental class $-⌢[M]$ is the Poincaré-duality isomorphism. Originator: Čech 1936 / Whitney 1938 / Lefschetz 1942; axiomatised in Eilenberg-Steenrod 1952.

  requires: topology.singular-homology, topology.cellular-homology

• topology.universal-coefficientopen unit 03.12.18 →

  Universal coefficient theorem (homology and cohomology)

  Homology UCT: split SES $0\to H_n(X)\otimes G\to H_n(X;G)\to \mathrm{Tor}(H_{n-1}(X),G)\to 0$. Cohomology UCT: $0\to \mathrm{Ext}(H_{n-1}(X),G)\to H^n(X;G)\to \mathrm{Hom}(H_n(X),G)\to 0$. Algebraic version for any chain complex of free abelian groups. Tor and Ext as derived functors. Field-coefficient case: Tor and Ext vanish in characteristic 0. Bockstein homomorphism via SES $0\to \mathbb{Z}\to \mathbb{Z}\to \mathbb{Z}/n\to 0$. Originator: Cartan-Eilenberg 1956 (algebraic); Eilenberg-Steenrod 1952 (topological).

  requires: topology.singular-homology, topology.cellular-homology

• symplectic-geometry.ags-convexityopen unit 05.04.03 →

  Atiyah-Guillemin-Sternberg convexity theorem

  For closed connected $(M,\omega )$ with Hamiltonian $T=T^k$ action and moment map $\mu :M\to {\mathfrak{t}}^{\ast }$: (i) $\mu (M)$ is a convex polytope; (ii) $\mu (M)=\mathrm{conv}(\mu (M^T))$; (iii) every fibre ${\mu }^{-1}(c)$ is connected. Atiyah's proof: Morse-Bott analysis of ${\mu }^{\xi }$, even index/coindex via weight-space decomposition, level-set connectedness via even-index attaching, induction on rank. Examples: $S^2$ height function, ${\mathbb{CP}}^n$ standard polytope, coadjoint orbits / Schur-Horn permutohedron. Originators: Atiyah 1982 Convexity and commuting Hamiltonians; Guillemin-Sternberg 1982 Convexity properties of the moment mapping.

  requires: symplectic-geometry.moment-map, symplectic-geometry.symplectic-manifold

• topology.hurewicz-theoremopen unit 03.12.19 →

  Hurewicz theorem

  Hurewicz map $h_n:{\pi }_n(X)\to H_n(X)$, $[\sigma ]\mapsto {\sigma }_{\ast }[S^n]$. Low-dim form (Hatcher 2A.1): $h_1:{\pi }_1\to H_1$ is abelianisation map. Higher-dim form (Hatcher 4.32): if $X$ is $(n-1)$-connected then $H_k(X)=0$ for $1\le k<n$ and $h_n$ is iso. Relative form (Hatcher 4.37). ${\pi }_n(S^n)=\mathbb{Z}$ as a basic computation. Hopf map shows non-injectivity of $h_3:{\pi }_3(S^2)\to H_3(S^2)=0$. Originator: Witold Hurewicz 1935-36 four-paper series.

  requires: topology.homotopy, topology.singular-homology

• topology.whitehead-theoremopen unit 03.12.20 →

  Whitehead's theorem

  A continuous map $f:X\to Y$ between CW complexes inducing iso on all ${\pi }_n$ is a homotopy equivalence (Hatcher 4.5). Proof via cellular approximation + HEP for CW pairs + skeleton-by-skeleton inductive obstruction-theoretic construction of homotopy inverse. Mapping-cylinder reformulation. Hurewicz + Whitehead corollary: simply-connected CW complexes with iso on all $H_n$ are homotopy-equivalent. Warsaw circle is the standard counterexample for the CW hypothesis. Originator: J.H.C. Whitehead 1949.

  requires: topology.cw-complex, topology.homotopy, topology.hurewicz-theorem

• symplectic-geometry.contact-manifoldopen unit 05.10.01 →

  Contact manifold

  Odd-dimensional manifold $M^{2n+1}$ with maximally non-integrable codimension-1 hyperplane field $\xi \subset TM$, locally $\xi =\ker \alpha$ with $\alpha \wedge (d\alpha {)}^n\ne 0$. Co-orientable (global $\alpha$) vs non-co-orientable. Reeb vector field $R$: ${\iota }_R\alpha =1,{\iota }_{R}d\alpha =0$. Standard examples: ${\mathbb{R}}^{2n+1}$ with $dz-\sum y_{i}d{x}_i$; $S^{2n+1}\subset {\mathbb{C}}^{n+1}$ Sasakian; $S(T^{\ast }M)$ unit-cotangent (Reeb = geodesic flow); $J^1(M,\mathbb{R})$. Contact Darboux via Moser. Symplectisation $(M\times \mathbb{R},d(e^t\alpha ))$. Legendrian = $n$-dim submanifold tangent to $\xi$. Gray's theorem (1959): isotopic contact structures are diffeomorphic. Tight/overtwisted dichotomy (Eliashberg 1989).

  requires: symplectic-geometry.symplectic-manifold, diffgeo.exterior-derivative

• symplectic-geometry.generating-functionsopen unit 05.05.03 →

  Generating functions for symplectomorphisms

  Graph of symplectomorphism $\varphi :T^{\ast }M\to T^{\ast }M$ as a Lagrangian in $T^{\ast }M\times T^{\ast }M\cong T^{\ast }(M\times M)$. Type-1 generating function $F:M\times M\to \mathbb{R}$. Four classical types (physics literature). Generating function for exact Lagrangian = scalar potential $S:M\to \mathbb{R}$. Hörmander 1971 GFQI: every closed Lagrangian admits a generating function quadratic at infinity. Sikorav-Viterbo theorem: GFQIs for Hamiltonian-isotopic Lagrangians are equivalent up to stabilisation/fibre-change/constants — gives spectral invariants. Discrete action principle: critical points = fixed points / Lagrangian intersections. Conley-Zehnder used GFQI for the torus Arnold conjecture (1983).

  requires: symplectic-geometry.lagrangian-submanifold, symplectic-geometry.weinstein-neighbourhood, symplectic-geometry.symplectic-manifold

• symplectic-geometry.delzant-theoremopen unit 05.04.04 →

  Delzant theorem (symplectic toric classification)

  Bijection between symplectic toric manifolds (closed, effective Hamiltonian half-dim torus action) and Delzant polytopes (simple + rational + smooth). Five-step construction: facet primitive normals $v_i$; surjection $\beta :{\mathbb{R}}^d\to {\mathbb{R}}^n$; kernel torus $K$; standard $T^d$-action on ${\mathbb{C}}^d$ with diagonal moment map; symplectic reduction $M_{\Delta }={\mu }_K^{-1}(0)/K$. Examples: standard simplex = ${\mathbb{CP}}^n$ (Fubini-Study); standard cube = $({\mathbb{CP}}^1{)}^n$; trapezoid = Hirzebruch ${\mathbb{F}}_n$. Originator: Delzant 1988.

  requires: symplectic-geometry.moment-map, symplectic-geometry.ags-convexity, symplectic-geometry.symplectic-reduction

• symplectic-geometry.duistermaat-heckmanopen unit 05.04.05 →

  Duistermaat-Heckman theorem

  Pushforward of Liouville volume ${\omega }^n/n!$ under moment map $\mu :M\to {\mathfrak{t}}^{\ast }$ has piecewise-polynomial density $\rho (t)$ of degree $\le n-k$ on chambers of $\mu (M)\setminus \mathcal{W}$, walls $\mathcal{W}$ being moment-map images of fixed-point components of proper subtori. Three formulations: (1) pushforward density polynomial; (2) equivariant integration ${\int }_M{e}^{i⟨t,\mu ⟩}{\omega }^n/n!={\sum }_F{\int }_F{e}^{i⟨t,\mu (F)⟩}/e_T({\nu }_F)$ over fixed-point components; (3) reduced symplectic volume $\mathrm{vol}(M_t,{\omega }_t)$ polynomial in $t$. Proof spine: linear variation $[{\omega }_t]=[{\omega }_{t_0}]+⟨t-t_0,c_T⟩$ in cohomology of reduced space; binomial-theorem expansion gives polynomial of degree $\le n-k$. Toric corollary: for symplectic toric $M$, ${\int }_M{\omega }^n/n!=(2\pi {)}^n{\mathrm{vol}}_{\mathrm{Leb}}(\Delta )$. Worked examples: $S^2$ rotation (constant $\rho =2\pi$); ${\mathbb{CP}}^n$ toric (constant $\rho =(2\pi {)}^n$); coadjoint orbit of $U(n)$ (Harish-Chandra-Itzykson-Zuber integral via Weyl-image fixed points). Prototype of equivariant localisation, subsumed by Atiyah-Bott / Berline-Vergne 1982-1984; non-Abelian generalisation Witten 1992. Originators: Duistermaat-Heckman 1982; Atiyah-Bott 1984 (abstract).

  requires: symplectic-geometry.moment-map, symplectic-geometry.symplectic-reduction, symplectic-geometry.ags-convexity

• symplectic-geometry.symplectic-blowupopen unit 05.04.06 →

  Symplectic blow-up and symplectic cut

  Symplectic analogue of the algebraic blow-up: replace a Darboux ball $B(p,\delta )\subset (M,\omega )$ around a point $p$ with a tubular neighbourhood of ${\mathbb{CP}}^{n-1}$ (the exceptional divisor, scaled by $\delta$). Two equivalent constructions: (i) Lerman's symplectic cut at level $\mu =\delta$ for the rotational $S^1$-action with moment map $\mu (z)=\pi \Vert z{\Vert }^2$ on the Darboux ball, giving $M_{\delta }={\mu }^{-1}([0,\delta ])/\sim$ where the $\delta$-level is collapsed by the $S^1$-action; (ii) explicit gluing via the tautological line bundle ${\mathcal{O}}_{{\mathbb{P}}^{n-1}}(-1)$ over ${\mathbb{CP}}^{n-1}$, with the Hopf-style projection making the symplectic form patch smoothly. Properties: $H^{\ast }(\tilde{M})=H^{\ast }(M)\oplus H^{\ast }({\mathbb{CP}}^{n-1}{)}_{\ge 1}$ additively; for surfaces ($n=2$) introduces a $(-1)$-curve, a symplectic 2-sphere of self-intersection $-1$; symplectic volume $\mathrm{vol}(\tilde{M})=\mathrm{vol}(M)-\mathrm{vol}(B(p,\delta ))+\mathrm{vol}({\mathbb{CP}}_{\delta }^{n-1})=\mathrm{vol}(M)-O({\delta }^{2n})$. Examples: blow-up of ${\mathbb{CP}}^2$ at a point = Hirzebruch surface ${\mathbb{F}}_1$; blow-up of ${\mathbb{CP}}^1\times {\mathbb{CP}}^1$ at a point; symplectic cut on toric varieties chops the Delzant polytope along a hyperplane. Inverse: Castelnuovo's contractibility theorem (a $(-1)$-curve in a smooth projective surface contracts to a smooth point) has a symplectic analogue. Applications: birational classification of complex surfaces (Castelnuovo-Beauville-Bombieri); symplectic-embedding flexibility / obstructions (McDuff-Polterovich 1994); compactification of Donaldson instanton moduli spaces (ideal instantons via blow-ups). Originator: algebraic side, Cremona / Italian school 19th c.; symplectic side, Gromov 1985 (sketched) and Lerman 1995 (cut construction made explicit); McDuff-Polterovich 1994 systematic embedding applications.

  requires: symplectic-geometry.symplectic-reduction, symplectic-geometry.delzant-theorem, symplectic-geometry.almost-complex

• symplectic-geometry.symplectisationopen unit 05.10.02 →

  Symplectisation of a contact manifold

  Symplectisation $(M\times \mathbb{R},d(e^t\alpha ))$ of a co-orientable contact manifold. Verification via $(e^t\alpha {)}^{n+1}=e^{(n+1)t}dt\wedge \alpha \wedge (d\alpha {)}^n$. Independence of contact-form choice via $(p,t)\mapsto (p,t-\log f(p))$. Liouville structure with primitive $\lambda =e^t\alpha$ and Liouville vector field ${\partial }_t$. Reeb-flow lift. Floer-theoretic relevance: SFT (Eliashberg-Givental-Hofer 2000) and ECH (Hutchings 2002). Worked example: $S^{2n+1}\subset {\mathbb{C}}^{n+1}$ symplectisation = ${\mathbb{C}}^{n+1}\setminus \{0\}$.

  requires: symplectic-geometry.contact-manifold, symplectic-geometry.symplectic-manifold

• symplectic-geometry.gray-theoremopen unit 05.10.03 →

  Gray's stability theorem

  A smooth path of contact structures on a closed manifold is induced by an isotopy. Proof: Moser-trick adaptation. Choose contact forms ${\alpha }_t$, seek ${\psi }_t^{\ast }{\alpha }_t={\lambda }_t{\alpha }_0$. The contact condition makes the tangent equation ${\iota }_{X_t}d{\alpha }_t{|}_{{\xi }_t}=-{\dot{\alpha }}_t{|}_{{\xi }_t}$ uniquely solvable for $X_t\in {\xi }_t$. Integrate. Consequences: classification up to homotopy = up to diffeomorphism on closed manifolds; Reeb-dynamics stability; foundation for SFT/ECH. 3D extra: tight/overtwisted dichotomy (Eliashberg 1989). Originator: J.W. Gray 1959.

  requires: symplectic-geometry.contact-manifold, symplectic-geometry.moser-trick

• symplectic-geometry.contact-topology-surveyopen unit 05.10.04 →

  Contact topology and Reeb dynamics (survey)

  Survey unit (Cannas P4 optional pointer expanded). Tight vs overtwisted dichotomy (Eliashberg 1989): on closed contact 3-manifolds, overtwisted structures are classified up to isotopy by homotopy classes of plane fields; tight structures are subtle (Bennequin 1983 standard ${\mathbb{R}}^3$ tight, distinct from the overtwisted same-homotopy-class form). Reeb dynamics: closed orbits of $R_{\alpha }$. Weinstein conjecture (1979): every Reeb vector field on a closed contact manifold has at least one closed orbit; proven dim 3 by Taubes 2007 via Seiberg-Witten Floer. Floer-theoretic invariants from pseudoholomorphic curves in symplectisations: Symplectic Field Theory (Eliashberg-Givental-Hofer 2000), Embedded Contact Homology (Hutchings 2002), cylindrical contact homology. Legendrian knots: classical invariants (Thurston-Bennequin, rotation number); refined invariants (Chekanov-Eliashberg DGA 1997, Legendrian contact homology). Convex surface theory (Giroux 1991): characteristic foliation gives discrete neighbourhood data; foundation for cut-and-paste. Open-book decompositions and Giroux correspondence (2002): closed oriented contact 3-manifolds ↔ stable equivalence classes of open books. Higher-dimensional contact topology: Borman-Eliashberg-Murphy 2015 existence h-principle for overtwisted; Cieliebak-Eliashberg 2012 Weinstein-domain technology. Open problems: cardinality of tight structures, ECH spectral invariants, Stein-vs-Weinstein boundary rigidity, contact mapping class groups. Originators: Bennequin 1983; Eliashberg 1989-92; Giroux 2002; Taubes 2007.

  requires: symplectic-geometry.contact-manifold, symplectic-geometry.symplectisation, symplectic-geometry.gray-theorem

• symplectic-geometry.kam-theoremopen unit 05.09.01 →

  Kolmogorov-Arnold-Moser theorem

  APEX UNIT. Persistence of invariant tori under Hamiltonian perturbation. Setup: integrable $H_0(I)$ with action-angle coords; perturbation $H_{ϵ}=H_0+ϵH_1$. Diophantine condition $|⟨k,{\omega }^{\ast }⟩|\ge \gamma /|k{|}^{\tau }$. Kolmogorov non-degeneracy $\det {\partial }^2{H}_0/\partial I^2\ne 0$. Theorem: Diophantine torus survives for small $ϵ$. Newton-iteration / KAM scheme: cohomological equation, Fourier decomposition, super-exponential convergence with controlled analyticity loss. Measure conclusion: $1-C\sqrt{ϵ}$ of phase space remains invariant. Refinements: Moser twist 1962, Pöschel $C^k$ 1982, lower-dim tori, Nekhoroshev exponential stability, Aubry-Mather. Applications: solar-system stability, magnetic confinement, beam dynamics. Originators: Kolmogorov 1954 (4-page Dokl. announcement); Arnold 1963 (full proof); Moser 1962 (smooth twist version).

  requires: symplectic-geometry.integrable-system, symplectic-geometry.action-angle-coordinates, symplectic-geometry.symplectic-manifold, symplectic-geometry.generating-functions

• classical-mechanics.galilean-newtonian-setupopen unit 05.00.06 →

  Galilean group and Newtonian mechanics

  Galilean spacetime as affine 4-space ${\mathbb{A}}^4$ with absolute time $t:{\mathbb{A}}^4\to \mathbb{R}$ and Euclidean structure on each time-fibre. Galilean group $\mathrm{Gal}(4)$ as 10-parameter group of affine transformations preserving time differences and fibre Euclidean structure: spatial translations ${\mathbb{R}}^3$, time translation $\mathbb{R}$, rotations $\mathrm{SO}(3)$, Galilean boosts ${\mathbb{R}}^3$, with semi-direct product structure. Inertial frames = Galilean coordinate systems; Galilean relativity principle. Newton's laws restated geometrically on a configuration manifold $Q$ with Newton's 2nd $m_i{\ddot{q}}_i=F_i(q,\dot{q},t)$ and determinism principle (state $(q,\dot{q},t)$ determines the future). Conservative forces $F=-\nabla V$; Lagrangian $L=T-V$ as the bridge to the Lagrangian formalism. Examples: free particle (Newton's 1st), two-body Kepler, $N$-body system with Galilean invariance giving energy / momentum / angular momentum / centre-of-mass conservation laws. Lie algebra $\mathfrak{gal}(4)$ 10-dim with brackets among translations / boosts / rotations. Inönü-Wigner contraction $c\to \infty$ from Poincaré group. Bargmann central extension by $\mathbb{R}$ governs projective QM representations; mass = central charge (Lévy-Leblond 1963). Originator: Galileo 1632 Dialogue (relativity principle); Newton 1687 Principia; modern group-theoretic framing Bargmann 1954, Souriau 1970.

  requires: manifolds.smooth-manifold

• classical-mechanics.lagrangian-on-tmopen unit 05.00.01 →

  Lagrangian mechanics on the tangent bundle

  Configuration space $Q$, tangent bundle $TQ=\{(q,\dot{q})\}$, Lagrangian $L:TQ\to \mathbb{R}$ (often $T-V$). Action $S[\gamma ]=\int Ldt$. Euler-Lagrange equations $\frac{d}{dt}(\partial L/\partial {\dot{q}}^i)-\partial L/\partial q^i=0$ as critical-point condition. Coordinate-free framing via Poincaré-Cartan one-form ${\theta }_L=(\partial L/\partial {\dot{q}}^i)d{q}^i$. Energy $E_L={\dot{q}}^i(\partial L/\partial {\dot{q}}^i)-L$ as Hamilton-prefiguration. Examples: free particle, particle-in-potential = Newton's 2nd law, geodesics, pendulum. Regularity / hyper-regularity controlling Legendre transform. Originator: Lagrange 1788; modern coordinate-free framing mid-20th-c.

  requires: manifolds.smooth-manifold

• classical-mechanics.hamilton-principleopen unit 05.00.02 →

  Hamilton's principle of least action

  Path is a physical trajectory iff it is a critical point of the action functional $S[\gamma ]=\int L(\gamma ,\dot{\gamma })dt$ among paths with fixed endpoints. First-variation formula: $\delta S=\int (\partial L/\partial q-d/dt(\partial L/\partial \dot{q}))\delta qdt$ + boundary, which vanishes by endpoint condition. Vanishing for arbitrary $\delta q$ ⇔ Euler-Lagrange. Equivalence with Newton's 2nd law for $L=T-V$. Maupertuis reparametrisation pitfall and Jacobi metric. D'Alembert principle for non-conservative forces. Holonomic constraints via Lagrange multipliers. Field-theory generalisation: Klein-Gordon, Maxwell, Yang-Mills, Einstein-Hilbert. Originator: Hamilton 1834; Maupertuis 1744 less-precise predecessor.

  requires: classical-mechanics.lagrangian-on-tm

• classical-mechanics.legendre-transformopen unit 05.00.03 →

  Legendre transform

  Convex transform $f^{\ast }(p)={\sup }_q(⟨p,q⟩-f(q))$. Fenchel-Moreau involution $(f^{\ast }{)}^{\ast }=f$. Differential form $\nabla f^{\ast }(p)=q(p)$ inverse of $\nabla f$. Fibre Legendre transform $\mathbb{F}L:TQ\to T^{\ast }Q$, $\mathbb{F}L(q,\dot{q})=(q,\partial L/\partial \dot{q})$. Regularity = local diffeomorphism (Hessian non-singular); hyper-regularity = global diffeomorphism. Hamiltonian $H(q,p)=⟨p,\dot{q}⟩-L$. Equivalence of EL on $TQ$ with Hamilton's equations on $T^{\ast }Q$. Cotangent bundle as natural symplectic phase space. Singular Lagrangians and Dirac-Bergmann constraint analysis (gauge theories, GR). Originator: Legendre 1787; mechanics application Hamilton 1834; modern framing Abraham-Marsden 1978.

  requires: classical-mechanics.lagrangian-on-tm

• classical-mechanics.noether-theoremopen unit 05.00.04 →

  Noether's theorem

  Every smooth one-parameter family of symmetries of the action $S$ gives a conserved quantity along EL flow. Setup: vector field $X$ on $Q$ with prolongation $X^{(1)}$ on $TQ$; invariance ${\mathcal{L}}_{X^{(1)}}L=0$. Noether charge $J_X=⟨\partial L/\partial \dot{q},X⟩$. Examples: time-translation → energy; space-translation → momentum; rotation → angular momentum. Hamiltonian-side: Poisson-commute condition $\{f,H\}=0$. Lifts to moment-map theory: $G$-action on $(M,\omega )$ with $\mu :M\to {\mathfrak{g}}^{\ast }$. Field-theory generalisation: Noether currents ${\partial }_{\mu }{J}_X^{\mu }=0$. Inverse Noether (Cartan-Lie). Originator: Emmy Noether 1918.

  requires: classical-mechanics.lagrangian-on-tm, classical-mechanics.hamilton-principle

• symplectic-geometry.geodesic-flow-hamiltonianopen unit 05.02.06 →

  Geodesic flow as a Hamiltonian flow

  Kinetic-energy Hamiltonian $H(q,p)=\frac{1}{2}{g}^{ij}{p}_i{p}_j$ on $T^{\ast }M$. Hamilton's equations recover the geodesic equation ${\ddot{q}}^i+{\Gamma }_{jk}^i{\dot{q}}^j{\dot{q}}^k=0$. Unit cotangent bundle $S(T^{\ast }M)$ is contact; Reeb = geodesic spray. Killing vector fields → Noether-conserved quantities. Maupertuis-Jacobi reformulation: mechanics with potential $V$ on energy level $H=E$ ↔ pure geodesic flow of Jacobi metric $\tilde{g}=(E-V)g$. Examples: flat ${\mathbb{R}}^n$ (straight lines), round $S^n$ (great circles, integrable), hyperbolic plane (Anosov), Jacobi-integrable ellipsoid. Originator: Jacobi 1837.

  requires: symplectic-geometry.hamiltonian-vector-field, symplectic-geometry.cotangent-bundle, manifolds.smooth-manifold

• symplectic-geometry.euler-arnold-equationsopen unit 05.09.05 →

  Euler-Arnold equations

  Body-frame projection of geodesic flow on a Lie group $G$ with left-invariant Riemannian metric to the dual Lie algebra ${\mathfrak{g}}^{\ast }$. Setup: inertia operator $I:\mathfrak{g}\to {\mathfrak{g}}^{\ast }$, kinetic-energy Hamiltonian $H({\xi }^{\ast })=\frac{1}{2}⟨{\xi }^{\ast },I^{-1}{\xi }^{\ast }⟩$. Equation: ${\dot{\xi }}^{\ast }=-{\mathrm{ad}}_{I^{-1}{\xi }^{\ast }}^{\ast }{\xi }^{\ast }$. Hamiltonian flow of $H$ for the Lie-Poisson bracket on ${\mathfrak{g}}^{\ast }$; coadjoint orbits are the symplectic leaves with KKS form. Conservation: energy + Casimirs (= ${\mathrm{Ad}}_G^{\ast }$-invariant functions). Examples: $\mathrm{SO}(3)$ rigid-body Euler equations (Euler 1758, Liouville-integrable, Poinsot construction); $\mathrm{SO}(n)$ rigid body (Manakov 1976, Lax pair with spectral parameter); $\mathrm{SDiff}(M)$ ideal-fluid Euler equations (Arnold 1966, $L^2$ inertia); $\mathrm{Diff}(S^1)$ Bott-Virasoro group with $L^2$ metric → KdV, with $H^1$ metric → Camassa-Holm (Misiołek 1998). Tennis-racket theorem: stability of long/short-axis rotation, instability of medium-axis. Geodesic completeness: full for compact finite-dim $G$, blow-up possible for infinite-dim (Beale-Kato-Majda 1984 vorticity criterion for 3D Euler). Mishchenko-Fomenko argument-shift method (1978) gives Liouville-integrability on semisimple $\mathfrak{g}$. Heavy top via semidirect product $\mathrm{SO}(3)⋉{\mathbb{R}}^3$. Originators: Euler 1758 (rigid body), Arnold 1966 (general Lie group + ideal fluid).

  requires: symplectic-geometry.coadjoint-orbit, symplectic-geometry.geodesic-flow-hamiltonian, lie-theory.lie-group

• topology.blakers-masseyopen unit 03.12.21 →

  Blakers-Massey theorem

  Homotopy excision for CW triads $X=A\cup B$ with $C=A\cap B$ path-connected and connectivity hypotheses on the inclusions. ${\pi }_k(A,C)\to {\pi }_k(X,B)$ is iso for $k<m+n$, surjective at $m+n$. Homology-excision-up-to-stable-range. Freudenthal suspension theorem as corollary: ${\pi }_k(X)\to {\pi }_{k+1}(\Sigma X)$ iso for $k<2n-1$ when $X$ is $(n-1)$-connected. Foundation of stable homotopy theory and Adams spectral sequence. ∞-categorical generalisation in modal homotopy type theory.

  requires: topology.homotopy, topology.cw-complex, topology.hurewicz-theorem

• topology.euler-characteristicopen unit 03.12.23 →

  Euler characteristic

  Cellular form $\chi (X)=\sum (-1{)}^n{c}_n$. Homological form $\chi (X)=\sum (-1{)}^n\mathrm{rk}{H}_n$. Cellular = homological proof. Multiplicativity via Künneth. Euler-Poincaré for fibre bundles $\chi (E)=\chi (F)\chi (B)$. Vanishes on closed odd-dim orientable manifolds (Poincaré duality + alternating sum). Gauss-Bonnet $\int KdA=2\pi \chi$ on closed surfaces; Chern-Gauss-Bonnet via Pfaffian in even dim. Poincaré-Hopf $\sum {\mathrm{ind}}_{p}V=\chi (M)$. Lefschetz fixed-point. Examples: $S^n,{\Sigma }_g,{\mathbb{RP}}^n,{\mathbb{CP}}^n,T^n$. Originator: Euler 1758 Elementa doctrinae solidorum.

  requires: topology.cw-complex, topology.cellular-homology, topology.poincare-duality

• classical-mechanics.hamilton-jacobiopen unit 05.05.04 →

  Hamilton-Jacobi equation

  $\partial S/\partial t+H(q,\partial S/\partial q,t)=0$ for action $S(q,t)$; time-indep $H(q,\partial W/\partial q)=E$. Generating-function-type-2 making $H\mapsto 0$. Method of characteristics: characteristics = Hamilton flow. Complete integrals giving full integration. Separation of variables (central potentials, Stäckel framework). Action-angle coordinates from HJ. Geometric: solutions = Lagrangian submanifolds of $T^{\ast }M$. WKB / eikonal limit of Schrödinger. Viscosity-solution theory (Crandall-Lions 1983) for caustics + optimal control. Originator: Hamilton 1834; Jacobi 1866 computational.

  requires: symplectic-geometry.generating-functions, symplectic-geometry.cotangent-bundle, classical-mechanics.legendre-transform

• classical-mechanics.liouville-volumeopen unit 05.02.07 →

  Liouville's volume theorem

  Hamiltonian flows preserve symplectic volume ${\omega }^n/n!$. Proof via Cartan: ${\mathcal{L}}_{X_H}\omega =0\Rightarrow {\mathcal{L}}_{X_H}{\omega }^n=0$. Darboux-coordinate divergence-free form. Liouville equation ${\partial }_t\rho +\{\rho ,H\}=0$ for phase-space density. Equilibrium measures as $f(H)$. Volume-rigid + length-flexible characterising symplectic geometry. Counterexample: gradient flows are NOT volume-preserving. Foundation for Poincaré recurrence and statistical mechanics. Originator: Liouville 1838.

  requires: symplectic-geometry.hamiltonian-vector-field, symplectic-geometry.symplectic-manifold

• classical-mechanics.poincare-recurrenceopen unit 05.02.08 →

  Poincaré recurrence theorem

  On finite measure space with measure-preserving $T:X\to X$, every measurable $A$ with $\mu (A)>0$ has a.e. point returning infinitely often. Pigeonhole proof on $T^{-n}(A)$ disjoint iterates. Mean recurrence time = $\mu (X)/\mu (A)$ (Kac's lemma). Hamiltonian application via Liouville volume. Boltzmann/Zermelo timescale resolution of H-theorem tension. Quantum recurrence in finite-dim Hilbert space; failure in infinite-dim. Foundation of ergodic theory (ergodicity, mixing, K-systems, Bernoulli). Originator: Poincaré 1890; Carathéodory 1919 abstract.

  requires: classical-mechanics.liouville-volume, symplectic-geometry.hamiltonian-vector-field

• symplectic-geometry.poincare-cartan-invariantsopen unit 05.02.09 →

  Poincaré-Cartan integral invariants

  Extended phase space $\tilde{M}=T^{\ast }Q\times \mathbb{R}$. Poincaré-Cartan one-form $\theta =pdq-Hdt$. Differential $\omega =d\theta =dp\wedge dq-dH\wedge dt$ has rank $2n$ and one-dimensional kernel; suspended Hamiltonian vector field ${\tilde{X}}_H=X_H+{\partial }_t$ is the kernel direction. First integral invariant ${\oint }_{\gamma }\theta$ flow-invariant on closed one-cycles (Poincaré 1890). Higher invariants ${\omega }^k$ for $k=1,\dots ,n$ — Cartan 1922 graded family with Liouville volume ${\omega }^n/n!$ as top degree. Cartan-formula proof: ${\iota }_{{\tilde{X}}_H}\omega =0$ plus closedness of $\omega$ gives ${\mathcal{L}}_{{\tilde{X}}_H}{\omega }^k=0$. Tube-of-trajectories reformulation: integrals on two rims of any tube agree by Stokes. Action variables $I_k=\frac{1}{2\pi }{\oint }_{{\gamma }_k}pdq$ on a Liouville torus are first invariants on basis loops; angles canonical-conjugate (Liouville-Arnold). Maupertuis principle on energy level $H=E$ as reduction of Hamilton's principle to spatial $\delta \int pdq=0$. Cartan's relative-versus-absolute distinction. Originator: Poincaré 1890/1899; Cartan 1922 systematic theory.

  requires: symplectic-geometry.symplectic-manifold, symplectic-geometry.hamiltonian-vector-field, classical-mechanics.liouville-volume

• symplectic-geometry.adiabatic-invariantsopen unit 05.09.02 →

  Adiabatic invariants

  Slowly-varying Hamiltonian $H(q,p;\lambda (ϵt))$. Adiabatic theorem (Burgers/Ehrenfest 1916, 1D): action $I=(1/2\pi )\oint pdq$ conserved up to $O(ϵ)$ over time $O({ϵ}^{-1})$. Geometric proof: average over fast angle. Higher-dim issues with resonant tori. Magnetic mirror $\mu =m{v}_{\perp }^2/(2B)$ adiabatic invariant for charged-particle motion (foundational for tokamaks). Quantum adiabatic theorem (Born-Fock 1928). Berry phase as adiabatic-correction holonomy. Nekhoroshev exponential-stability extension. Connection to KAM. Originator: Ehrenfest 1913-16.

  requires: symplectic-geometry.action-angle-coordinates, symplectic-geometry.integrable-system, symplectic-geometry.symplectic-manifold

• symplectic-geometry.birkhoff-normal-formopen unit 05.09.03 →

  Birkhoff normal form

  Hamiltonian system near an elliptic equilibrium $z_0$ with $H_2=\frac{1}{2}\sum {\omega }_j(q_j^2+p_j^2)$. If $\omega =({\omega }_1,\dots ,{\omega }_n)$ is non-resonant up to order $N$ — no integer relation $\sum k_j{\omega }_j=0$ with $|k|\le N$ — then a sequence of canonical (Lie-series) transformations puts $H=H_2+Z_4+Z_6+\cdots +Z_N+R_{N+1}$ where each $Z_{2k}$ depends only on the actions $I_j=(q_j^2+p_j^2)/2$ and $R_{N+1}=O(\Vert z{\Vert }^{N+1})$. Construction: at each order $k$, generating function $F_k$ with $\{H_2,F_k\}$ killing the non-resonant part of $H_k$; resonant terms become $Z_k$. Small-divisor / Poincaré non-integrability: full series generally diverges; Diophantine condition gives Gevrey-class smoothness. Examples: 1D (action-angle), 2D non-resonant (full Birkhoff), 2D 1:1 resonance (focus/saddle/centre classification), Lyapunov families near triangular Lagrange points in restricted three-body problem. Refinements: Birkhoff-Gustavson algorithm; KAM as the convergent island in the Birkhoff scheme; Nekhoroshev stability bounds derived from finite-order Birkhoff truncation. Originator: Birkhoff 1927; Siegel 1942 small divisors; Moser 1956 analytic refinement; Pöschel 1989 Gevrey.

  requires: symplectic-geometry.hamiltonian-vector-field, symplectic-geometry.action-angle-coordinates, symplectic-geometry.symplectic-manifold

• symplectic-geometry.nekhoroshevopen unit 05.09.06 →

  Nekhoroshev estimates

  Near-integrable Hamiltonian $H=H_0(I)+ϵH_1(I,\theta )$ with $H_0$ analytic and steep (a generic non-degeneracy condition stronger than non-resonance, weaker than convexity). Theorem (Nekhoroshev 1977): for all initial conditions and all $|t|\le T(ϵ)\asymp \exp (c{ϵ}^{-a})$, $|I(t)-I(0)|\le C{ϵ}^b$ for explicit Nekhoroshev exponents $a,b$ depending on $H_0$ and dimension. Comparison with KAM. KAM gives $|I(t)-I(0)|=0$ on a positive-measure Cantor set of Diophantine initial conditions for all time; Nekhoroshev gives $|I(t)-I(0)|=O({ϵ}^b)$ for every initial condition but only over an exponentially long interval. Steepness condition. $H_0$ is steep at $I^{\ast }$ if for every linear subspace $L\subseteq {\mathbb{R}}^n$ through $I^{\ast }$, the restriction $H_0{|}_L$ has only a degenerate isolated extremum at $I^{\ast }$ along directions in some lower-dim sublattice. Generic in $C^{\infty }$ topology, weaker than convexity, but excludes degenerate $H_0$. Proof outline (Lochak 1992 / Pöschel 1993). Block Birkhoff normal form on resonance regions: each resonance class ${\mathcal{M}}_K=\{⟨k,\omega ⟩=0:k\in K\}$ gets its own normal form, the non-resonant part is killed up to order $\sim 1/ϵ$, and steepness controls geometric diffusion across resonance regions. Iteration over resonance lattices gives the exponential time bound. Examples and exponents. Convex $H_0$ (Lochak): $a=1/(2n)$, $b=1/(2n)$. Quasi-convex (Pöschel 1993): same exponents — convex on energy levels suffices. Generic steep: smaller exponents, same shape. Solar system (Niederman 2007): explicit Nekhoroshev-type stability bounds; Arnold conjectured Lyapunov stability of the planetary motion at the perihelion-precession level over Hubble timescales. Arnold diffusion. For non-convex $H_0$ in dimensions $n\ge 3$, Nekhoroshev is essentially optimal: there exist initial conditions with $|I(t)-I(0)|\to \infty$. Arnold 1964 Instability of dynamical systems with several degrees of freedom (Doklady Akad. Nauk SSSR 156) gave the first explicit example via heteroclinic chains of whiskered tori. Mather, Berti-Bolle, Cheng-Yan made the diffusion mechanism rigorous in various settings. Combined picture (Bourgain-Kuksin and beyond). KAM gives positive-measure permanent stability; Nekhoroshev gives global polynomial stability over exponentially long times; Arnold diffusion gives the exceptional unstable orbits in the resonance gaps. Originator: Nekhoroshev 1977; Lochak 1992 simplification via simultaneous Diophantine approximation; Pöschel 1993 explicit constants.

  requires: symplectic-geometry.kam-theorem, symplectic-geometry.action-angle-coordinates, symplectic-geometry.birkhoff-normal-form

• symplectic-geometry.williamson-normal-formopen unit 05.09.04 →

  Williamson normal form for quadratic Hamiltonians

  Symplectic congruence classification of quadratic Hamiltonians $H(z)=\frac{1}{2}{z}^{T}Az$ on $({\mathbb{R}}^{2n},{\omega }_0)$ with $A$ real symmetric. Positive-definite case (the headline statement): there exists $S\in \mathrm{Sp}(2n,\mathbb{R})$ with $S^{T}AS=\mathrm{diag}({\omega }_1,\dots ,{\omega }_n,{\omega }_1,\dots ,{\omega }_n)$, where the positive numbers $\{{\omega }_1,\dots ,{\omega }_n\}$ — the symplectic eigenvalues or Williamson invariants of $A$ — are read off as the absolute values of the eigenvalues of $JA$ (which are $\pm i{\omega }_k$, all purely imaginary). The unordered multiset $\{{\omega }_k\}$ is a complete symplectic-conjugacy invariant. Proof: $JA$ is skew-self-adjoint for the inner product $⟨u,v{⟩}_A=u^{T}Av$ (using $A^T=A$, $J^T=-J$), so its spectrum is purely imaginary; pair complex eigenvectors with their conjugates to build a real symplectic basis that simultaneously diagonalises $A$. Indefinite case (full Williamson 1936 classification): $JA$-spectrum decomposes under joint $\pm$ and $\bar{\cdot }$ symmetries into orbits of four types — purely imaginary pair (elliptic, $H=\omega (q^2+p^2)/2$ with a $\pm$ Krein sign), real pair (saddle, $H=\omega qp$), complex quadruple (loxodromic), and zero (parabolic / nilpotent). Long 1971 refinement at degenerate strata (Williamson-Long form). Connections: quadratic linearisation underlying the Birkhoff normal form near an elliptic equilibrium; symplectic eigenvalues are the frequency vector entering the KAM Diophantine condition; symplectic capacity of an ellipsoid $\{z^{T}Az\le 1\}$ is $\pi /{\omega }_{\max }$; metaplectic quantisation has spectrum $\sum (n_k+1/2){\omega }_k$; Robertson-Schrödinger uncertainty principle bounds each symplectic eigenvalue of a Gaussian covariance matrix below by $\hslash /2$; Krein theory of strong stability identifies elliptic-block sign collisions as the linear birth of parametric resonance instabilities (e.g. triangular Lagrange points at the critical mass ratio of the restricted three-body problem). Originator: Williamson 1936 On the algebraic problem concerning the normal forms of linear dynamical systems (Amer. J. Math. 58); Long 1971 refinement; Arnold App. 6 mechanics-flavoured exposition.

  requires: symplectic-geometry.symplectic-vector-space, symplectic-geometry.symplectic-manifold, symplectic-geometry.symplectic-group

• topology.cofibrationopen unit 02.01.08 →

  Cofibration and homotopy extension property

  Map $i:A\to X$ with the homotopy extension property: any homotopy $H:A\times [0,1]\to Z$ compatible with $f:X\to Z$ extends to $\tilde{H}:X\times [0,1]\to Z$. Equivalent retract characterisation: $X\times \{0\}{\cup }_{A\times \{0\}}A\times [0,1]$ is a retract of $X\times [0,1]$. Mapping-cylinder factorisation $X\hookrightarrow M_f\stackrel{\simeq }{\to }Y$. CW pair inclusions are cofibrations. Eckmann-Hilton dual to fibration; cofibre sequence $A\to X\to X/A\to \Sigma A\to \cdots$. Originator: Borsuk 1931 (ANR theory); modern HEP framing Strom 1968 / Steenrod 1967.

  requires: topology.topological-space, topology.continuous-map, topology.quotient-topology, topology.homotopy

• topology.compact-open-topologyopen unit 02.01.09 →

  Compact-open topology and function spaces

  Compact-open topology on $C(X,Y)$: subbasis $W(K,U)=\{f:f(K)\subseteq U\}$ for $K$ compact, $U$ open. Evaluation $C(X,Y)\times X\to Y$ continuous when $X$ is locally compact Hausdorff. Exponential law $C(X\times Y,Z)\cong C(X,C(Y,Z))$ as a homeomorphism. Compactly-generated weak Hausdorff (CGWH) = Steenrod's convenient category. Loop space $\Omega X$, suspension-loop adjunction $[\Sigma X,Y]\cong [X,\Omega Y]$. Originator: Fox 1945; Steenrod 1967 A convenient category of topological spaces.

  requires: topology.topological-space, topology.continuous-map

• topology.cw-complexopen unit 03.12.10 →

  CW complex

  Inductive skeleton construction $X^{(n)}=X^{(n-1)}{\cup }_{\Phi }{⨆}_{\alpha }{D}_{\alpha }^n$ with attaching map $\Phi :{⨆}_{\alpha }{S}_{\alpha }^{n-1}\to X^{(n-1)}$. Weak (colimit) topology on $X=\mathrm{colim}{X}^{(n)}$. Cellular pushout square. Standard examples: $S^n$, ${\mathbb{RP}}^n$, ${\mathbb{CP}}^n$, classifying spaces, Lie groups. CW pair inclusions are cofibrations. Cellular approximation theorem; Whitehead's theorem (homotopy-equivalence detection by ${\pi }_n$). Originator: J.H.C. Whitehead 1949 Combinatorial homotopy I.

  requires: topology.topological-space, topology.quotient-topology, topology.homotopy

• topology.fibrationopen unit 02.01.07 →

  Fibration (Hurewicz and Serre)

  Hurewicz fibration: HLP for all spaces. Serre fibration: HLP for CW pairs / discs. Standard examples: covering maps, fibre bundles over paracompact bases, path-space fibration $PX\to X$ with fibre $\Omega X$. Long exact sequence of homotopy groups ${\pi }_n(F)\to {\pi }_n(E)\to {\pi }_n(B)\to {\pi }_{n-1}(F)$. Fibration replacement: any map factors through a Hurewicz fibration. Hopf fibration $S^1\to S^3\to S^2$ as the originator example. Loop-space adjunction: ${\pi }_n(\Omega X)\cong {\pi }_{n+1}(X)$. Connection to Leray-Serre spectral sequence.

  requires: topology.topological-space, topology.continuous-map, topology.homotopy

• topology.quotient-topologyopen unit 02.01.06 →

  Quotient and identification topology

  Identification topology on $Y=X/\sim$: $U$ open iff $q^{-1}(U)$ open in $X$. Universal property: continuous maps $Y\to Z$ are continuous maps $X\to Z$ constant on equivalence classes. Standard quotients: cone $CX$, suspension $SX$ ($\Sigma S^n=S^{n+1}$), mapping cylinder $M_f$, mapping cone $C_f$, adjunction space $Y{\cup }_{f}X$, wedge $X\vee Y$, smash $X\wedge Y$. Cellular pushout for CW complex skeleton attachments. Quotient by group action gives covering spaces when properly discontinuous.

  requires: topology.topological-space, topology.continuous-map

• topology.seifert-van-kampenopen unit 03.12.09 →

  Seifert-van Kampen theorem

  Classical group form: ${\pi }_1(U\cup V,x_0)={\pi }_1(U,x_0){\ast }_{{\pi }_1(U\cap V,x_0)}{\pi }_1(V,x_0)$ when $U$, $V$, $U\cap V$ path-connected. Brown's groupoid form: pushout in $\mathbf{Groupoid}$ for ${\pi }_1(X,A)$ with $A$ meeting every path-component of $U$, $V$, $U\cap V$ — no connectedness hypothesis on $U\cap V$. Lebesgue-number subdivision argument. Key computations: figure eight ($F_2$), sphere ($0$), genus-$g$ surface ($⟨a_i,b_i\mid \prod [a_i,b_i]⟩$).

  requires: topology.homotopy, topology.fundamental-groupoid, topology.topological-space

• rep-theory.cartan-weyl-classificationopen unit 07.04.01 →

  Cartan-Weyl classification

  Bijection {irreducible Dynkin diagrams} ↔ {simple complex Lie algebras}. Four infinite families $A_n,B_n,C_n,D_n$ and five exceptionals $G_2,F_4,E_6,E_7,E_8$. Root systems with crystallographic condition. Cartan matrix and Serre relations. Compact real forms. Langlands duality on root systems. Connections to Kac-Moody algebras, buildings (Tits), McKay correspondence, $E_8$ in lattice theory and string theory.

  requires: rep-theory.group-representation, rep-theory.highest-weight-representation, lie.lie-algebra

• rep-theory.highest-weight-representationopen unit 07.03.01 →

  Highest weight representation

  Cartan + root decomposition $\mathfrak{g}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}\oplus {\mathfrak{n}}^{-}$. Weight space decomposition. Highest weight vector annihilated by ${\mathfrak{n}}^{+}$. Verma modules $M(\lambda )$, irreducible quotient $L(\lambda )$. Bijection between dominant integral weights and finite-dim irreducibles. Weyl character formula and Weyl dimension formula. Borel-Weil(-Bott) realisation. Category $\mathcal{O}$, Kazhdan-Lusztig polynomials, crystals, quantum groups.

  requires: rep-theory.group-representation, rep-theory.schur-lemma, lie.lie-algebra

• rep-theory.schur-lemmaopen unit 07.01.02 →

  Schur's lemma

  (1) An equivariant linear map between irreducibles is zero or an isomorphism. (2) Over an algebraically closed field, ${\mathrm{End}}_G(V)=k\cdot {\mathrm{id}}_V$ for irreducible $V$. Powers character orthogonality, dimension formula $\sum (\dim V_i{)}^2=|G|$. Generalises to simple modules over algebras and to abelian categories. von Neumann's commutant theorem extends to unitary representations. Schur-Weyl duality and categorical Schur.

  requires: rep-theory.group-representation, linalg.vector-space

• rep-theory.group-representationopen unit 07.01.01 →

  Group representation

  Homomorphism $\rho :G\to \mathrm{GL}(V)$ equivalently a $kG$-module structure on $V$. Subrepresentations, irreducibility, semisimplicity, intertwiners. Direct sum, tensor product, dual, Hom. Maschke's theorem (complete reducibility in char 0 or coprime to $|G|$). Characters, orthogonality relations. Regular representation. Frobenius reciprocity. Connections to modular and categorical representation theory.

  requires: groups.group, linalg.vector-space

• rep-theory.symmetric-group-representationopen unit 07.05.01 →

  Symmetric group representation

  Bijection {partitions of $n$} ↔ {irreducible $S_n$-reps over $\mathbb{C}$}. Frobenius character formula via Vandermonde × power-sum determinant. Young symmetriser $c_{\lambda }\in \mathbb{C}[S_n]$ realising $V^{\lambda }=\mathbb{C}[S_n]\cdot c_{\lambda }$ as a left ideal. Hook length formula $\dim V^{\lambda }=n!/\prod h(i,j)$. Murnaghan-Nakayama border-strip rule. Frobenius characteristic map to symmetric functions. Schur-Weyl duality with ${\mathrm{GL}}_d$. RSK correspondence. Connections to Hecke algebras, Brauer algebras, crystal bases, Schubert calculus.

  requires: rep-theory.group-representation, rep-theory.schur-lemma

• rep-theory.young-diagramopen unit 07.05.02 →

  Young diagram and tableau

  Young diagram $[\lambda ]$ as left-justified array of cells encoding partition $\lambda ⊢n$. Standard Young tableau (SYT): bijective filling strictly increasing along rows and columns; $f^{\lambda }=|\mathrm{SYT}(\lambda )|$. Semistandard Young tableau (SSYT): weak rows, strict columns, with content. Hook length formula $f^{\lambda }=n!/\prod h(i,j)$ (Frame-Robinson-Thrall 1954); Greene-Nijenhuis-Wilf hook-walk proof; Novelli-Pak-Stoyanovskii bijective proof. Schur polynomials $s_{\lambda }$ as generating functions of SSYTs; basis of $\Lambda$. RSK correspondence $S_n\leftrightarrow {⨆}_{\lambda }\mathrm{SYT}(\lambda {)}^2$. Littlewood-Richardson rule. Plancherel measure asymptotics (Vershik-Kerov-Logan-Shepp; Baik-Deift-Johansson Tracy-Widom). Crystal bases (Kashiwara). Schubert calculus on Grassmannians.

  requires: rep-theory.symmetric-group-representation

• rep-theory.specht-moduleopen unit 07.05.03 →

  Specht module

  Permutation module $M^{\lambda }=k\{\text{tabloids}\}={\mathrm{Ind}}_{S_{\lambda }}^{S_n}(\mathrm{triv})$. Polytabloid $e_T={\sum }_{\sigma \in C(T)}\mathrm{sgn}(\sigma )\{\sigma T\}$ (column antisymmetrisation). Specht module $S^{\lambda }=k⟨e_T⟩\subseteq M^{\lambda }$. Standard polytabloids $\{e_T:T\in \mathrm{SYT}(\lambda )\}$ form a $\mathbb{Z}$-basis. Theorem: in char 0, $\{S^{\lambda }{\}}_{\lambda ⊢n}$ are exactly the irreducible $k{S}_n$-modules; in char $p$, $D^{\lambda }=S^{\lambda }/\mathrm{rad}(S^{\lambda })$ for $p$-regular $\lambda$ are the irreducible modular representations (James 1976). James submodule theorem: any submodule $U\subseteq M^{\lambda }$ contains $S^{\lambda }$ or sits in $(S^{\lambda }{)}^{\perp }$. Garnir relations. Branching rule via removable corners. Mullineux involution $D^{\lambda }\otimes \mathrm{sgn}\cong D^{m_p(\lambda )}$ (Ford-Kleshchev 1997). Connections to Hecke algebras, $q$-Schur algebras, LLT algorithm, KLR categorification, Cellular algebras (Graham-Lehrer).

  requires: rep-theory.symmetric-group-representation, rep-theory.young-diagram

• complex-analysis.riemann-roch-compact-rsopen unit 06.04.01 →

  Riemann-Roch theorem for compact Riemann surfaces

  $\ell (D)-i(D)=\mathrm{deg}(D)-g+1$ on compact Riemann surface of genus $g$. Analytic version of algebraic Riemann-Roch [04.04.01]; equivalent by Serre's GAGA. Index of speciality $i(D)=\dim \{\omega \in {\Omega }_{\mathrm{hol}}^1:(\omega )\ge D\}$. Serre duality identifies $i(D)=\ell (K-D)$. Hodge decomposition powers the analytic proof. Riemann-Hurwitz, Brill-Noether, Clifford generalisations. Hirzebruch-Riemann-Roch and Grothendieck-Riemann-Roch generalisations.

  requires: complex-analysis.holomorphic-function, complex-analysis.riemann-surface, alg-geom.riemann-roch-curves

• complex-analysis.riemann-surfaceopen unit 06.03.01 →

  Riemann surface

  1-dimensional complex manifold (real dimension 2). Charts to $\mathbb{C}$ with holomorphic transitions. Examples: $\mathbb{C}$, Riemann sphere $\hat{\mathbb{C}}$, tori $\mathbb{C}/\Lambda$, smooth complex projective curves. Genus and Euler characteristic. Uniformization: every simply connected Riemann surface is $\hat{\mathbb{C}}$, $\mathbb{C}$, or $\mathbb{H}$. GAGA: compact Riemann surfaces $=$ smooth complex projective curves. Sheaves ${\mathcal{O}}_X$, ${\omega }_X$, $\mathcal{O}(D)$. Hodge decomposition, Abel-Jacobi, Teichmüller and moduli spaces.

  requires: complex-analysis.holomorphic-function, manifolds.smooth-manifold, topology.topological-space

• complex-analysis.holomorphic-functionopen unit 06.01.01 →

  Holomorphic function

  Complex differentiability $f^{\prime }(z_0)={\lim }_{h\to 0}(f(z_0+h)-f(z_0))/h$ direction-independently. Cauchy-Riemann equations $u_x=v_y$, $u_y=-v_x$ equivalent to $\bar{\partial }f=0$. Analyticity (local power series). Cauchy's theorem, Cauchy integral formula, residue theorem, Liouville, maximum modulus, identity theorem. Conformal maps, open mapping theorem, Schwarz lemma, Riemann mapping theorem, Picard, Mittag-Leffler.

  requires: topology.topological-space, topology.continuous-map

• alg-geom.cartier-divisoropen unit 04.05.04 →

  Cartier divisor

  Section of ${\mathcal{K}}^{\times }/{\mathcal{O}}^{\times }$ — system of local equations $\{(U_i,f_i)\}$ with $f_i/f_j\in {\mathcal{O}}^{\times }$. $\mathrm{CaCl}(X)=\mathrm{Pic}(X)$ exactly. On smooth/locally-factorial schemes, $\mathrm{CaDiv}=\mathrm{Div}$. Effective Cartier divisor = locally principal codimension-1 closed subscheme. Failure of Weil = Cartier on cone over conic. Base-change-friendly, the natural relative-setting divisor. Cartier dual of group schemes. Arakelov arithmetic.

  requires: alg-geom.weil-divisor, alg-geom.line-bundle-scheme, alg-geom.coherent-sheaf

• alg-geom.line-bundle-schemeopen unit 04.05.03 →

  Line bundle on a scheme

  Locally free coherent sheaf of rank 1. Transition functions in ${\mathcal{O}}^{\times }$. $\mathcal{L}\otimes {\mathcal{L}}^{-1}={\mathcal{O}}_X$. Picard group $\mathrm{Pic}(X)=H^1(X;{\mathcal{O}}^{\times })$ via Čech cohomology. Equivalence with Cartier divisors. On locally factorial schemes, $\mathrm{Pic}=\mathrm{Cl}$. Twisting sheaves $\mathcal{O}(d)$ on ${\mathbb{P}}^n$. Exponential sequence and first Chern class. Picard scheme. Ample/nef/big positivity classes.

  requires: alg-geom.coherent-sheaf, alg-geom.quasi-coherent-sheaf, alg-geom.weil-divisor

• alg-geom.weil-divisoropen unit 04.05.01 →

  Weil divisor

  Formal $\mathbb{Z}$-linear combinations of codimension-1 prime divisors. $\mathrm{div}(f)=\sum v_Y(f)[Y]$ for rational $f$. Principal divisors and divisor class group $\mathrm{Cl}(X)$. Linear equivalence. $\mathrm{Cl}({\mathbb{P}}^n)=\mathbb{Z}$ via degree. ${\mathrm{Cl}}^0$ for curves equals Jacobian. Connection to ideal class group of number fields. Equality $\mathrm{Cl}=\mathrm{Pic}$ for locally factorial schemes. Excision sequence for class groups. Foundation for Chow rings.

  requires: alg-geom.scheme, alg-geom.affine-scheme, alg-geom.coherent-sheaf

• alg-geom.coherent-sheafopen unit 04.06.02 →

  Coherent sheaf

  Quasi-coherent + locally finitely generated. On Noetherian $\mathrm{Spec}(R)$: ${\mathbf{Mod}}^{\mathrm{fg}}(R)\cong \mathbf{Coh}(\mathrm{Spec}(R))$. Closed under kernel/cokernel/tensor/Hom/pullback. Pushforward coherent under proper morphisms (Grothendieck EGA III). Finiteness theorem: $H^i$ on projective scheme is finitely generated. Hilbert polynomial, Castelnuovo-Mumford regularity. Resolution by locally free sheaves on regular schemes.

  requires: alg-geom.quasi-coherent-sheaf, alg-geom.scheme

• alg-geom.quasi-coherent-sheafopen unit 04.06.01 →

  Quasi-coherent sheaf

  On $\mathrm{Spec}(R)$, quasi-coherent sheaves $\tilde{M}$ correspond to $R$-modules $M$ via $\tilde{M}(D(f))=M[f^{-1}]$. Equivalence $\mathbf{Mod}(R)\cong \mathbf{QCoh}(\mathrm{Spec}(R))$. Closed under kernels, cokernels, tensor product, pullback, pushforward (q-c-q-s). Serre's vanishing on affines: $H^i(\mathrm{Spec}(R);\tilde{M})=0$ for $i\ge 1$. On Proj: $\mathbf{grMod}(S)/\mathbf{Tor}\cong \mathbf{QCoh}(\mathrm{Proj}(S))$.

  requires: alg-geom.sheaf, alg-geom.scheme, alg-geom.affine-scheme

• alg-geom.affine-schemeopen unit 04.02.02 →

  Affine scheme

  $\mathrm{Spec}(R)=\{\text{prime ideals of }R\}$ with Zariski topology and structure sheaf $\mathcal{O}$. Stalks are localisations $R_{\mathfrak{p}}$. Locally ringed space. Spec/Γ adjunction: ${\mathbf{CRing}}^{\mathrm{op}}\cong \mathbf{AffSch}$. Distinguished opens $D(f)$, sections $\mathcal{O}(D(f))=R_f$. Hilbert Nullstellensatz reformulation. Generic vs closed points. Non-reduced schemes via nilpotents. $\mathrm{Spec}(\mathbb{Z})$ as arithmetic line.

  requires: alg-geom.sheaf, alg-geom.scheme, algebra.associative-algebra, algebra.ideal

• alg-geom.projective-schemeopen unit 04.02.03 →

  Projective scheme

  $\mathrm{Proj}(S)$ for graded commutative ring $S$; homogeneous primes excluding the irrelevant ideal $S_{+}$. Affine cover by distinguished opens $D(f)\cong \mathrm{Spec}((S[f^{-1}]{)}_0)$. Twisting sheaves $\mathcal{O}(d)$. Every projective $k$-scheme embeds in ${\mathbb{P}}_k^N$. Veronese and Segre embeddings. Closed subschemes correspond to homogeneous ideals. Coherent sheaves correspond to graded modules modulo torsion. Hilbert polynomial / regularity / Hilbert scheme.

  requires: alg-geom.sheaf, alg-geom.scheme, alg-geom.affine-scheme, alg-geom.projective-space

• alg-geom.projective-spaceopen unit 04.07.01 →

  Projective space

  ${\mathbb{P}}_k^n=(k^{n+1}\setminus 0)/k^{\times }=\mathrm{Proj}k[x_0,\dots ,x_n]$. Open affine cover by $U_i=\{x_i\ne 0\}$. Twisting sheaves $\mathcal{O}(d)$, foundational cohomology computation. $\mathrm{Pic}({\mathbb{P}}^n)=\mathbb{Z}$. Canonical $\omega =\mathcal{O}(-n-1)$. Functor of points: surjections $R^{n+1}\to L$. Toric perspective. Plücker embedding of Grassmannians. Bézout's theorem.

  requires: alg-geom.scheme, linalg.vector-space, linalg.field

• alg-geom.riemann-roch-curvesopen unit 04.04.01 →

  Riemann-Roch theorem for curves

  $\ell (D)-\ell (K-D)=\mathrm{deg}(D)-g+1$ on smooth projective curve of genus $g$. Equivalently $\chi (\mathcal{L})=\mathrm{deg}(\mathcal{L})+1-g$. Serre duality $H^1(C;\mathcal{L})\cong H^0(C;{\omega }_C\otimes {\mathcal{L}}^{-1}{)}^{\vee }$. Canonical divisor of degree $2g-2$. Inductive proof via skyscraper short exact sequences. Hirzebruch and Grothendieck generalisations. Brill-Noether theory and Clifford's theorem for special divisors.

  requires: alg-geom.sheaf, alg-geom.scheme, alg-geom.sheaf-cohomology

• alg-geom.hurwitz-formulaopen unit 04.04.02 →

  Hurwitz formula

  $2g_Y-2=n(2g_X-2)+\mathrm{deg}R$ for a finite separable morphism $\pi :Y\to X$ of smooth projective curves of degree $n$, with ramification divisor $R={\sum }_y(e_y-1)y$ in the tame case ($\mathrm{char}k\nmid e_y$). Proof via ${\pi }^{\ast }{K}_X=K_Y-R$ and degree comparison. Worked examples: $t\mapsto t^n$ on ${\mathbb{P}}^1$, hyperelliptic double cover ramified at $2g+2$ points, elliptic double cover with four branch points. Wild ramification in positive characteristic replaces $\sum (e_y-1)$ with the different ${\mathfrak{d}}_{Y/X}$. Castelnuovo-Severi inequality bounds $g_Y$ in terms of $g_X$, $n$, and ramification.

  requires: alg-geom.riemann-roch-curves

• alg-geom.elliptic-curvesopen unit 04.04.03 →

  Elliptic curves

  A smooth projective curve of genus 1 over $k$ equipped with a $k$-rational point $O$. Equivalently a 1-dimensional abelian variety over $k$. Weierstrass form $y^2=x^3+ax+b$ with $\Delta =-16(4a^3+27b^2)\ne 0$ (in $\mathrm{char}k\ne 2,3$); $j$-invariant $j(E)=1728\cdot 4a^3/(4a^3+27b^2)$ classifies $E$ over $\bar{k}$. Group law via chord-and-tangent construction or via $E(k)\cong {\mathrm{Pic}}^0(E)$ from Riemann-Roch on a genus-1 curve. Mordell-Weil theorem: $E(K)$ is finitely generated for every number field $K$ (Mordell 1922 for $\mathbb{Q}$, Weil 1929 for general number fields). Mazur's torsion theorem 1977: $E(\mathbb{Q}{)}_{\mathrm{tors}}$ is one of 15 explicit groups. Modularity theorem (Wiles-Taylor-Breuil-Conrad-Diamond 1995-2001): every $E/\mathbb{Q}$ is modular. BSD conjecture: ${\mathrm{ord}}_{s=1}L(E,s)=\mathrm{rank}(E(\mathbb{Q}))$. Hasse bound over ${\mathbb{F}}_q$: $|\#E({\mathbb{F}}_q)-q-1|\le 2\sqrt{q}$. Tate's algorithm for reduction types. CM theory and class field theory of imaginary quadratic fields. Heegner points and Gross-Zagier-Kolyvagin for rank-$\le 1$ BSD. Modular curves $X_0(N),X_1(N)$ as moduli of elliptic curves with level structure.

  requires: alg-geom.riemann-roch-curves, alg-geom.hurwitz-formula, alg-geom.picard-group

• alg-geom.sheaf-cohomologyopen unit 04.03.01 →

  Sheaf cohomology

  $H^i(X;\mathcal{F})=R^i\Gamma (X,-)$ as right-derived functors of global sections. Long exact sequence in cohomology from short exact sequence of sheaves. Čech cohomology and comparison theorem. Acyclic resolutions: flabby, soft, fine. Serre's vanishing theorem on affine schemes. Grothendieck vanishing in degrees above dimension. Leray spectral sequence. Hodge decomposition for Kähler manifolds. Coherent cohomology of ${\mathbb{P}}^n$ via $\mathcal{O}(d)$.

  requires: alg-geom.sheaf, alg-geom.scheme, topology.de-rham-cohomology

• alg-geom.schemeopen unit 04.02.01 →

  Scheme

  Locally ringed space locally isomorphic to $\mathrm{Spec}(R)$. Anti-equivalence ${\mathbf{CRing}}^{\mathrm{op}}\cong \mathbf{AffSch}$. Zariski topology, structure sheaf, projective and affine schemes. Functor of points. Reduced vs non-reduced (nilpotents).

  requires: alg-geom.sheaf, algebra.associative-algebra, algebra.ideal, topology.topological-space

• alg-geom.sheafopen unit 04.01.01 →

  Sheaf

  Presheaf as contravariant functor from open sets; sheaf axioms (locality + gluing); sheafification; stalks; morphisms; pushforward and pullback adjunction; étale space construction; sheaves form a topos.

  requires: topology.topological-space, topology.continuous-map

• alg-geom.picard-groupopen unit 04.05.02 →

  Picard group

  $\mathrm{Pic}(X)=\{\text{line bundles on }X\}/\cong$ under tensor product. Three descriptions: $H^1(X;{\mathcal{O}}_X^{\times })$ via Čech cocycles; Cartier divisor classes $\mathrm{CaCl}(X)$; Weil divisor classes $\mathrm{Cl}(X)$ on locally factorial schemes. Functorial pullback. Picard scheme ${\mathrm{Pic}}_{X/S}$ representing the relative Picard functor (Grothendieck FGA 1962). ${\mathrm{Pic}}^0(X)$ as connected component, abelian variety for smooth projective $X$. Néron-Severi $\mathrm{NS}(X)=\mathrm{Pic}(X)/{\mathrm{Pic}}^0(X)$, finitely generated. Jacobian of a curve as ${\mathrm{Pic}}^0(C)$. Examples: $\mathrm{Pic}({\mathbb{P}}^n)=\mathbb{Z}$, $\mathrm{Pic}(\mathrm{Spec}{\mathcal{O}}_K)=\mathrm{Cl}({\mathcal{O}}_K)$. Picard scheme of an abelian variety as dual abelian variety with Poincaré bundle (Mumford).

  requires: alg-geom.weil-divisor, alg-geom.line-bundle-scheme, alg-geom.cartier-divisor

• alg-geom.ample-line-bundleopen unit 04.05.05 →

  Ample and very ample line bundle

  $\mathcal{L}$ very ample if global sections give a closed immersion $X\hookrightarrow {\mathbb{P}}^N$; ample if some power ${\mathcal{L}}^{\otimes m}$ is very ample. Cartan-Serre-Grothendieck: $\mathcal{L}$ ample iff for every coherent $\mathcal{F}$, $\mathcal{F}\otimes {\mathcal{L}}^{\otimes m}$ is globally generated for $m\gg 0$ iff $H^i(X;\mathcal{F}\otimes {\mathcal{L}}^{\otimes m})=0$ for $i\ge 1$, $m\gg 0$. Numerical (Nakai-Moishezon): on a complete scheme, $\mathcal{L}$ ample iff ${\mathcal{L}}^{\dim Y}\cdot Y>0$ for every irreducible closed $Y$. Kleiman criterion for nef. Ample cone in $\mathrm{NS}(X)\otimes \mathbb{R}$, dual to Mori cone of curves. Kodaira embedding theorem (analytic ample = positive line bundle). MMP and the cone theorem.

  requires: alg-geom.line-bundle-scheme, alg-geom.coherent-sheaf, alg-geom.projective-space

• alg-geom.stalk-of-sheafopen unit 04.01.02 →

  Stalk of a sheaf

  ${\mathcal{F}}_x={\underrightarrow{\lim }}_{U\ni x}\mathcal{F}(U)$, the colimit over open neighbourhoods of $x$. Equivalently equivalence classes of pairs $(U,s)$ with $s\in \mathcal{F}(U)$, modulo agreement on a smaller neighbourhood. Stalks of ${\mathcal{O}}_X$ are local rings. Sheafification produces a sheaf with the same stalks as the input presheaf. Morphisms of sheaves are isomorphisms iff each stalk map is an isomorphism. Skyscraper sheaves. Étale space $\acute{\mathrm{E}}\mathrm{t}(\mathcal{F})={⨆}_x{\mathcal{F}}_x$ with local-section topology. Leray's 1946 introduction in Oflag XVII-A.

  requires: alg-geom.sheaf

• alg-geom.sheafificationopen unit 04.01.03 →

  Sheafification

  Left adjoint $(-{)}^{+}:\mathbf{PSh}(X)\to \mathbf{Sh}(X)$ to the inclusion. Construction via étale-space: ${\mathcal{F}}^{+}(U)=\{$continuous sections $s:U\to \acute{\mathrm{E}}\mathrm{t}(\mathcal{F})\}$. Universal property: for any sheaf $\mathcal{G}$ and presheaf morphism $\mathcal{F}\to \mathcal{G}$, factors uniquely through ${\mathcal{F}}^{+}$. Stalks unchanged: ${\mathcal{F}}_x^{+}={\mathcal{F}}_x$ for all $x$. Sheafification is exact. For sheaf categories: kernels are presheaf kernels, cokernels and images require sheafification. Foundational tool throughout sheaf theory. Cartan-Serre exposés 1948–55; systematised in Godement 1958.

  requires: alg-geom.sheaf, alg-geom.stalk-of-sheaf

• alg-geom.direct-inverse-imageopen unit 04.01.04 →

  Direct and inverse image of sheaves

  For $f:X\to Y$ continuous, pushforward $f_{\ast }\mathcal{F}(V)=\mathcal{F}(f^{-1}(V))$. Inverse image $f^{-1}\mathcal{G}$ is the sheafification of $V\mapsto {\underrightarrow{\lim }}_{U\supseteq f(V)}\mathcal{G}(U)$. Adjunction $f^{-1}⊣f_{\ast }$ on sheaves of sets. For ringed spaces: pullback $f^{\ast }\mathcal{G}=f^{-1}\mathcal{G}{\otimes }_{f^{-1}{\mathcal{O}}_Y}{\mathcal{O}}_X$, adjoint to $f_{\ast }$ on $\mathcal{O}$-modules. $f^{-1}$ exact; $f_{\ast }$ left exact only. Right derived $R^i{f}_{\ast }$ measure failure. Six functor formalism precursor: Grothendieck Tôhoku 1957. Proper base change for $f$ proper, projection formula. Modern: Lurie's $\infty$-categorical six functors; Grothendieck duality $f^{!}$.

  requires: alg-geom.sheaf

• alg-geom.morphism-of-schemesopen unit 04.02.04 →

  Morphism of schemes

  Morphism of locally ringed spaces $(f,f^{♯}):(X,{\mathcal{O}}_X)\to (Y,{\mathcal{O}}_Y)$ with $f$ continuous and $f^{♯}:{\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$ a sheaf-of-rings map inducing local-ring maps on stalks. For affine schemes $\mathrm{Hom}(\mathrm{Spec}A,\mathrm{Spec}B)=\mathrm{Hom}(B,A)$ (anti-equivalence). Properties (EGA IV): finite type, finite presentation, separated, proper, flat, smooth, étale, finite, affine, projective, quasi-finite, quasi-compact. Closed/open immersions. Base change $X{\times }_{Y}Z$. Diagonal $\Delta :X\to X{\times }_{Y}X$. Valuative criteria for separatedness and properness. Functor of points: $h_X(T)=\mathrm{Hom}(T,X)$. Galois descent and faithfully flat descent for morphisms.

  requires: alg-geom.scheme, alg-geom.affine-scheme

• alg-geom.smooth-etale-unramifiedopen unit 04.02.05 →

  Smooth, étale, and unramified morphisms

  Three local properties of $f:X\to Y$ at $x\in X$, $y=f(x)$. Unramified: $f$ locally of finite type with ${\Omega }_{X/Y,x}^1=0$, equivalently ${\mathfrak{m}}_y{\mathcal{O}}_{X,x}={\mathfrak{m}}_x$ and $\kappa (x)/\kappa (y)$ finite separable. Smooth (relative dim $n$): $f$ locally of finite presentation, flat at $x$, with regular geometric fibre $X_{\bar{y}}$ at $\bar{x}$; equivalently ${\Omega }_{X/Y}^1$ locally free of rank $n$ near $x$ (EGA IV.17.5.1). Étale: smooth of relative dimension $0$ = flat + locally finitely presented + unramified. Jacobian criterion: for $X=\mathrm{Spec}(A[t_1,\dots ,t_n]/(g_1,\dots ,g_c))$ over $\mathrm{Spec}(A)$, smooth at $x$ of relative dim $n-c$ iff $\mathrm{rank}(\partial g_i/\partial t_j)(x)=c$. Stable under composition and base change. Examples: $\mathrm{Spec}(k[x,y]/(y^2-x))\to \mathrm{Spec}(k[x])$ étale on $x\ne 0$ (char $\ne 2$); Frobenius $F:X\to X$ in char $p$ never étale (purely inseparable); $z\mapsto z^n$ on ${\mathbb{G}}_m$ étale iff $n$ invertible in base. Local structure (EGA IV.18.4.6): every unramified morphism factors étale-locally as a closed immersion into an étale cover. Formal characterisations: smooth = formally smooth + locally finitely presented (lifting along square-zero ideals surjective); étale = formally étale (bijective lifting); unramified = formally unramified (injective lifting). Étale fundamental group ${\pi }_1^{\text{eˊt}}(X,\bar{x})$ classifies finite étale covers (SGA 1 Exposé V); profinite completion of ${\pi }_1^{\text{top}}(X^{\text{an}})$ for smooth complex $X$ (SGA 1 Exposé XII, Riemann existence). Henselisation / strict henselisation as étale-local rings; cotangent complex $L_{X/Y}$ as derived refinement (Illusie 1971–72); Kunz 1969: regular ⟺ Frobenius flat in char $p$. Originators: Auslander-Buchsbaum-Serre 1956–58 (regularity criterion); Grothendieck-Dieudonné EGA IV (1965–67); Grothendieck SGA 1 (1960–61, étale fundamental group).

  requires: alg-geom.morphism-of-schemes, alg-geom.sheaf-of-differentials

• alg-geom.nullstellensatz-dimensionopen unit 04.02.07 →

  Nullstellensatz and dimension theory

  Over algebraically closed $k$ and $R=k[x_1,\dots ,x_n]$: Weak Nullstellensatz — every proper ideal $\mathfrak{a}⊊R$ has $V(\mathfrak{a})\ne \varnothing$. Strong Nullstellensatz — $I(V(\mathfrak{a}))=\sqrt{\mathfrak{a}}$, equivalently $f\in I(V(\mathfrak{a}))\iff f^N\in \mathfrak{a}$. Maximal-ideal form — every maximal ideal $\mathfrak{m}\subset R$ has the form ${\mathfrak{m}}_a=(x_1-a_1,\dots ,x_n-a_n)$ for a unique $a\in k^n$. Proof: weak via Zariski lemma (a field finitely generated as a $k$-algebra equals $k$ when $k$ is algebraically closed) and Noether normalisation; weak ⇒ strong via the Rabinowitsch trick (introduce $y$ and use $(\mathfrak{a},1-yf)$). Krull dimension $\dim R=\sup$ length of prime chains ${\mathfrak{p}}_0⊊\cdots ⊊{\mathfrak{p}}_d$. For irreducible affine variety $X$ over $k$: $\dim X={\dim }_{\mathrm{Krull}}k[X]={tr.deg}_{k}k(X)$. For noetherian local $(R,\mathfrak{m})$: $\dim R=\mathrm{deg}{P}_{\mathfrak{q}}(n)$ (Hilbert-Samuel polynomial of $\mathrm{length}(R/{\mathfrak{q}}^n)$ for $\mathfrak{q}$ $\mathfrak{m}$-primary) = min generators of an $\mathfrak{m}$-primary ideal (system of parameters). Krull's Hauptidealsatz / height theorem (1928): in noetherian $R$, every minimal prime over $(f)$ has height $\le 1$ (= 1 if $f$ is a non-zero-divisor non-unit); minimal primes over $(f_1,\dots ,f_r)$ have height $\le r$. Geometrically: hypersurface in ${\mathbb{A}}^n$ has dimension $n-1$. Cohen-Seidenberg going-up / going-down for integral extensions; catenary rings and the dimension formula $\mathrm{ht}(\mathfrak{p})+\dim (R/\mathfrak{p})=\dim R$; finitely generated $k$-algebras universally catenary; Nagata 1956 non-catenary domain. Cohen-Macaulay rings (depth = dimension); Krull's intersection theorem; effective Nullstellensatz (Brownawell 1987 with $N\le n{d}^n$, Kollár 1988 sharpening to $d^n$); real Nullstellensatz / Positivstellensatz (Stengle 1974, Krivine 1964); arithmetic Nullstellensatz over $\mathbb{Z}$; model-theoretic content as quantifier elimination for ${\mathrm{ACF}}_p$; Stillman's conjecture (Ananyan-Hochster 2020). Originators: Hilbert 1893 (Math. Ann. 42, appendix); Lasker 1905 (primary decomposition); Noether 1921 (noetherian foundations); Krull 1928 (Hauptidealsatz); Rainich/Rabinowitsch 1929 (auxiliary-variable trick); Zariski 1947 (modern proof).

  requires: alg-geom.affine-scheme, alg-geom.morphism-of-schemes

• alg-geom.blowupopen unit 04.07.02 →

  Blowup

  ${\mathrm{Bl}}_{Z}X={\mathrm{Proj}}_X{⨁}_{n\ge 0}{\mathcal{I}}^n$, the relative Proj of the Rees algebra. Universal property: minimal surgery making the ideal sheaf invertible. Exceptional divisor as effective Cartier divisor; projective bundle $\mathbb{P}({\mathcal{N}}_{Z/X}^{\vee })$ for smooth centres. Hironaka 1964 resolution theorem in characteristic 0; weak factorisation; Castelnuovo contraction for surfaces. Open in characteristic $p$ for dimension $\ge 4$.

  requires: alg-geom.scheme, alg-geom.projective-space, alg-geom.cartier-divisor

• alg-geom.sheaf-of-differentialsopen unit 04.08.01 →

  Sheaf of differentials

  ${\Omega }_{X/Y}^1$ defined by the universal ${\mathcal{O}}_Y$-linear derivation $d:{\mathcal{O}}_X\to {\Omega }_{X/Y}^1$. Corresponds to Kähler differentials ${\Omega }_{R/S}^1$ on affines. Conormal/normal sheaf exact sequences for closed immersions and smooth morphisms. First fundamental exact sequence $f^{\ast }{\Omega }_{Y/Z}^1\to {\Omega }_{X/Z}^1\to {\Omega }_{X/Y}^1\to 0$. Locally free of rank equal to relative dimension on smooth morphisms. Foundation for canonical sheaf and Serre duality.

  requires: alg-geom.scheme, alg-geom.coherent-sheaf, alg-geom.morphism-of-schemes

• alg-geom.canonical-sheafopen unit 04.08.02 →

  Canonical sheaf

  ${\omega }_X=\det {\Omega }_X^1={\bigwedge }^n{\Omega }_X^1$ for smooth variety of dimension $n$. Canonical divisor class $K_X$. $\mathrm{deg}(K_C)=2g-2$ on a smooth projective curve. Adjunction formula ${\omega }_Y=({\omega }_X\otimes \mathcal{O}(D)){|}_Y$ for smooth divisor $Y\subset X$. Dualising sheaf and Serre duality. Riemann's implicit canonical divisor (1857) via everywhere-holomorphic 1-forms. Kodaira dimension classifies birational geometry by canonical-sheaf positivity.

  requires: alg-geom.sheaf-of-differentials, alg-geom.line-bundle-scheme

• alg-geom.serre-dualityopen unit 04.08.03 →

  Serre duality

  For smooth projective $X$ of dimension $n$ over a field, $H^i(X;\mathcal{F})\cong H^{n-i}(X;{\mathcal{F}}^{\vee }\otimes {\omega }_X{)}^{\vee }$ for locally free $\mathcal{F}$. Trace map $H^n(X;{\omega }_X)\to k$. Serre 1955 Un théorème de dualité. Grothendieck duality (1966) generalises to proper morphisms via dualising complex ${\omega }_X^{\bullet }$. On curves: $H^1(\mathcal{L})\cong H^0(\omega \otimes {\mathcal{L}}^{-1}{)}^{\vee }$, powering Riemann-Roch.

  requires: alg-geom.sheaf-cohomology, alg-geom.canonical-sheaf, alg-geom.coherent-sheaf

• alg-geom.serre-duality-curvesopen unit 06.04.04 →

  Serre duality on a curve

  For a line bundle $L$ on a smooth projective curve $X$ of genus $g$ over an algebraically closed field $k$, the residue trace $\mathrm{tr}:H^1(X;K_X)\to k$ pairs $H^1(X;L)\times H^0(X;K_X\otimes L^{-1})\to k$ non-degenerately, giving $H^1(X;L)\cong H^0(X;K_X\otimes L^{-1}{)}^{\ast }$. Combined with Riemann-Roch ($\dim H^0(L)-\dim H^1(L)=\mathrm{deg}L+1-g$) yields $\dim H^0(L)-\dim H^0(K\otimes L^{-1})=\mathrm{deg}L+1-g$. Vanishing $H^1(L)=0$ for $\mathrm{deg}L>2g-2$. Specialisation of Grothendieck duality to dimension 1; the dualising sheaf is $K_X={\omega }_X$. Geometric content: $H^0(K\otimes L^{-1})$ measures specialty (Brill-Noether). Originator: Serre 1955 (Comm. Math. Helv. 29).

  requires: complex-analysis.holomorphic-line-bundle, complex-analysis.riemann-roch-compact-rs, alg-geom.canonical-sheaf

• alg-geom.hodge-decompositionopen unit 04.09.01 →

  Hodge decomposition

  For compact Kähler $X$, $H^n(X;\mathbb{C})={⨁}_{p+q=n}{H}^{p,q}(X)$ with $H^{p,q}=H^q(X;{\Omega }^p)$ and $\overline{H^{p,q}}=H^{q,p}$. Hodge 1941 Theory and Applications of Harmonic Integrals — harmonic representatives. Hodge-to-de-Rham degeneration. Algebraic proof: Deligne-Illusie 1987 via reduction mod $p$. Polarised Hodge structures and period domains. Bridges algebra and topology.

  requires: alg-geom.sheaf-cohomology, topology.de-rham-cohomology

• hodge.hodge-riemann-bilinearopen unit 04.09.08 →

  Hodge-Riemann bilinear relations

  On a compact Kähler $n$-fold $X$ with Kähler class $\omega$, the Hodge-Riemann bilinear form on $H^k(X,\mathbb{C})$ is $Q(\alpha ,\beta )=(-1{)}^{k(k-1)/2}{\int }_X\alpha \wedge \beta \wedge {\omega }^{n-k}$. The Hodge-Riemann bilinear relations (Hodge 1941) state: (HR1) $Q$ vanishes on $H^{p,q}\times H^{p^{\prime },q^{\prime }}$ unless $(p^{\prime },q^{\prime })=(k-p,k-q)$ — orthogonality by bidegree counting; and (HR2) $i^{p-q}Q(\alpha ,\bar{\alpha })>0$ for non-zero primitive $\alpha \in P^{p,q}$ — positivity on primitive Hodge pieces. The proof of HR2 reduces to Weil's identity $\ast \alpha =i^{p-q}(-1{)}^{k(k-1)/2}{L}^{n-k}\alpha /(n-k)!$ on primitive forms (Weil 1958 Variétés Kähleriennes), which converts $i^{p-q}Q(\alpha ,\bar{\alpha })$ into the squared $L^2$-norm of $\alpha$. The combination of Hodge decomposition + HR1 + HR2 makes $(H^k(X,\mathbb{Q}),F^{\bullet },Q)$ a polarised Hodge structure of weight $k$ — the canonical algebraic-geometric example of the abstract notion (Deligne 1971-74). Surface case ($n=k=2$): HR specialises to the Hodge index theorem (signature $(1,h^{1,1}-1)$ on $H^{1,1}(X,\mathbb{R})$); the algebraic Néron-Severi version has signature $(1,\rho -1)$. Hard Lefschetz follows from HR2 by a primitive-decomposition + positivity argument: $L^k:H^{n-k}\to H^{n+k}$ is an isomorphism. Combinatorial applications: Stanley 1980 proved McMullen's g-conjecture on simplicial polytopes via HR on the cohomology of a smooth projective toric variety; Kahn-Saks 1984 deduced log-concavity correlation inequalities for poset linear extensions; Adiprasito-Huh-Katz 2018 proved the Kähler package (Poincaré duality + Hard Lefschetz + Hodge-Riemann) for the Chow ring of any matroid, settling Rota's 1971 log-concavity conjecture (Huh Fields Medal 2022). Mixed-Hodge generalisation: Cattani-Kaplan-Schmid 1986-87 extend the bilinear relations to limit Hodge structures of nilpotent orbits in degenerations of polarised Hodge structures. The bilinear relations are the load-bearing positivity statement of modern Hodge theory and combinatorial Hodge theory. Lean status: partial, module Codex.AlgGeom.Hodge.HodgeRiemannBilinear records HR1 + HR2 + surface specialisation with sorry proof bodies pending Mathlib Kähler-manifold and primitive-cohomology infrastructure.

  requires: alg-geom.hodge-decomposition` (04.09.01), alg-geom.hodge-index-theorem` (04.05.09)

• alg-geom.hodge-decomposition-curvesopen unit 06.04.03 →

  Hodge decomposition on a compact Riemann surface

  For a compact Riemann surface $X$ (= smooth projective complex curve) of genus $g$, complex conjugation on $C^{\infty }$ forms intertwines the Dolbeault bidegree decomposition with the Kähler-harmonic theorem to give $H^1(X;\mathbb{C})=H^{1,0}(X)\oplus H^{0,1}(X)$, with $H^{1,0}(X)=H^0(X;{\Omega }_X^1)$ (holomorphic 1-forms), $H^{0,1}(X)=H^1(X;{\mathcal{O}}_X)$, and $\overline{H^{1,0}}=H^{0,1}$. Dimensions $h^{1,0}=h^{0,1}=g$, total $2g$ matching topology. Period matrix $\Pi =(A\mid B)$ over a symplectic homology basis satisfies Riemann's bilinear relations $A{B}^T=B{A}^T$, $\mathrm{Im}(B{A}^{-1})>0$, exhibiting the Jacobian $\mathrm{Jac}(X)=H^0(X;{\Omega }_X^1{)}^{\ast }/H_1(X;\mathbb{Z})$ as a principally polarised abelian variety. Curve case of the general Kähler Hodge $(p,q)$-decomposition (Hodge 1941); originator content traces to Riemann's 1857 bilinear relations. Direct prerequisite for Serre duality on curves and for the bilinear-relations and Schottky machinery on Jacobians.

  requires: complex-analysis.holomorphic-line-bundle, complex-analysis.riemann-roch-compact-rs, alg-geom.canonical-sheaf, topology.de-rham-cohomology

• alg-geom.jacobi-inversionopen unit 06.06.06 →

  Jacobi inversion theorem

  For a smooth projective compact Riemann surface $X$ of genus $g\ge 1$, fix a basis ${\omega }_1,\dots ,{\omega }_g$ of $H^0(X;{\Omega }_X^1)$, a symplectic basis of $H_1(X;\mathbb{Z})$, and a reference point $p_0\in X$. The Abel-Jacobi map $\alpha :{\mathrm{Sym}}^g(X)\to \mathrm{Jac}(X)$, $\alpha (p_1+\cdots +p_g)=({\sum }_i{\int }_{p_0}^{p_i}{\omega }_j{)}_j\mathrm{mod}\Lambda$, is surjective and birational (Jacobi 1834 — Considerationes generales de transcendentibus Abelianis, Crelle 9). Riemann 1857 refines this: the exceptional locus, after translation by the Riemann constant $\kappa$, coincides with the theta divisor $\Theta =\alpha (W_{g-1})\subset \mathrm{Jac}(X)$, equivalently with the zero locus of the Riemann theta function $\theta (z;\tau )={\sum }_{n\in {\mathbb{Z}}^g}\exp (\pi i{n}^T\tau n+2\pi i{n}^{T}z)$ (Riemann's vanishing theorem). Proof via four steps: image is closed (compactness of ${\mathrm{Sym}}^g(X)$), image has dimension $g$ (differential = Brill-Noether matrix, full-rank by Riemann-Roch + Serre duality on non-special divisors), image equals the full Jacobian (closed + full-dim in connected target), generic fibre is a point (Riemann-Roch on a generic line bundle of degree $g$ gives $h^0=1$). Combined with Abel's theorem yields the structural identification ${\mathrm{Pic}}^0(X)\cong \mathrm{Jac}(X)$ as complex Lie groups. Brill-Noether stratification $W_d^r\subset {\mathrm{Pic}}^d(X)$ extends the inversion to all degrees: $\dim W_d^r\ge g-(r+1)(g-d+r)$ when non-negative (Kempf 1971, Kleiman-Laksov 1972), with equality on a general curve (Griffiths-Harris 1980, Gieseker 1982). Torelli's theorem (Torelli 1913) recovers $X$ from the principally polarised Jacobian $(\mathrm{Jac}(X),\Theta )$. Schottky problem characterises Jacobi loci inside ${\mathcal{A}}_g$; Schottky 1888 ($g=4$), Welters trisecant (Krichever 2006), Novikov-Shiota KP characterisation (Shiota 1986).

  requires: complex-analysis.jacobian-variety, alg-geom.serre-duality-curves, alg-geom.hodge-decomposition-curves

• alg-geom.riemann-bilinearopen unit 06.06.07 →

  Riemann's bilinear relations

  For a compact Riemann surface $X$ of genus $g$, fix a symplectic basis $a_1,\dots ,a_g,b_1,\dots ,b_g$ of $H_1(X;\mathbb{Z})$ ($a_i\cdot b_j={\delta }_{ij}$, $a_i\cdot a_j=b_i\cdot b_j=0$) and a basis ${\omega }_1,\dots ,{\omega }_g$ of $H^0(X;{\Omega }_X^1)$ normalised so that ${\int }_{a_j}{\omega }_i={\delta }_{ij}$. The period matrix ${\tau }_{ij}={\int }_{b_j}{\omega }_i$ satisfies (RB1) ${\tau }_{ij}={\tau }_{ji}$ (symmetry) and (RB2) $\mathrm{Im}\tau$ positive definite. Equivalently, in non-normalised form $(A\mid B)$, $A{B}^T=B{A}^T$ and $\mathrm{Im}(B{A}^{-1})>0$. Proof: Riemann's bilinear identity ${\int }_X\omega \wedge \eta ={\sum }_k[{\int }_{a_k}\omega \cdot {\int }_{b_k}\eta -{\int }_{b_k}\omega \cdot {\int }_{a_k}\eta ]$ (cut $X$ along the symplectic basis to a $4g$-gon, apply Stokes); for $\omega ,\eta$ holomorphic, $\omega \wedge \eta \in {\Omega }^{2,0}(X)=0$ on a curve, giving (RB1). For (RB2), positivity of $\frac{i}{2}{\int }_X\omega \wedge \bar{\omega }$ (the Kähler form $\frac{i}{2}fdz\wedge d\bar{z}=|f{|}^{2}dV$) plus the bilinear identity gives $\mathrm{Im}\tau$ as the Gram matrix of a positive-definite Hermitian form on $H^0(X;{\Omega }_X^1)$. Geometric content: the period matrix lies in the Siegel upper half space ${\mathfrak{H}}_g=\{$symmetric $g\times g$ complex matrices with positive-definite imaginary part$\}$; conversely every $\tau \in {\mathfrak{H}}_g$ defines a principally polarised abelian variety (PPAV). The Schottky problem (Schottky 1888 for $g=4$ via a quartic modular relation; Welters-Shiota for the KP / Novikov characterisation; Krichever 2006 for the trisecant conjecture) asks which points of ${\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ come from curves: the Jacobi locus has dimension $3g-3$ vs. $g(g+1)/2$ for ${\mathfrak{H}}_g$, so the inclusion is far from surjective for $g\ge 4$. Riemann theta function $\theta (z;\tau )$ on ${\mathbb{C}}^g\times {\mathfrak{H}}_g$ uses the bilinear relations directly in its quasi-periodicity. Foundation of Riemann's 1857 theta-function theory and of the modern theory of Abelian integrals, complex multiplication, Heegner points, modular curves.

  requires: complex-analysis.jacobi-inversion, alg-geom.hodge-decomposition-curves, complex-analysis.period-matrix

• alg-geom.schottky-problemopen unit 06.06.08 →

  Schottky problem

  Identify the period matrices of compact Riemann surfaces among all symmetric positive-imaginary $g\times g$ complex matrices, modulo $\mathrm{Sp}(2g,\mathbb{Z})$. Setup: Siegel upper half-space ${\mathfrak{H}}_g=\{\tau \in M_g(\mathbb{C}):\tau ={\tau }^T,\mathrm{Im}\tau >0\}$ of dimension $g(g+1)/2$; quotient ${\mathcal{A}}_g={\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ = moduli space of principally polarised abelian varieties; period mapping $\mathrm{Per}:{\mathcal{M}}_g\to {\mathcal{A}}_g$, $X\mapsto \mathrm{Jac}(X)$; Jacobi locus ${\mathcal{J}}_g=\mathrm{Per}({\mathcal{M}}_g)\subset {\mathcal{A}}_g$ of dimension $3g-3$ for $g\ge 2$; codimension $(g-2)(g-3)/2$, so ${\mathcal{J}}_g={\mathcal{A}}_g$ iff $g\le 3$ and strict subvariety for $g\ge 4$. Five characterisations of ${\mathcal{J}}_g$: (1) Schottky 1888 — single explicit polynomial $S(\theta )=0$ in even theta-constants of degree $4$ on $12$ specific characteristics, irreducible hypersurface (Igusa 1980); Schottky-Jung 1909 extends via Pryms; (2) Andreotti-Mayer 1967 — ${\mathcal{J}}_g\subseteq {\mathcal{N}}_{g-4}$ where ${\mathcal{N}}_k=\{\dim \mathrm{Sing}(\Theta )\ge k\}$, with the reverse containment conjectural and false in general (Beauville-Debarre 1986, Debarre 1992 identify additional components); (3) Novikov-Shiota 1979/1986 — $(A,\Theta )$ is a Jacobian iff $u(x,y,t)=2{\partial }_x^2\log \theta (xV+yU+tW+z_0;\tau )$ solves the KP equation $\frac{3}{4}{u}_{yy}=(u_t-\frac{3}{2}u{u}_x-\frac{1}{4}{u}_{xxx}{)}_x$, descending from Krichever's 1977 construction of KP solutions from algebraic curves via the Baker-Akhiezer function; alternative algebro-geometric proof in Arbarello-De Concini 1987; (4) Welters 1984 / Krichever 2010 — $(A,\Theta )$ is a Jacobian iff its Kummer variety $K(\tau )\subset {\mathbb{P}}^{2^g-1}$ admits a trisecant line, descending from the Fay 1973 trisecant identity for Jacobians; (5) modular forms / Siegel-Igusa-Tsuyumine ring of theta-constants $\theta [m,m^{\prime }]$ of weight $1/2$ on $\Gamma (2)$. Generalisations: Prym Schottky problem (Beauville 1977, Donagi 1981); hyperelliptic Schottky (Mumford Tata II §IIIb); $k$-gonal strata; $p$-adic Schottky uniformisation (Mumford 1972, Mumford curves). Modern surveys: Donagi 1988, Grushevsky 2009. Foundation for the integration of curve geometry, theta-function theory, Sato-Grassmannian / Hirota tau-function integrable systems, projective geometry of Kummer varieties, and Siegel modular forms.

  requires: alg-geom.riemann-bilinear, alg-geom.vhs-jacobian, alg-geom.jacobi-inversion

• alg-geom.gauss-maninopen unit 06.08.01 →

  Gauss-Manin connection

  For a smooth proper morphism $\pi :\mathcal{X}\to S$ with compact Kähler fibres $X_s$, the cohomology bundle ${\mathcal{H}}^k=R^k{\pi }_{\ast }{\mathbb{C}}_{\mathcal{X}}$ is a local system on $S$ with fibre $H^k(X_s;\mathbb{C})$, and the Gauss-Manin connection $\nabla :{\mathcal{H}}^k\to {\mathcal{H}}^k\otimes {\Omega }_S^1$ is the canonical flat connection whose horizontal sections are locally-constant cycles. Three equivalent formulations: (i) trivial connection on the local system encoding monodromy; (ii) Katz-Oda algebraic construction on the de Rham cohomology bundle of a smooth proper morphism; (iii) Čech-cocycle level differentiation of locally lifted cocycles. Periods $\Pi (s)={\int }_{{\gamma }_s}\omega$ are multivalued holomorphic functions on $S$, solutions of an algebraic Picard-Fuchs ODE with regular singularities at boundary divisors. Worked examples: family of elliptic curves with Picard-Fuchs = Gauss hypergeometric / Legendre equation (Euler-Gauss); quintic threefold $\sum z_i^5=5\psi z_1\cdots z_5$ whose Picard-Fuchs is the mirror-symmetry quintic equation (Candelas-de la Ossa-Green-Parkes 1991, encoding genus-zero Gromov-Witten invariants of the mirror via Givental-Kontsevich); family of Riemann surfaces over ${\mathcal{M}}_g$ producing the variation of Hodge structure (VHS) on $H^1$. Hodge filtration $F^p{\mathcal{H}}^k$ varies holomorphically but is not preserved by $\nabla$; instead Griffiths transversality $\nabla F^p\subseteq F^{p-1}\otimes {\Omega }_S^1$ holds. Period mapping $\mathcal{P}:S\to \mathcal{D}/\Gamma$ to the period domain modulo monodromy is horizontal in the transversality distribution. Modern context: mirror-symmetry identification with quantum cohomology connection; Shimura-variety automorphic theory; $p$-adic Hodge comparison theorems (Berthelot-Ogus-Faltings); Dubrovin's Frobenius-manifold framework. Originator: Manin 1958 (Izv. Akad. Nauk SSSR 22), with Picard-Fuchs antecedents in Gauss's hypergeometric work and Picard-Fuchs 1891-1928; modern algebraic framework Katz-Oda 1968 (J. Math. Kyoto Univ. 8); Hodge-theoretic interpretation Griffiths 1968.

  requires: alg-geom.hodge-decomposition-curves, alg-geom.serre-duality-curves, alg-geom.sheaf-cohomology, topology.de-rham-cohomology

• alg-geom.vhs-jacobianopen unit 06.08.02 →

  Variation of Hodge structure on the Jacobian

  For a smooth proper family $\pi :\mathcal{X}\to S$ of compact Riemann surfaces of genus $g$, each fibre cohomology $H^1(X_s;\mathbb{C})$ carries a polarised pure Hodge structure of weight $1$ with Hodge filtration $F^1{H}^1=H^{1,0}(X_s)\subset H^1(X_s;\mathbb{C})$ of rank $g$ inside the rank-$2g$ total space; the Hodge subbundle $F^1{\mathcal{H}}^1={\pi }_{\ast }{\Omega }_{\mathcal{X}/S}^1$ varies holomorphically over $S$. The relative Jacobian $\mathrm{Jac}(\mathcal{X}/S)=(F^1{\mathcal{H}}^1{)}^{\ast }/R^1{\pi }_{\ast }\mathbb{Z}$ is a smooth proper family of principally polarised abelian varieties (PPAV) over $S$, and the variation of Hodge structure is the data $(R^1{\pi }_{\ast }\mathbb{Z},F^1{\mathcal{H}}^1,Q)$ with the symplectic intersection form $Q$ providing the polarisation. The period mapping $\mathcal{P}:S\to {\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ to the Siegel upper half-space modulo the symplectic monodromy is holomorphic and horizontal: Griffiths transversality $\nabla F^1\subseteq F^0\otimes {\Omega }_S^1={\mathcal{H}}^1\otimes {\Omega }_S^1$ is the differential constraint that holds automatically in weight $1$, while the substantive content is the Cauchy-Riemann compatibility of the period matrix $\Pi (s)$ via the Gauss-Manin connection (Griffiths 1968 Amer. J. Math. 90). The period domain $\mathcal{D}$ for weight-1 PHS is the symmetric domain $\mathrm{Sp}(2g,\mathbb{R})/U(g)$ realised as ${\mathfrak{H}}_g$, the Siegel upper half-space; the quotient ${\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ is the moduli space ${\mathcal{A}}_g$ of PPAV. Torelli's theorem (Andreotti 1958) asserts the period mapping ${\mathcal{M}}_g\to {\mathcal{A}}_g$ is injective: a curve is determined by its principally polarised Jacobian. The Schottky problem (Schottky 1888 for $g=4$ via a single explicit modular relation; Shiota 1986 / Novikov conjecture: KP-equation characterisation) asks for the image — the Schottky locus ${\mathcal{J}}_g$ of dimension $3g-3$ inside ${\mathcal{A}}_g$ of dimension $g(g+1)/2$, of codimension $g(g-1)/2-(3g-3)$ for $g\ge 4$. The Riemann theta function $\theta (z;\tau )$ provides quasi-periodic sections of the principal polarisation line bundle on $\mathrm{Jac}(X_s)$; Jacobian theta functions satisfy the KP hierarchy (Shiota's theorem). Modular interpretation: $g=1$ recovers the upper half-plane ${\mathfrak{H}}_1$ and modular forms for $\mathrm{SL}(2,\mathbb{Z})$, the bridge to number theory. Generalisations: VHS of higher weight (multi-step Hodge filtration with substantive Griffiths transversality), mixed Hodge structures (Deligne 1971-74) for non-compact / singular fibres, $\mathbb{R}$-VHS on homogeneous period domains $G(\mathbb{R})/V$, and $p$-adic VHS (Faltings, Berthelot rigid cohomology). Originator: Griffiths 1968-70 four-paper series Periods of integrals on algebraic manifolds (Inventiones, Amer. J. Math. 90, Publ. Math. IHÉS 38); Andreotti 1958 (Torelli for curves); Schottky 1888 (genus-4 Schottky); Shiota 1986 (general Schottky via KP, Invent. Math. 83).

  requires: alg-geom.gauss-manin, alg-geom.jacobi-inversion, alg-geom.hodge-decomposition-curves

• alg-geom.moduli-of-riemann-surfacesopen unit 06.08.03 →

  Moduli of Riemann surfaces

  The moduli space ${\mathcal{M}}_g$ of compact Riemann surfaces of genus $g$ is the parameter space of all isomorphism classes of smooth projective complex curves of fixed genus. It is naturally a smooth Deligne-Mumford stack of complex dimension $3g-3$ for $g\ge 2$ (Riemann's count, rigorously realised in Mumford 1965 GIT), $\dim {\mathcal{M}}_1=1$ (the $j$-line), $\dim {\mathcal{M}}_0=0$ (a single point with stack structure $B\mathrm{PGL}(2)$). The coarse moduli space $M_g$ is a quasi-projective complex variety; the stack structure records non-trivial automorphisms of curves (hyperelliptic involutions, Klein quartic $\mathrm{PSL}(2,{\mathbb{F}}_7)$ etc.) that prevent ${\mathcal{M}}_g$ from being a scheme. Three non-tautological constructions: Teichmüller-theoretic (${\mathcal{T}}_g$ contractible, $\cong {\mathbb{R}}^{6g-6}$, ${\mathcal{M}}_g={\mathcal{T}}_g/\mathrm{MCG}(g)$ with $\mathrm{MCG}(g)={\pi }_0({\mathrm{Diff}}^{+}({\Sigma }_g))$); algebraic-geometric (Mumford GIT on the Hilbert scheme of pluri-canonically embedded curves); period-mapping ($\mathcal{P}:{\mathcal{M}}_g\to {\mathcal{A}}_g$ via the Jacobian, image the Schottky locus ${\mathcal{J}}_g$, Torelli injectivity by Andreotti 1958). For $g\ge 2$ a fourth description by complete hyperbolic metrics (Fenchel-Nielsen length-twist coordinates) realises ${\mathcal{T}}_g\cong {\mathbb{R}}^{6g-6}$. Compactification: ${\overline{\mathcal{M}}}_g$ adds stable nodal curves (Deligne-Mumford 1969 Publ. IHÉS 36); a smooth proper Deligne-Mumford stack of dimension $3g-3$ with normal-crossings boundary $\partial {\overline{\mathcal{M}}}_g={\Delta }_0\cup {\Delta }_1\cup \cdots \cup {\Delta }_{\lfloor g/2\rfloor }$. Marked-point version ${\mathcal{M}}_{g,n}$ has dimension $3g-3+n$ (when $2g-2+n>0$). Tautological ring: classes ${\kappa }_i,{\psi }_i,{\lambda }_i\in H^{\ast }({\overline{\mathcal{M}}}_{g,n};\mathbb{Q})$, generated by Mumford-Morita-Miller / Hodge / cotangent-line classes; Mumford's relation ${\lambda }_1={\kappa }_1/12$ via Grothendieck-Riemann-Roch on the universal curve; Faber's conjecture (1999) on the Gorenstein structure of $R^{\ast }({\mathcal{M}}_g)$ (open in general). Witten conjecture (1990) / Kontsevich theorem (1992 Comm. Math. Phys. 147): the generating function of $\psi$-class intersection numbers on ${\overline{\mathcal{M}}}_{g,n}$ is a tau-function of the KdV hierarchy, proved via the matrix Airy integral and Strebel ribbon-graph combinatorics. Madsen-Weiss 2007 Ann. of Math. 165: ${\mathrm{colim}}_g{H}^{\ast }({\mathcal{M}}_g;\mathbb{Q})\cong \mathbb{Q}[{\kappa }_1,{\kappa }_2,\dots ]$ (Mumford's conjecture), via cobordism-category methods (Galatius-Madsen-Tillmann-Weiss 2009 Acta Math. 202). Applications: Gromov-Witten invariants integrate over ${\overline{\mathcal{M}}}_{g,n}(X,\beta )$ pushing to $\psi$-classes on ${\overline{\mathcal{M}}}_{g,n}$ (Kontsevich-Manin 1994); closed-bosonic-string scattering amplitudes are integrals of vertex operators against the Weil-Petersson measure on ${\overline{\mathcal{M}}}_{g,n}$ (Polyakov 1981); Hurwitz numbers via the ELSV formula (Ekedahl-Lando-Shapiro-Vainshtein 2001) translate combinatorial enumeration of branched covers to $\lambda$- and $\psi$-class integrals. Arithmetic moduli: ${\mathcal{M}}_g$ has a model over $\mathrm{Spec}\mathbb{Z}$, with Galois action of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on $\widehat{{\pi }_1({\mathcal{M}}_g\times \overline{\mathbb{Q}})}$ and the Grothendieck-Teichmüller programme (Grothendieck Esquisse 1984; Drinfeld 1990). Open problems: Faber's conjecture, Schottky problem in higher genus, explicit description of $H^{\ast }({\overline{\mathcal{M}}}_g)$ for large $g$, gauge-theoretic / quantum-field-theoretic interpretations. Originator: Riemann 1857 (parameter count); Mumford 1965 (algebraic-geometric construction); Deligne-Mumford 1969 (stack and stable-curves compactification); Teichmüller 1939-44 (foundations of ${\mathcal{T}}_g$).

  requires: alg-geom.vhs-jacobian, alg-geom.jacobi-inversion, alg-geom.moduli-of-curves

• alg-geom.kodaira-vanishingopen unit 04.09.02 →

  Kodaira vanishing theorem

  $H^i(X;{\omega }_X\otimes L)=0$ for $i>0$, $L$ ample on smooth projective complex $X$. Kodaira 1953 (transcendental). Akizuki-Nakano generalisation to ${\Omega }^p$. Algebraic proof via Deligne-Illusie reduction mod $p$. Kawamata-Viehweg generalisation to nef + big. Kollár's injectivity. Foundational for the minimal model program.

  requires: alg-geom.sheaf-cohomology, alg-geom.ample-line-bundle, alg-geom.hodge-decomposition

• hodge.kodaira-embeddingopen unit 04.09.11 →

  Kodaira embedding theorem

  A compact Kähler manifold $X$ is projective iff $X$ admits a positive holomorphic line bundle $L$, equivalently iff $X$ carries an integral Kähler class (Hodge class of type $(1,1)$ with positive representative). The embedding is via global sections of $L^{\otimes k}$ for $k$ sufficiently large. Proof factors through Kodaira vanishing: $H^1(L^{\otimes k}\otimes {\mathcal{I}}_{p,q})=0$ and $H^1(L^{\otimes k}\otimes {\mathfrak{m}}_p^2)=0$ for $k$ large give separation of points and tangent vectors. Hodge-class equivalence via the Lefschetz $(1,1)$-theorem and the exponential exact sequence. Sharpness: generic K3 surfaces are Kähler but non-projective (Picard rank $0$ at the generic point of the period domain); the Noether-Lefschetz locus is the dense union of codimension-1 strata where projectivity holds. Moduli consequence: polarised moduli are algebraic (Hilbert scheme + GIT), while unpolarised analytic moduli need not be. Originator: Kodaira 1954 On Kähler varieties of restricted type (Ann. Math. 60, 28–48). Modern anchor: Voisin Vol I §7.4; Demailly's Complex Analytic and Differential Geometry. Companion: Moishezon 1966 (Kähler-Moishezon = projective); Chow 1949 (analytic in ${\mathbb{P}}^N$ = algebraic). Used downstream in moduli of polarised varieties, Calabi-Yau moduli, K3 period domains, and the minimal model programme.

  requires: alg-geom.hodge-decomposition, alg-geom.kodaira-vanishing

• hodge.akizuki-nakano-vanishingopen unit 04.09.10 →

  Akizuki-Nakano vanishing theorem

  $H^q(X,{\Omega }_X^p\otimes L)=0$ for all $(p,q)$ with $p+q>n$, $L$ ample on smooth projective complex $X$ of dimension $n$. Proof via the Bochner-Kodaira-Nakano identity ${\Delta }_{{\bar{\partial }}_L}-{\Delta }_{D_L^{\prime }}=[i{\Theta }_h(L),{\Lambda }_{\omega }]$ on $L$-valued $(p,q)$-forms; pointwise positivity of the curvature commutator on a unitary diagonalising frame yields strict positivity for $p+q>n$; harmonic-form representation forces vanishing. Case $p=n$ recovers Kodaira vanishing. Hierarchy of refinements: Kodaira (ample, canonical-twist only) $\Rightarrow$ Akizuki-Nakano (ample, full upper-right triangle) $\Rightarrow$ Kawamata-Viehweg (nef + big, canonical-twist corner) $\Rightarrow$ Nadel-Demailly-Peternell-Schneider (pseudoeffective, multiplier-ideal correction). Failure on non-Kähler manifolds (Hopf surface, Kodaira 1964); partial Bogomolov-Sommese replacement for logarithmic differentials on smooth pairs. Raynaud 1978 counter-example in characteristic $p>0$. Foundational for the Kodaira embedding theorem (separation of points and tangent vectors via high tensor powers), the Lefschetz hyperplane theorem on Dolbeault cohomology, the structure of complete-intersection Hodge numbers, and the minimal model program (BCHM 2010 via Kawamata-Viehweg). Lean status: partial, module Codex.AlgGeom.Hodge.AkizukiNakanoVanishing states the theorem, its Kodaira specialisation, and its Serre-dual form with sorry proof bodies pending Mathlib's Bochner-Kodaira-Nakano and Dolbeault-cohomology infrastructure.

  requires: alg-geom.ample-line-bundle, alg-geom.hodge-decomposition, alg-geom.kodaira-vanishing

• alg-geom.lefschetz-1-1-theoremopen unit 04.09.09 →

  Lefschetz (1,1)-theorem

  For $X$ a smooth projective complex variety, the first Chern class map $c_1:\mathrm{Pic}(X)\to H^2(X;\mathbb{Z})$ has image equal to $H^{1,1}(X,\mathbb{Z}):=H^2(X;\mathbb{Z})\cap H^{1,1}(X)$, and kernel equal to ${\mathrm{Pic}}^0(X)=H^1(X;{\mathcal{O}}_X)/H^1(X;\mathbb{Z})$, a complex torus of dimension $h^{0,1}(X)$. Equivalently, the Néron-Severi group $\mathrm{NS}(X)=\mathrm{Pic}(X)/{\mathrm{Pic}}^0(X)$ equals the integer-Hodge $(1,1)$-classes, finitely generated of rank $\rho (X)\le h^{1,1}(X)$ (the Picard number). Proof via the exponential sheaf sequence $0\to \mathbb{Z}\to {\mathcal{O}}_X\to {\mathcal{O}}_X^{\ast }\to 0$: the connecting map $\delta =c_1$, and its image is the kernel of $H^2(X;\mathbb{Z})\to H^2(X;{\mathcal{O}}_X)=H^{0,2}(X)$, identified via complex conjugation with $H^{1,1}(X,\mathbb{Z})$. This is the codimension-one case of the Hodge conjecture — known since Lefschetz 1924, with the modern cohomological proof due to Hodge 1941 and Griffiths-Harris 1978. The codimension $\ge 2$ case is the open Hodge conjecture (one of the Clay Millennium Prize Problems). Examples: $\mathrm{Pic}({\mathbb{P}}^n)=H^2({\mathbb{P}}^n;\mathbb{Z})=\mathbb{Z}$ (every class is algebraic); K3 surfaces $\rho \in \{1,\dots ,20\}=\{1,\dots ,h^{1,1}\}$; abelian varieties $\rho \le g^2$ with Riemann forms as the algebraic classes.

  requires: alg-geom.hodge-decomposition, alg-geom.picard-group, alg-geom.line-bundle-scheme, alg-geom.sheaf-cohomology

• alg-geom.intersection-pairing-surfacesopen unit 04.05.06 →

  Intersection pairing on a surface

  Symmetric bilinear pairing $\mathrm{Pic}(X)\times \mathrm{Pic}(X)\to \mathbb{Z}$, $(D,D^{\prime })\mapsto D\cdot D^{\prime }$, on a smooth projective surface $X$ over algebraically closed $k$. Three equivalent definitions: geometric (transverse intersection count after moving lemma), cohomological ($D\cdot D^{\prime }=\chi ({\mathcal{O}}_X)-\chi (\mathcal{O}(-D))-\chi (\mathcal{O}(-D^{\prime }))+\chi (\mathcal{O}(-D-D^{\prime }))$, Hartshorne V.1.4), cup-product (cycle-class map $\mathrm{Pic}(X)\to H^2(X,\mathbb{Z})$ intertwines intersection with $⌣$). Self-intersection $D^2$. Examples: line $L^2=1$ on ${\mathbb{P}}^2$, exceptional divisor $E^2=-1$, ruling $f^2=0$ on ${\mathbb{P}}^1\times {\mathbb{P}}^1$. Italian school (Castelnuovo-Enriques 1880–1910); Hartshorne V scheme-theoretic framing; Fulton's $A^{\ast }(X)$-deepening. Load-bearing for adjunction, Riemann-Roch on surfaces, Hodge index, and the Castelnuovo-Beauville-Bombieri-Kodaira classification of surfaces.

  requires: alg-geom.scheme, alg-geom.weil-divisor, alg-geom.cartier-divisor, alg-geom.line-bundle-scheme, alg-geom.picard-group

• alg-geom.adjunction-formulaopen unit 04.05.07 →

  Adjunction formula on a surface

  For a smooth projective curve $C$ on a smooth projective surface $X$ over algebraically closed $k$, the adjunction formula in canonical-restriction form: ${\omega }_C\cong ({\omega }_X\otimes {\mathcal{O}}_X(C)){|}_C$. Genus form: $2g_C-2=K_X\cdot C+C^2$, equivalently $g_C=1+\frac{1}{2}(K_X\cdot C+C^2)$. Proof via the conormal exact sequence $0\to {\mathcal{N}}_{C/X}^{\vee }\to {\Omega }_X^1{|}_C\to {\Omega }_C^1\to 0$, taking determinants, and using ${\mathcal{N}}_{C/X}^{\vee }\cong {\mathcal{O}}_X(-C){|}_C$. Codim-$1$ case of the general adjunction ${\omega }_Y\cong ({\omega }_X\otimes \det {\mathcal{N}}_{Y/X}){|}_Y$. Recovers Plücker's plane-curve genus formula $g=(d-1)(d-2)/2$, the bidegree-$(a,b)$ genus $(a-1)(b-1)$ on ${\mathbb{P}}^1\times {\mathbb{P}}^1$, the K3 inequality $C^2\ge -2$, the Castelnuovo $(-1)$-curve diagnostic, and Noether's formula $K_X^2+e(X)=12\chi ({\mathcal{O}}_X)$. Picard 1897 originator-text; Castelnuovo 1892 and Severi 1921 Italian-school synthesis; Hartshorne V.1.5 modern scheme-theoretic framing.

  requires: alg-geom.intersection-pairing-surfaces, alg-geom.canonical-sheaf, alg-geom.sheaf-of-differentials, alg-geom.riemann-roch-curves

• alg-geom.riemann-roch-surfacesopen unit 04.05.08 →

  Riemann-Roch theorem for surfaces

  Theorem (Hartshorne V.1.6). For a smooth projective surface $S$ over algebraically closed $k$ and a divisor $D$ on $S$, $\chi ({\mathcal{O}}_S(D))=\chi ({\mathcal{O}}_S)+\frac{1}{2}D\cdot (D-K_S)$, where $K_S$ is the canonical divisor and $\chi ({\mathcal{O}}_S)=1-q+p_g$ with $q=h^1({\mathcal{O}}_S)$ the irregularity and $p_g=h^0({\omega }_S)=h^2({\mathcal{O}}_S)$ the geometric genus. Noether's formula (Max Noether 1883): $12\chi ({\mathcal{O}}_S)=K_S^2+c_2(S)$ couples the holomorphic and topological Euler characteristics on $S$. Hirzebruch-Riemann-Roch derivation: surface Riemann-Roch is the degree-two specialisation of $\chi (X,\mathcal{F})={\int }_X\mathrm{ch}(\mathcal{F})\cdot \mathrm{td}(T_X)$ at $\mathcal{F}={\mathcal{O}}_S(D)$, with $\mathrm{ch}({\mathcal{O}}_S(D))=1+D+D^2/2$ and $\mathrm{td}(T_S)=1-K_S/2+(K_S^2+c_2)/12$ truncated on a surface. Direct (Italian-school) proof: short exact sequence $0\to {\mathcal{O}}_S(D-C)\to {\mathcal{O}}_S(D)\to {\mathcal{O}}_C(D{|}_C)\to 0$ for a smooth curve $C\subset S$, take Euler characteristics, apply curve Riemann-Roch on $C$, and use adjunction to identify the genus. Worked examples. (1) $S={\mathbb{P}}^2$ with $K_S=-3H$, $H^2=1$: $\chi (\mathcal{O}(d))=d(d+3)/2+1=\left(\genfrac{}{}{0ex}{}{d+2}{2}\right)$, recovering the dimension count for plane curves of degree $d$. (2) $S={\mathbb{P}}^1\times {\mathbb{P}}^1$ with $K_S=-2H_1-2H_2$ and bidegree $(a,b)$: $\chi =(a+1)(b+1)$. Hodge index theorem (Hartshorne V.1.9): the intersection form on $\mathrm{NS}(S)\otimes \mathbb{R}$ has signature $(1,\rho -1)$, equivalently the intersection form on $H^{1,1}(S,\mathbb{R})$ has signature $(1,h^{1,1}-1)$. Castelnuovo's contractibility criterion (Hartshorne V.5.7): a smooth rational curve $E$ with $E^2=-1$ on a smooth projective surface is the exceptional divisor of a blow-up at a smooth point, with adjunction giving $E\cdot K_S=-1$ as the dual diagnostic. Geography of surfaces: Noether inequality $K_S^2\ge 2p_g-4$ (Noether 1875); Bogomolov-Miyaoka-Yau inequality $K_S^2\le 9\chi ({\mathcal{O}}_S)$ (Yau 1977, Miyaoka 1977) for surfaces of general type; the strip $2\chi -6\le K_S^2\le 9\chi$ is the realisation region. Enriques-Kodaira classification: every smooth projective surface is birational to a unique minimal model with no $(-1)$-curves, and minimal models fall into four Kodaira-dimension classes: $\kappa =-\infty$ (rational/ruled), $\kappa =0$ (K3, Enriques, abelian, bielliptic), $\kappa =1$ (properly elliptic), $\kappa =2$ (general type). Castelnuovo-Enriques-Severi (1900s-1920s) for char zero, Bombieri-Mumford (1976-77) for char $p$. Originators: Max Noether 1883 (the topological correction term); Castelnuovo, Enriques, Severi 1890s-1920s (Italian-school synthesis); Severi 1926 Trattato (refined version); Enriques 1949 Le superficie algebriche (synthesis); Hirzebruch 1956 (modern HRR proof); Hartshorne 1977 §V.1 (canonical scheme-theoretic framing).

  requires: alg-geom.intersection-pairing-surfaces, alg-geom.adjunction-formula, alg-geom.riemann-roch-curves, alg-geom.serre-vanishing-finiteness

• alg-geom.hirzebruch-riemann-roch-generalopen unit 04.05.10 →

  Hirzebruch-Riemann-Roch theorem (general dimension)

  Theorem (Hirzebruch 1954 / Hirzebruch 1956 §IV.21). For a smooth projective complex algebraic variety $X$ of complex dimension $n$ and a holomorphic vector bundle (or coherent sheaf) $\mathcal{E}$ on $X$, $\chi (X,\mathcal{E})={\int }_X\mathrm{ch}(\mathcal{E})\cdot \mathrm{td}(T_X)$, with the integrand a class in $H^{2\ast }(X,\mathbb{Q})$ and the integral extracting the degree-$2n$ part against the fundamental class. Chern character: $\mathrm{ch}(\mathcal{E})={\sum }_i{e}^{x_i}$ in formal Chern roots $x_i$ of $\mathcal{E}$, additive and multiplicative on bundles. Todd class: $\mathrm{td}(\mathcal{E})={\prod }_i{x}_i/(1-e^{-x_i})$ in formal Chern roots, the multiplicative sequence attached to $Q(x)=x/(1-e^{-x})$. Truncated expansions. Curves $(n=1)$: $\mathrm{td}(T_X)=1+c_1/2=1-K_X/2$ recovers $\chi (L)=\mathrm{deg}L+1-g$. Surfaces $(n=2)$: $\mathrm{td}(T_X)=1-K_X/2+(K_X^2+c_2)/12$ recovers $\chi ({\mathcal{O}}_S(D))=\chi ({\mathcal{O}}_S)+D\cdot (D-K_S)/2$ together with Noether's formula $12\chi ({\mathcal{O}}_S)=K_S^2+c_2(S)$ at $\mathcal{E}={\mathcal{O}}_S$. Threefolds $(n=3)$: ${\mathrm{td}}_3=c_1{c}_2/24$ contributes the arithmetic-genus term. Worked examples. (1) ${\mathcal{O}}_{{\mathbb{P}}^n}(d)$ gives $\chi =\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$, the standard projective-space dimension table. (2) Smooth complete intersection in ${\mathbb{P}}^n$: arithmetic genus via Todd-class restriction along the conormal bundle. (3) K3 surface: $\chi (X,E)=2\mathrm{rk}(E)+c_1(E{)}^2/2-c_2(E)$ since $K_X=0$ and $\chi ({\mathcal{O}}_X)=2$. (4) Tangent bundle on ${\mathbb{P}}^n$: arithmetic genus computation via the Euler sequence. Proof outline. Originator route (Hirzebruch 1956 §IV.20): the holomorphic Euler characteristic $\chi (X,\mathcal{E})$ and the integral ${\int }_X\mathrm{ch}(\mathcal{E})\cdot \mathrm{td}(T_X)$ are both additive on short exact sequences and both stable under products; Thom's theorem identifies the complex cobordism ring ${\Omega }_{\ast }^U\otimes \mathbb{Q}$ with a polynomial ring on $[{\mathbb{P}}^n]$; verifying the identity on the polynomial generators $[{\mathbb{P}}^n]$ pins down the Todd class as the unique multiplicative sequence with $\mathrm{td}({\mathbb{P}}^n)=1$ for all $n$. Modern route: Atiyah-Singer index theorem applied to the Dolbeault complex of $\mathcal{E}$; the topological index of the $\bar{\partial }$-operator twisted by $\mathcal{E}$ is ${\int }_X\mathrm{ch}(\mathcal{E})\cdot \mathrm{td}(T_X)$, the analytic index is $\chi (X,\mathcal{E})$, and Atiyah-Singer asserts their equality. Algebraic route: Grothendieck-Riemann-Roch (Grothendieck 1957) generalises HRR to proper morphisms via $\mathrm{ch}(f_{\ast }\mathcal{E})\cdot \mathrm{td}(T_Y)=f_{\ast }(\mathrm{ch}(\mathcal{E})\cdot \mathrm{td}(T_X))$ in the Chow ring; HRR is the case $Y=\mathrm{Spec}k$. Specialisations. Classical Riemann-Roch on curves (Riemann 1857 / Roch 1865) reproduced from HRR at $n=1$ via ${\mathrm{td}}_1=1+c_1/2$; Noether's formula on surfaces (Noether 1883) reproduced from HRR applied to ${\mathcal{O}}_S$ at $n=2$; arithmetic-genus formula $\chi ({\mathcal{O}}_X)={\int }_X\mathrm{td}(T_X)$ for every $n$. Connections. Atiyah-Singer index theorem (Atiyah-Singer 1963) as the index-theoretic generalisation to elliptic operators on closed manifolds; Grothendieck-Riemann-Roch (1957) as the relative-Euler-characteristic generalisation to proper morphisms; Hirzebruch signature theorem (1956) as the parallel application of the $L$-genus to oriented $4k$-manifolds; Hodge index theorem (Hodge 1937) for the signature constraint on the surface intersection form coupling to Noether's formula. Originator chain: J. A. Todd 1937 (combinatorial Todd class on subvarieties); F. Hirzebruch 1954 PNAS (four-page announcement); F. Hirzebruch 1956 Neue topologische Methoden (full proof via cobordism); A. Grothendieck 1957 (GRR generalisation, written up by Borel-Serre 1958 Bull. SMF); M. F. Atiyah and F. Hirzebruch 1959 Bull. AMS (K-theoretic differentiable-manifold version); Atiyah-Singer 1963 Bull. AMS and 1968 Annals series (index-theoretic generalisation). Lean status: none. Mathlib has the Euler-characteristic formalism, partial Chern-class infrastructure, and a Mathlib.AlgebraicGeometry.RiemannRoch namespace, but no general-dimension HRR theorem; the Todd class as a named multiplicative sequence, the Chern character ring isomorphism $\mathrm{ch}:K(X)\otimes \mathbb{Q}\stackrel{\sim }{\to }{H}^{\mathrm{ev}}(X,\mathbb{Q})$, the integration pairing ${\int }_X(\cdot )$ on smooth projective varieties, and the cobordism identification ${\Omega }_{\ast }^U\otimes \mathbb{Q}\cong \mathbb{Q}[{\mathbb{P}}^n]$ are all absent as named theorems.

  requires: alg-geom.riemann-roch-surfaces, alg-geom.adjunction-formula, alg-geom.cohomology-projective, char-classes.chern-character, char-classes.multiplicative-sequences-genera

• alg-geom.moduli-of-curvesopen unit 04.10.01 →

  Moduli of curves

  ${\mathcal{M}}_g$ moduli space of smooth genus-$g$ curves; quasi-projective scheme of dimension $3g-3$ for $g\ge 2$ (Riemann's count, Mumford's GIT construction). Deligne-Mumford compactification $\overline{{\mathcal{M}}_g}$ via stable curves (1969). Tautological classes $\kappa ,\psi ,\lambda$. Witten's conjecture (Kontsevich theorem). ${\mathcal{M}}_{g,n}$ with marked points. Connection to teichmüller theory and string topology.

  requires: alg-geom.riemann-roch-curves, alg-geom.coherent-sheaf

• alg-geom.gitopen unit 04.10.02 →

  Geometric invariant theory

  Mumford 1965 Geometric Invariant Theory. GIT quotient $X//G=\mathrm{Proj}{⨁}_n{H}^0(X;L^n{)}^G$ for linearised ample $L$. Stable, semistable, unstable points; Hilbert-Mumford numerical criterion. Symplectic reduction correspondence (Kempf-Ness). Foundational for moduli of vector bundles, varieties, sheaves. Kirwan's stratification of unstable locus. Variation of GIT.

  requires: alg-geom.scheme, lie-groups.lie-group, algebra.group-action

• moduli.kempf-ness-symplectic-dictionaryopen unit 04.10.04 →

  Kempf-Ness theorem and the GIT-symplectic dictionary

  Kempf-Ness 1979 Length of vectors in representation spaces (Springer LNM 732, 233-243): for a complex reductive group $G$ with maximal compact $K$ acting linearly on a Hermitian vector space $V$ with induced moment map $\mu :V\to {\mathfrak{k}}^{\ast }$, the algebraic GIT quotient $V/{/}_G$ is canonically homeomorphic to the symplectic reduction ${\mu }^{-1}(0)/K$ at level zero. Four-part theorem (KN1-KN4): (KN1) $v$ semistable iff $\overline{G\cdot v}$ contains a minimum-norm vector; (KN2) minimum-norm vectors in $G\cdot v$ are exactly ${\mu }^{-1}(0)\cap G\cdot v$; (KN3) inside a polystable orbit the moment-zero set is a single $K$-orbit; (KN4) the inclusion ${\mu }^{-1}(0)\cap V^{ss}\hookrightarrow V^{ss}$ descends to the homeomorphism ${\mu }^{-1}(0)/K\cong V/{/}_G$. Proof rests on the variational identity $df(\rho (i\xi )v)=-2⟨\mu (v),\xi ⟩$ along the non-compact $i\mathfrak{k}$-direction of the Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus i\mathfrak{k}$, plus convexity of the length-squared along the orbit (sum of exponentials with real exponents is convex). Worked example: $V={\mathbb{C}}^2$, $G={\mathbb{C}}^{\times }$ diagonal, $K=U(1)$ — algebraic GIT quotient ${\mathbb{C}}^2//{\mathbb{C}}^{\times }={\mathbb{P}}^1$ matches symplectic reduction $S^3/U(1)={\mathbb{P}}^1$ via the Hopf fibration. Kirwan stratification (Kirwan 1984 Ch. 4): the unstable locus decomposes as $V\setminus V^{ss}={⨆}_{\beta }{S}_{\beta }$ via the stable manifolds of critical sets of $\Vert \mu {\Vert }^2$, indexed by Hilbert-Mumford-most-destabilising directions $\beta \in {\mathfrak{k}}^{\ast }/K$; Hesselink-Kirwan-Ness equivalence identifies the algebraic Hilbert-Mumford stratification with the analytic gradient-flow stratification. Atiyah-Bott 1983 Phil. Trans. R. Soc. 308: infinite-dimensional Kempf-Ness for the gauge-group action on the space of unitary connections on a Riemann surface — moment map equals curvature, moduli of polystable holomorphic bundles equals moduli of flat unitary connections (this is Narasimhan-Seshadri 1965 viewed as infinite-dimensional Kempf-Ness); Atiyah-Bott Morse theory on $\Vert F{\Vert }^2$ stratifies by Harder-Narasimhan filtrations giving closed-form Poincaré polynomial of $\mathcal{N}(r,d)$. Kobayashi-Hitchin correspondence (Donaldson 1985 / Uhlenbeck-Yau 1986): holomorphic vector bundle $E\to X$ over compact Kähler $X$ is $\omega$-polystable iff $E$ admits a Hermitian-Einstein metric. Hitchin moduli space (Hitchin 1987): hyperkähler reduction producing the Higgs-bundle moduli space. Yau-Tian-Donaldson conjecture (Chen-Donaldson-Sun 2015): Fano manifold admits Kähler-Einstein metric iff K-stable. Originator chain: Hilbert 1890 → Mumford 1965 → Marsden-Weinstein 1974 → Narasimhan-Seshadri 1965 → Kempf-Ness 1979 → Kirwan 1984 → Atiyah-Bott 1983 → Donaldson-Uhlenbeck-Yau 1985-86 → Hitchin 1987 → Chen-Donaldson-Sun 2015.

  requires: alg-geom.git, symplectic-geometry.moment-map, symplectic-geometry.symplectic-reduction

• moduli.hilbert-mumford-criterionopen unit 04.10.03 →

  Hilbert-Mumford numerical criterion

  For a reductive group $G$ acting on a projective variety $X$ with $G$-linearised ample line bundle $L$, the Hilbert-Mumford numerical criterion characterises GIT stability via one-parameter subgroups: $x\in X$ is GIT-semistable iff ${\mu }^L(x,\lambda )\ge 0$ for every one-parameter subgroup $\lambda :{\mathbb{G}}_m\to G$, where ${\mu }^L(x,\lambda )$ is the weight of $\lambda$ on the line-bundle fibre at the limit point ${\lim }_{t\to 0}\lambda (t)\cdot x$. Equivalently, on the affine cone ${\mu }^L(\tilde{x},\lambda )=-\min \{m_i:{\tilde{x}}_i\ne 0\}$ where $\tilde{x}={\sum }_i{\tilde{x}}_i$ is the weight decomposition. Proof architecture: (1) reduction from $G$ to a maximal torus $T$ via conjugation; (2) convex-hull lemma — ${\mu }^L\ge 0$ for all ${\lambda }_T\in X_{\ast }(T)$ iff $0\in \mathrm{conv}({\mathrm{wt}}_T(\tilde{x}))$ in the character lattice; (3) Reynolds-operator averaging from $T$-invariant to $G$-invariant, requiring reductivity. Worked example: ${\mathrm{SL}}_2$ acting on binary forms of degree $d$ — the diagonal one-parameter subgroup gives weight $d-2i$ on the monomial $x^{d-i}{y}^i$, and Hilbert's classical theorem (1893, modern form) follows: a binary form is unstable iff some root has multiplicity strictly greater than $d/2$. Kempf 1978 refinement: for an unstable point, a unique optimal destabilising one-parameter subgroup ${\lambda }^{\ast }$ minimising ${\mu }^L/\Vert \lambda \Vert$ exists (up to stabiliser-conjugacy), driving the Kirwan stratification of the unstable locus. Applications: ${\mathcal{M}}_g$ (Mumford 1965), moduli of vector bundles with slope semistability (Seshadri 1967), Gieseker semistability of sheaves (1977), K-stability of Fano varieties (Tian, Donaldson, Chen-Donaldson-Sun 2015), derived GIT (Halpern-Leistner 2014). Originator: Hilbert 1893 one-parameter heuristic; Mumford 1965 rigorous formulation.

  requires: alg-geom.git, alg-geom.scheme, rep-theory.lie-algebra-representation

• moduli.vector-bundles-curve-slope-stabilityopen unit 04.10.06 →

  Moduli of vector bundles on a curve and slope stability

  Mumford 1963 introduced the slope $\mu (E)=\mathrm{deg}(E)/\mathrm{rk}(E)$ and slope stability for vector bundles on a smooth projective curve. Stable: $\mu (F)<\mu (E)$ for every proper sub-bundle $F$; semistable: $\le$. The moduli space $M_C(r,d)$ is a quasi-projective variety of dimension $r^2(g-1)+1$, projective when $\gcd (r,d)=1$ (every semistable bundle then stable). Construction as GIT quotient of a Quot scheme by ${\mathrm{PGL}}_h$. Harder-Narasimhan filtration (1975, Math. Ann. 212): every bundle has a unique filtration by sub-bundles with semistable subquotients of strictly decreasing slope; existence by maximal-slope, maximal-rank sub-bundle argument; functorial under bundle morphisms. Narasimhan-Seshadri 1965 identifies $M_C(r,d{)}^{\mathrm{stable}}$ with irreducible unitary $r$-dimensional representations of ${\pi }_1(C)$ of fixed determinant character ${\zeta }_d$; bridge is the essentially unique projectively-flat Hermitian-Einstein connection on the underlying smooth bundle. Atiyah-Bott 1983 reinterprets $M_C(r,d)\cong {\mu }^{-1}(c)/\mathcal{G}$ as the symplectic reduction of the affine space of unitary connections by the gauge group at the central curvature level $c=-2\pi i(d/r)\cdot \omega$; Yang-Mills functional $\Vert F_A{\Vert }^2$ as equivariant Morse function recovers the Poincaré polynomial of the moduli via the Harder-Narasimhan stratification. Donaldson 1983 J. Diff. Geom. 18 gives an analytic proof via convergence of the Donaldson functional on Hermitian metrics. Schur lemma for stable bundles: $\mathrm{Hom}(E,F)=0$ when $\mu (E)>\mu (F)$ and $E,F$ semistable; $\mathrm{End}(E)=k$ for $E$ stable. Deformation-theoretic dimension at a stable point: $\dim T_{[E]}M=\dim {\mathrm{Ext}}^1(E,E)=r^2(g-1)+1$ via Riemann-Roch on $\mathcal{E}nd(E)$. Generalises to Higgs bundles (Hitchin 1987, Simpson 1992), to higher-dimensional Kähler manifolds (Donaldson-Uhlenbeck-Yau), to Gieseker stability of sheaves (Gieseker 1977), and to Bridgeland stability (2007). Foundational for the Verlinde formula, geometric Langlands, and non-abelian Hodge theory.

  requires: alg-geom.moduli-of-curves, alg-geom.git, alg-geom.coherent-sheaf, riemann-surfaces.holomorphic-line-bundle

• moduli.variation-of-git-vgitopen unit 04.10.09 →

  Variation of GIT (VGIT)

  For a reductive group $G$ acting on a projective variety $X$, the GIT quotient $X/{/}_{L}G$ depends on the choice of $G$-linearised ample line bundle $L$. The set of linearisations forms an open cone ${\mathrm{Amp}}^G(X)\subset {\mathrm{Pic}}^G(X)\otimes \mathbb{Q}$ in the rational equivariant Picard group. Dolgachev-Hu 1998 and Thaddeus 1996 prove independently that ${\mathrm{Amp}}^G(X)$ admits a finite decomposition into rational polyhedral chambers $C_1,\dots ,C_r$ — the GIT chambers — such that (i) inside each chamber the semistable locus $X^{ss}(L)$, the stable locus $X^s(L)$, and the GIT quotient $X/{/}_{L}G$ are constant; (ii) the chamber count is finite (via Kempf's 1978 finiteness of numerical types); (iii) crossing a wall $W=\overline{C_{-}}\cap \overline{C_{+}}$ between adjacent chambers gives a birational map $Q_{-}⇢Q_{+}$ realised by a master space $M$ together with projective birational morphisms $Q_{-}\stackrel{{\varphi }_{-}}{\leftarrow }M\stackrel{{\varphi }_{+}}{\to }{Q}_{+}$ contracting the loci of strictly semistable orbits at the wall. When the codimensions are $\ge 2$ on both sides, this is a flip in the Mori-theoretic sense. The simplest example — the Atiyah flop — is the ${\mathbb{C}}^{\times }$-action on ${\mathbb{C}}^4$ with weights $(+1,+1,-1,-1)$; the two chambers give the two small resolutions of the conifold $\{XZ-YW=0\}$. The proof of chamber-constancy rests on the linearity of the Hilbert-Mumford function ${\mu }^L(x,\lambda )$ in $L$: each numerical type of one-parameter subgroup contributes one linear half-space inequality, and the chambers are the connected components of the resulting hyperplane arrangement. Worked example — binary quartics: for $G={\mathrm{SL}}_2$ acting on $V_4={\mathrm{Sym}}^4(k^2)$, the rational equivariant Picard group is one-dimensional, giving a single chamber and the unique GIT quotient $\mathbb{P}(V_4)//{\mathrm{SL}}_2\cong {\mathbb{A}}^1$ — the $j$-line of moduli of elliptic curves. Hu-Keel theorem (2000) identifies Mori dream spaces as exactly the projective varieties $Y$ admitting a torus-GIT presentation $Y=\mathrm{Spec}\mathrm{Cox}(Y)/{/}_{L}T$ in which the Mori chamber decomposition of $\overline{\mathrm{Eff}}(Y)$ coincides with the VGIT chamber decomposition of ${\mathrm{Pic}}^T(\mathrm{Spec}\mathrm{Cox}(Y))\otimes \mathbb{Q}$. Castravet-Tevelev 2015 showed ${\overline{\mathcal{M}}}_{0,n}$ fails to be a Mori dream space for $n\ge 7$. Modern extensions: Halpern-Leistner 2014 magic windows lift VGIT to derived categories, with wall-crossings becoming derived equivalences subsuming Bondal-Orlov's flop derived equivalences; Chen-Donaldson-Sun 2015 prove the Yau-Tian-Donaldson conjecture via infinite-dimensional VGIT (K-stability with test configurations as one-parameter subgroups); Bérczi-Doran-Hawes-Kirwan develop non-reductive VGIT. Thaddeus 1994 used VGIT chains to derive the Verlinde formula by induction over walls. Lean status: partial, module Codex.AlgGeom.Moduli.VariationOfGIT declares an abstract VGITChamberData structure, the chamber_constancy and wall_crossing_flip theorems as sorry-stubbed statements, the chamber_finiteness corollary, the Hu-Keel hu_keel_mds_iff_torus_git theorem, and the binary-quartics worked example as a named theorem pending the equivariant Picard group + GIT quotient functor + Cox ring infrastructure in Mathlib.

  requires: alg-geom.git` (04.10.02), alg-geom.scheme` (04.02.01), alg-geom.ample-line-bundle` (04.05.05)

• moduli.hilbert-schemeopen unit 04.10.05 →

  Hilbert scheme Hilb^P(X)

  ${\mathrm{Hilb}}_{X/S}^P$ is the projective $S$-scheme parametrising closed subschemes $Z\subset X$ flat over the base with fibrewise Hilbert polynomial $P$. Grothendieck's 1962 existence theorem (FGA, Séminaire Bourbaki 221): for $X$ projective over Noetherian $S$ with chosen relatively very ample line bundle, the Hilbert functor is representable. Construction goes through (i) uniform Mumford-Castelnuovo $m_0$-regularity, (ii) Grassmannian embedding via $W=H^0({\mathcal{I}}_Z(m_0))$, (iii) closedness by rank conditions on multiplication maps, (iv) flatness of the universal family by cohomology-and-base-change. Tangent space at $[Z]$ is $H^0(Z,N_{Z/X})$ and obstruction space is $H^1(Z,N_{Z/X})$ (Grothendieck-Mumford). Schlessinger inequality $h^0-h^1\le {\dim }_{[Z]}{\mathrm{Hilb}}^P\le h^0$. Fogarty 1968 Amer. J. Math. 90: ${\mathrm{Hilb}}^n$ of a smooth surface is smooth and irreducible of dimension $2n$. Hartshorne 1966 Publ. Math. IHES 29 proved connectedness; Piene-Schlessinger 1985 Amer. J. Math. 107 the twisted-cubic reducibility. Hilbert-Chow morphism ${\mathrm{Hilb}}^n(X)\to {\mathrm{Sym}}^n(X)$ is a resolution of singularities on surfaces, crepant for K3 (Beauville 1983 — hyperkähler ${\mathrm{Hilb}}^n(K3)$). Göttsche 1990 generating function for Betti numbers. Nakajima 1997 Heisenberg-algebra representation on ${⨁}_n{H}^{\ast }({\mathrm{Hilb}}^n(X))$. McKay correspondence via $G$-Hilbert schemes (Ito-Nakamura 1996; Bridgeland-King-Reid 2001). Donaldson-Thomas invariants (Donaldson-Thomas 1998; Behrend 2009) and the MNOP conjecture (Maulik-Nekrasov-Okounkov-Pandharipande 2006). Generalised by the Quot scheme (Grothendieck 1961). Foundational for [04.10.01] moduli of curves (Mumford GIT on tri-canonically embedded curves), [04.10.06] moduli of vector bundles (via Quot scheme), and modern enumerative geometry of surfaces and Calabi-Yau threefolds.

  requires: alg-geom.scheme, alg-geom.projective-scheme, alg-geom.sheaf-cohomology, alg-geom.moduli-of-curves

• topology.spectrumopen unit 03.12.04 →

  Spectrum

  Sequential spectra with structure maps $\Sigma E_n\to E_{n+1}$. $\Omega$-spectra. Stable homotopy groups indexed over $\mathbb{Z}$. Brown representability. Stable homotopy category as triangulated, symmetric monoidal. Examples: sphere, Eilenberg-MacLane, K-theory, bordism. Connection to generalised cohomology theories.

  requires: topology.topological-space, topology.homotopy, topology.suspension

• topology.suspensionopen unit 03.12.03 →

  Suspension

  Unreduced and reduced suspension. $\Sigma S^n\cong S^{n+1}$. Suspension-loop adjunction. Freudenthal suspension theorem. Stable homotopy as the limit of iterated suspension. Smash product: $\Sigma X=S^1\wedge X$.

  requires: topology.topological-space, topology.continuous-map, topology.homotopy

• topology.covering-spaceopen unit 03.12.02 →

  Covering space

  Local-product covering maps with discrete fibres. Path-lifting, homotopy-lifting, deck transformation group. Universal cover. Galois correspondence between subgroups of ${\pi }_1$ and connected covers. Spin double cover $\mathrm{Spin}(n)\to \mathrm{SO}(n)$ as a fundamental example.

  requires: topology.topological-space, topology.continuous-map, topology.homotopy

• topology.metric-spaceopen unit 02.01.05 →

  Metric space

  Sets equipped with a distance function satisfying positive-definiteness, symmetry, and triangle inequality. Metric topology, Cauchy sequences and completeness, equivalence of metrics generating the same topology, completion. Banach fixed-point theorem, Heine-Borel, Arzelà-Ascoli, Stone-Weierstrass, Baire category. Bridge from topology to functional analysis.

  requires: topology.topological-space

• functional-analysis.banach-spaceopen unit 02.11.04 →

  Banach space

  Complete normed vector spaces; foundation for bounded linear operators, compact operators, Fredholm theory, and PDE estimates.

  requires: linalg.vector-space

• functional-analysis.normed-vector-spaceopen unit 02.11.06 →

  Normed vector space

  Norm axioms, induced metric, norm topology, continuous linear maps, equivalent norms in finite dimension, and foundation for Banach spaces.

  requires: linalg.vector-space, topology.metric-space

• functional-analysis.inner-product-spaceopen unit 02.11.07 →

  Inner product space

  Inner products, induced norm, Cauchy-Schwarz inequality, orthogonality, projection geometry, parallelogram identity, and foundation for Hilbert spaces.

  requires: functional-analysis.normed-vector-space, linalg.bilinear-form

• functional-analysis.hilbert-spaceopen unit 02.11.08 →

  Hilbert space

  Complete inner-product spaces; supplies orthogonality, projection, adjoints, and state-space language for quantum theory and CFT.

  requires: linalg.vector-space, linalg.bilinear-form, functional-analysis.banach-space

• functional-analysis.unbounded-self-adjointopen unit 02.11.03 →

  Unbounded self-adjoint operators

  Densely defined unbounded operators on Hilbert spaces, adjoints, symmetric versus self-adjoint operators, closed graph, deficiency spaces, and spectral calculus.

  requires: functional-analysis.hilbert-space

• functional-analysis.bounded-operatorsopen unit 02.11.01 →

  Bounded linear operators

  Linear operators between normed spaces, operator norm, equivalence of boundedness and continuity, Banach-space structure of $\mathcal{B}(X,Y)$, submultiplicativity, Banach algebra structure. Banach-Steinhaus, open mapping, closed graph. Adjoints in Hilbert space and the C*-identity. Spectrum and spectral radius.

  requires: functional-analysis.banach-space, linalg.vector-space

• functional-analysis.compact-operatorsopen unit 02.11.05 →

  Compact operators

  Bounded linear operators sending bounded sets to relatively compact sets. Closed two-sided ideal in $\mathcal{B}(X)$. Density of finite-rank operators in $\mathcal{K}(\mathcal{H})$ (Hilbert spaces). Schauder's theorem (compactness of adjoint). Riesz-Schauder spectral theorem. Hilbert-Schmidt and trace-class refinements (Schatten ideals). Calkin algebra $\mathcal{B}/\mathcal{K}$ as the home of Fredholm theory.

  requires: functional-analysis.bounded-operators, functional-analysis.banach-space, linalg.vector-space

• functional-analysis.fredholm.operatorsopen unit 03.09.06 →

  Fredholm operators

  Bounded operators with finite-dimensional kernel, closed range, finite-dimensional cokernel. Index = $\dim \ker -\dim \mathrm{coker}$ is the central invariant. Atkinson: Fredholm iff invertible modulo compacts. Stable under compact and small-norm perturbations. Foundation for the Atiyah-Singer index theorem.

  requires: functional-analysis.bounded-linear-operators, functional-analysis.banach-space, functional-analysis.compact-operators

• char-classes.chern-weil.homomorphismopen unit 03.06.06 →

  Chern-Weil homomorphism

  Invariant polynomials on a Lie algebra evaluated on curvature forms. Produces closed de Rham classes independent of connection; natural under pullback. Gateway from connections and gauge curvature to characteristic classes.

  requires: bundle.principal-bundle, diffgeo.connection.connection, diffgeo.connection.curvature, lie-algebra.lie-algebra, topology.de-rham-cohomology

• bundle.principal-bundleopen unit 03.05.01 →

  Principal bundle

  Smooth principal right $G$-bundles, free transitive fiber action, local product charts, transition functions and cocycles, associated bundles, pullbacks, reductions of structure group, and gauge transformations. Connections and curvature are separate units.

  requires: topology.topological-space, diffgeo.smooth-manifold, lie-groups.lie-group

• bundle.vector-bundleopen unit 03.05.02 →

  Vector bundle

  Finite-rank real and complex vector bundles, local linear product charts, sections, transition functions into ${\mathrm{GL}}_n$, frame bundles, associated-bundle equivalence, pullback, direct sum, tensor product, duals, and relation to K-theory and characteristic classes.

  requires: linalg.vector-space, topology.topological-space, diffgeo.smooth-manifold

• bundle.sphere-bundle-hopf-indexno unit yet

  Sphere bundle, the global angular form, and the Hopf index theorem

  Oriented sphere bundle as the unit-sphere bundle of an oriented rank-$r$ real vector bundle with $SO(r)$ structure group. Global angular form $\psi \in {\Omega }^{r-1}(S(E))$ characterised by fibre-integral $1$ and $d\psi =-{\pi }^{\ast }e(E)$. Euler class of an oriented sphere bundle equals the Euler class of the associated vector bundle. Hopf index theorem ${\sum }_p{\mathrm{ind}}_p(V)=\chi (M)$ for a vector field with isolated zeros. Poincaré-Hopf as the Morse-function specialisation. Worked examples: $e(T{S}^{2n})$ as twice the generator, Hopf bundle Euler class $-h$ on $\mathbb{C}{P}^1$. Bott-Tu §11 originator-text for the global-angular-form derivation; Hopf 1926 Vektorfelder in $n$-dimensionalen Mannigfaltigkeiten (Math. Ann. 96) for the original index theorem.

  requires: bundle.vector-bundle, bundle.frame-bundle.orthonormal, char-classes.pontryagin-chern.definitions, diffgeo.de-rham.thom-cv-cohomology

• diffgeo.connection.vector-bundle-connectionopen unit 03.05.04 →

  Connection on a vector bundle

  Covariant derivatives on vector bundles, Leibniz rule, local connection forms, affine space modeled on End(E)-valued one-forms, induced connections, and curvature.

  requires: bundle.vector-bundle, manifold.smooth, diffgeo.differential-forms

• bundle.complex-vector-bundleopen unit 03.05.08 →

  Complex vector bundle

  Smooth complex vector bundles via $J^2=-\mathrm{id}$ structure or via ${\mathrm{GL}}_n(\mathbb{C})$ cocycles. Hermitian metrics and reduction to $U(n)$. Direct sum, tensor, dual, conjugate, complexification, realification. Self-conjugacy of $V_{\mathbb{C}}$ and 2-torsion of odd Chern classes (motivating Pontryagin classes). Holomorphic bundles, Chern connections, K-theory.

  requires: bundle.vector-bundle, linalg.vector-space

• bundle.connection.curvatureopen unit 03.05.09 →

  Curvature of a connection

  Curvature 2-form on a principal bundle, Cartan structure equation, Bianchi identity. Vector-bundle curvature as $\mathrm{End}(E)$-valued 2-form. Gauge covariance of curvature. Frobenius integrability of horizontal distribution. Ambrose-Singer holonomy theorem. Riemann tensor / Ricci / scalar curvature as the metric-affine special case.

  requires: bundle.principal-bundle, diffgeo.connection.vector-bundle-connection, diffgeo.connection.connection, lie-algebra.lie-algebra, topology.de-rham-cohomology

• diffgeo.connection.connectionopen unit 03.05.07 →

  Principal bundle with connection

  Principal connections as Lie-algebra-valued one-forms or equivariant horizontal distributions; local gauge potentials, gauge transformations, associated vector-bundle connections, and curvature.

  requires: bundle.principal-bundle, lie-groups.lie-group, lie-algebra.lie-algebra, diffgeo.differential-forms

• topology.de-rham-cohomologyopen unit 03.04.06 →

  De Rham cohomology

  Defines $H_{\mathrm{dR}}^k(M)=\ker d/\mathrm{im}d$, exact forms, closed forms, functorial pullback, wedge product, Poincare lemma, de Rham theorem, and integration pairing with cycles. Provides the cohomology target for Chern-Weil representatives.

  requires: diffgeo.smooth-manifold, diffgeo.differential-forms, diffgeo.exterior-derivative, diffgeo.stokes-theorem

• diffgeo.integration-on-manifoldsopen unit 03.04.03 →

  Integration on manifolds

  Integration of compactly supported top-degree forms on oriented manifolds, partitions of unity, change of variables, orientation reversal, boundary orientation, and Stokes-compatible formalism.

  requires: diffgeo.smooth-manifold

• diffgeo.variational-calculusopen unit 03.04.08 →

  Variational calculus on manifolds

  Action functionals, variations, first variation, Euler-Lagrange equations, integration by parts, boundary terms, and gauge-theoretic variational formulas.

  requires: diffgeo.integration-on-manifolds, diffgeo.smooth-manifold

• char-classes.pontryagin-chern.definitionsopen unit 03.06.04 →

  Pontryagin and Chern classes

  Chern classes for complex vector bundles, Pontryagin classes for real bundles via complexification, splitting principle, Whitney product formula, Chern-Weil representatives, and standard examples such as $T{\mathbb{CP}}^n$.

  requires: char-classes.chern-weil.homomorphism, bundle.vector-bundle, topology.de-rham-cohomology, linalg.vector-space

• char-classes.hirzebruch-signatureopen unit 03.06.11 →

  Hirzebruch signature theorem

  For a closed oriented smooth $4k$-manifold $M$, the cup-product intersection form on $H^{2k}(M;\mathbb{R})$ is symmetric and non-degenerate; its signature $\mathrm{sig}(M)=b_{+}(M)-b_{-}(M)$ equals the Pontryagin number $⟨L_k(p_1,\dots ,p_k),[M]⟩$ where $L_k$ is the degree-$k$ component of the Hirzebruch multiplicative sequence associated to $Q(x)=\sqrt{x}/\tanh \sqrt{x}$. The first three $L$-polynomials are $L_1=p_1/3$, $L_2=(7p_2-p_1^2)/45$, $L_3=(62p_3-13p_1{p}_2+2p_1^3)/945$. Proof structure: signature and Pontryagin numbers are both oriented bordism invariants, and the rational oriented bordism ring ${\Omega }_{\ast }^{\mathrm{SO}}\otimes \mathbb{Q}$ is a polynomial ring on $[{\mathbb{CP}}^{2k}]$; multiplicativity reduces the formula to verification on the projective-space generators where it is direct. Atiyah-Singer 1968 reformulates the theorem as the index identity for the signature operator $D^{+}=d+d^{\ast }$. Canonical examples: $\mathrm{sig}({\mathbb{CP}}^2)=1$ with $p_1=3$; $\mathrm{sig}(K3)=-16$ with $p_1=-48$; $\mathrm{sig}({\mathbb{CP}}^4)=1$ with $(p_1,p_2)=(5,10)$; $\mathrm{sig}({\mathbb{HP}}^2)=1$ with $(p_1,p_2)=(2,7)$.

  requires: char-classes.pontryagin-chern.definitions, char-classes.chern-weil.homomorphism, topology.singular-cohomology

• char-classes.signature-4k-intersection-formopen unit 03.06.19 →

  Signature of a $4k$-manifold and the intersection form

  For a closed oriented smooth $4k$-manifold $M$, the cup-product pairing $Q_M(\alpha ,\beta )=⟨\alpha \cup \beta ,[M]⟩$ on $H^{2k}(M;\mathbb{Z})$ is symmetric and (after passing to the torsion-free quotient) unimodular by integral Poincaré duality, with $\det Q_M=\pm 1$. The signature $\mathrm{sig}(M)=b_{+}(M)-b_{-}(M)\in \mathbb{Z}$ is the difference of positive and negative eigenvalue counts of the real extension. Four basic properties pin down the invariant: symmetric, unimodular, orientation-reversal-flips-sign, additive on disjoint union and multiplicative on products. In dimension four the intersection form is the central topological invariant. Freedman 1982 classifies closed simply-connected topological 4-manifolds by the pair $(Q_M,\mathrm{ks}(M))$ where $\mathrm{ks}(M)\in \mathbb{Z}/2$ is the Kirby-Siebenmann invariant, subject to the parity constraint $\mathrm{ks}\equiv \mathrm{sig}(Q_M)/8(\mathrm{mod}2)$ when $Q_M$ is even; every unimodular form is realised. Donaldson 1983 proves the diagonalisation theorem: a positive-definite intersection form arising from a smooth 4-manifold is $\mathbb{Z}$-isomorphic to $⟨+1{⟩}^{\oplus b_2(M)}$, ruling out the $E_8$ lattice and exhibiting the smooth-vs-topological category gap in dimension four. Cobordism invariance: signature vanishes on null-bordant manifolds via the Lagrangian-half-space argument. Rokhlin congruence on spin 4-manifolds: $\mathrm{sig}(M)\equiv 0(\mathrm{mod}16)$ when $w_2(M)=0$. Canonical examples: $\mathrm{sig}({\mathbb{CP}}^2)=1$, $\mathrm{sig}(S^2\times S^2)=0$, $\mathrm{sig}(E_8)=8$, $\mathrm{sig}(K3)=-16$. Connects to Hirzebruch 03.06.11 (signature as L-genus), Poincaré duality 03.12.16 (unimodularity input), Hodge index 04.05.09 (signature-$(1,\rho -1)$ on algebraic surfaces), Atiyah-Singer 03.09.10 (signature as index of $d+d^{\ast }$), Yang-Mills moduli 03.07.05 ($b_{+}$ in the dimension formula).

  requires: char-classes.hirzebruch-signature, topology.poincare-duality, alg-geom.hodge-index-theorem

• char-classes.multiplicative-sequences-generaopen unit 03.06.15 →

  Multiplicative sequences and the $L$-, $\hat{A}$-, Todd genera

  Multiplicative sequence: given a formal power series $f(z)=1+a_{1}z+a_2{z}^2+\cdots$, the polynomials $K_n({\sigma }_1,\dots ,{\sigma }_n)$ defined by ${\prod }_{j}f(z_j)={\sum }_n{K}_n({\sigma }_1,\dots ,{\sigma }_n)$, with ${\sigma }_i$ the elementary symmetric polynomials. Hirzebruch's reduction theorem identifies multiplicative sequences with formal power series with $f(0)=1$ via a bijection. Three canonical instances: the $L$-sequence from $Q_L(z)=\sqrt{z}/\tanh \sqrt{z}$ producing $L_1=p_1/3$, $L_2=(7p_2-p_1^2)/45$, $L_3=(62p_3-13p_1{p}_2+2p_1^3)/945$, with ${\phi }_L[M]=\mathrm{sig}(M)$ for closed oriented $4k$-manifolds; the $\hat{A}$-sequence from $Q_{\hat{A}}(z)=(\sqrt{z}/2)/\sinh (\sqrt{z}/2)$ producing ${\hat{A}}_1=-p_1/24$, ${\hat{A}}_2=(7p_1^2-4p_2)/5760$, with $\hat{A}[M]=\mathrm{index}(\text{slashed}{D}^{+})$ on closed spin manifolds (Atiyah-Singer); the Todd class from $Q_{\mathrm{Td}}(z)=z/(1-e^{-z})$ producing ${\mathrm{Td}}_1=c_1/2$, ${\mathrm{Td}}_2=(c_1^2+c_2)/12$, ${\mathrm{Td}}_3=c_1{c}_2/24$, with $⟨{\mathrm{Td}}_n,[X]⟩=\chi ({\mathcal{O}}_X)$ for compact complex $n$-manifolds (Hirzebruch-Riemann-Roch). Each canonical genus is normalised by its evaluation on projective spaces: ${\phi }_L[{\mathbb{CP}}^{2k}]=1$, ${\phi }_{\mathrm{Td}}[{\mathbb{CP}}^n]=1$. Lichnerowicz vanishing connects positive scalar curvature to $\hat{A}[M]=0$.

  requires: char-classes.pontryagin-chern.definitions, char-classes.hirzebruch-signature, char-classes.chern-character

• char-classes.chern-characteropen unit 03.06.18 →

  Chern character $\mathrm{ch}(E)$ as a ring homomorphism

  The Chern character $\mathrm{ch}(E)={\sum }_j{e}^{x_j}$ is a ring homomorphism from topological K-theory to rational even cohomology, additive on direct sums and multiplicative on tensor products via the splitting principle. The Atiyah-Hirzebruch theorem promotes the rationalisation $\mathrm{ch}\otimes \mathbb{Q}:K(X)\otimes \mathbb{Q}\to H^{\mathrm{even}}(X;\mathbb{Q})$ to a ring isomorphism on every finite CW complex. The Chern character is the universal coefficient appearing in Hirzebruch-Riemann-Roch and Atiyah-Singer, and is compatible with Adams operations via ${\mathrm{ch}}_n({\psi }^k(x))=k^n{\mathrm{ch}}_n(x)$.

  requires: char-classes.pontryagin-chern.definitions, k-theory.vector-bundles, k-theory.adams-operations, bundle.complex-vector-bundle

• char-classes.borel-hirzebruch-g-topen unit 03.06.20 →

  Borel-Hirzebruch and the cohomology of $G/T$

  For a compact connected Lie group $G$ with maximal torus $T$ of rank $r$ and Weyl group $W$, the rational cohomology of the flag variety $G/T$ is the coinvariant algebra $H^{\ast }(G/T;\mathbb{Q})=\mathbb{Q}[t_1,\dots ,t_r]/((S^{+}{)}^W)$ of total $\mathbb{Q}$-dimension $|W|$ (Borel 1953). The Poincaré polynomial is ${\prod }_i(1-q^{2d_i})/(1-q^2)$ where $d_i$ are the fundamental degrees of $W$, with $\prod d_i=|W|$. Borel's proof uses the Serre spectral sequence collapse of $G/T\to BT\to BG$ together with Chevalley's reflection-group invariant theorem identifying $H^{\ast }(BG;\mathbb{Q})=S({\mathfrak{t}}^{\ast }{)}_{\mathbb{Q}}^W$ as a polynomial ring on $r$ generators. Bernstein-Gelfand-Gelfand 1973 and Demazure 1974 produced the Schubert basis $\{{\sigma }_w{\}}_{w\in W}$ as an integral additive basis with $\mathrm{deg}{\sigma }_w=2\ell (w)$, obtained from the Bruhat cell decomposition; the BGG divided-difference operators ${\partial }_{i}f=(f-s_{i}f)/(t_i-s_i{t}_i)$ produce polynomial representatives via ${\sigma }_w={\partial }_{i_1}\cdots {\partial }_{i_k}({\sigma }_{w_0})$ for any reduced word. Kleiman 1974 generic transversality gives non-negativity of Schubert structure constants $c_{u,v}^w\ge 0$ in ${\sigma }_u\cdot {\sigma }_v=\sum c_{u,v}^w{\sigma }_w$; in type $A$ these are the Littlewood-Richardson coefficients, closed off by the Knutson-Tao saturation conjecture 1999. The splitting principle for a rank-$n$ complex vector bundle $E\to X$ (Borel-Hirzebruch III 1960; Milnor-Stasheff §14) is the geometric form: the flag bundle $\mathrm{Fl}(E)\to X$ has ${\pi }^{\ast }$ injective on cohomology and the pullback splits as $L_1\oplus \cdots \oplus L_n$, identifying ${\pi }^{\ast }c(E)=\prod (1+x_i)$ as elementary symmetric polynomials in Chern roots; equivalently $H^{\ast }(BU(n);\mathbb{Q})=H^{\ast }(BT(n);\mathbb{Q}{)}^{S_n}=\mathbb{Q}[c_1,\dots ,c_n]$ via the maximal-torus restriction. Borel-Weil-Bott (Bott 1957) computes $H^p(G/T;L_{\lambda })$ of any integral-weight line bundle: zero if $\lambda +\rho$ is singular, concentrated in degree $p=\ell (w)$ if $\lambda +\rho$ is regular (where $w\in W$ is the unique reflection moving $\lambda +\rho$ into the dominant chamber), and there equal to $V_{w\cdot \lambda }$. The Borel-Weil specialisation $w=e$ gives Borel-Weil [07.06.09] and recovers the Weyl character formula via Atiyah-Bott localisation on $G/T$ with fixed-point set $W$. The equivariant refinement of Goresky-Kottwitz-MacPherson 1998 gives the GKM presentation of $H_T^{\ast }(G/T;\mathbb{Q})$ as a Steinberg double-polynomial ring, fitting into the Atiyah-Bott / Duistermaat-Heckman localisation framework. Lean status: partial, module Codex.Modern.CharClasses.BorelHirzebruchGT declares Borel's coinvariant isomorphism, the dimension equality $\dim H^{\ast }(G/T;\mathbb{Q})=|W|$, the Schubert basis cardinality, and the splitting-principle anchor with sorry proof bodies pending the root-datum / reflection-group invariant theory plus bundle-level characteristic-class infrastructure in Mathlib.

  requires: char-classes.pontryagin-chern.definitions` (03.06.04), rep-theory.weyl-group` (07.06.04), rep-theory.borel-weil-theorem` (07.06.09), rep-theory.weyl-character-formula` (07.06.07)

• char-classes.modularity-elliptic-genusopen unit 03.06.23 →

  Modularity of the elliptic genus

  The Ochanine elliptic genus ${\phi }_{\mathrm{Och}}:{\Omega }_{\ast }^{SO}\to \mathbb{Z}[1/2][[q]]$ is the multiplicative-sequence genus attached to the characteristic power series $Q_{\mathrm{Och}}(x,q)=(x/2)/\sinh (x/2)\cdot {\prod }_{n\ge 1}(1-q^n{)}^2/((1-q^n{e}^x)(1-q^n{e}^{-x}))$, equivalently the theta-function quotient $x{\theta }_1^{\prime }(0,\tau )/(2\pi i{\theta }_1(x/(2\pi i),\tau ))$ (Ochanine 1987, Topology 26, 143-151). The modularity theorem (Ochanine 1987; Landweber-Stong 1988; Zagier 1988): for $M$ a closed spin manifold of real dimension $4k$, ${\phi }_{\mathrm{Och}}(M)(\tau )$ is the $q$-expansion of a modular form of weight $2k$ for the congruence subgroup ${\Gamma }_0(2)\subset {\mathrm{SL}}_2(\mathbb{Z})$ with $\mathbb{Z}$-integral Fourier coefficients. The Witten genus ${\phi }_W:{\Omega }_{\ast }^{\mathrm{String}}\to \mathbb{Z}[[q]]$ (Witten 1987 CMP 109; 1988 LNM 1326) is the level-$1$ refinement defined on string manifolds ($p_1/2=0$); for $M$ a closed string $4k$-manifold, ${\phi }_W(M)(\tau )$ is a modular form of weight $2k$ for the full modular group ${\mathrm{SL}}_2(\mathbb{Z})$. Cuspidal limits: ${\phi }_{\mathrm{Och}}(M)(i\infty )=\hat{A}(M)$ (the Dirac index, by collapsing $q\to 0$ to recover the $\hat{A}$-power series $(x/2)/\sinh (x/2)$); the Fricke involution $W_2:\tau \mapsto -1/(2\tau )$ on ${\Gamma }_0(2)$ exchanges the two cusps, and $(W_2\cdot {\phi }_{\mathrm{Och}}(M))(i\infty )=\mathrm{sig}(M)$ (the signature, recovering the $L$-genus). The Witten genus has one cusp, with ${\phi }_W(M)(i\infty )=\hat{A}(M)$. Topological characterisation (Landweber-Stong 1988, Topology 27, 145-161): the Ochanine genus is the unique multiplicative genus on ${\Omega }_{\ast }^{SO}\otimes \mathbb{Z}[1/2]$ vanishing on quaternionic-projective-line bundles with reduced structure group. Physical interpretation (Witten 1987 CMP 109): ${\phi }_{\mathrm{Och}}$ is the partition function of an $N=1$ supersymmetric 2-D sigma model with target $M$ on a worldsheet torus of modulus $\tau$; modular invariance reflects worldsheet ${\mathrm{SL}}_2(\mathbb{Z})$-symmetry. Hopkins-Miller / TMF refinement (Hopkins-Miller 1994; Ando-Hopkins-Strickland 2001 Invent. Math. 146, 595-687; Hopkins 2002 ICM Vol. I, 291-317): the Witten genus lifts to a ring-spectrum map $\mathrm{MString}\to \mathrm{TMF}$, where $\mathrm{TMF}$ is the topological-modular-form spectrum constructed via Goerss-Hopkins obstruction theory on the moduli stack of elliptic curves. Equivariant rigidity (Bott-Taubes 1989 JAMS 2, 137-186; Liu 1995 J. Diff. Geom. 41, 343-396): for $M^{4k}$ closed spin with $S^1$-action, ${\phi }_{\mathrm{Och}}^{S^1}(M)(q,t)$ is constant in the equivariant parameter $t$; Liu's proof identifies the equivariant genus with an index-$0$ Jacobi form on $\mathbb{H}\times \mathbb{C}$. Lean status: none; Mathlib lacks the modular-forms ring $M_{\ast }({\Gamma }_0(N))$ for congruence subgroups, the Jacobi theta functions ${\theta }_1,\dots ,{\theta }_4$ on $\mathbb{H}$, the oriented and string bordism rings as named graded $\mathbb{Z}$-algebras, and the multiplicative-sequence functor ${\Omega }_{\ast }^{SO}\to \mathbb{Q}[[q]]$ attached to a formal power series; each piece is a green-field formalisation target.

  requires: char-classes.multiplicative-sequences-genera` (03.06.15), char-classes.hirzebruch-signature` (03.06.11), char-classes.oriented-bordism-pontryagin-thom` (03.06.13), index-theory.atiyah-singer.index-theorem` (03.09.10), number-theory.modular-forms-sl2-z` (21.04.01)

• k-theory.classifying-spacesopen unit 03.08.04 →

  Classifying space

  Universal principal bundles, pullback classification, homotopy invariance, numerable bundles, and the role of $BU(n)$ and $BO(n)$ in characteristic classes and K-theory.

  requires: bundle.principal-bundle, topology.homotopy, topology.topological-space

• k-theory.vector-bundlesopen unit 03.08.01 →

  Topological K-theory

  Grothendieck group of vector bundles, direct sum, virtual bundles, reduced K-theory, pullback functoriality, and preparation for Bott periodicity.

  requires: bundle.vector-bundle, k-theory.classifying-spaces, topology.topological-space

• topology.eilenberg-maclaneopen unit 03.12.05 →

  Eilenberg-MacLane space

  Spaces $K(A,n)$ with ${\pi }_n=A$ and other ${\pi }_k=0$. Uniqueness up to weak equivalence. Loop-space characterisation. Representability of ordinary cohomology: $H^n(X;A)\cong [X,K(A,n)]$. Postnikov towers. Cohomology operations and the Steenrod algebra.

  requires: topology.topological-space, topology.homotopy, topology.spectrum

• homotopy.stable-homotopyopen unit 03.08.06 →

  Stable homotopy

  Suspension, stable homotopy groups, spectra-level intuition, Freudenthal stabilization, stable phenomena behind generalized cohomology theories, and the stable classical-group context for Bott periodicity.

  requires: topology.homotopy, topology.suspension, topology.sphere

• k-theory.bott.periodicityopen unit 03.08.07 →

  Bott periodicity

  Complex K-theory has period $2$ and real K-theory has period $8$. The unit treats the classifying-space form, coefficient tables, Bott elements, Clifford-module source of real periodicity, and the role of Bott periodicity in the topological index.

  requires: k-theory.vector-bundles, k-theory.classifying-spaces, spin-geometry.clifford.clifford-algebra, char-classes.pontryagin-chern.definitions

• k-theory.adams-operationsopen unit 03.08.02 →

  Adams operations ${\psi }^k$ on K-theory

  ${\psi }^k$ as ring homomorphisms on $K(X)$; Newton-power-sum interpretation via $\lambda$-ring structure; the Adams 1962 application — non-existence of $H$-space structures on $S^{2n-1}$ beyond $n=1,2,4$ — and Adams' theorem on vector fields on spheres, which Atiyah-Singer-style index theory recovers through Real K-theory.

  requires: k-theory.vector-bundles, char-classes.pontryagin-chern.definitions

• topology.kr-theoryopen unit 03.08.12 →

  KR-theory (K-theory with reality)

  Atiyah's bigraded $K{R}^{p,q}(X)$ for a Real space $(X,\tau )$, unifying $KU$, $KO$, $KSp$, and $KSC$ in one ring. The (1, 1)-periodicity $K{R}^{p+1,q+1}\cong K{R}^{p,q}$ is the K-theoretic shadow of the Clifford bridging identity ${\mathrm{Cl}}_{r+1,s+1}\cong {\mathrm{Cl}}_{r,s}\otimes {\mathrm{Cl}}_{1,1}$. Combined with the complex two-fold periodicity it produces the eight-fold real Bott periodicity. KR hosts the Atiyah-Singer index theorem for Real elliptic operators, with $\mathbb{Z}/2$-valued indices in dimensions $\equiv 1,2(\mathrm{mod}8)$ detecting positive-scalar-curvature obstructions. The KR-refinement of Adams' division-algebra argument gives a unified proof that the only normed division-algebra spheres are $S^0,S^1,S^3,S^7$. Distinct from spin-geometry.kr-theory (same theory, complementary entry point).

  requires: k-theory.vector-bundles, k-theory.bott.periodicity, spin-geometry.clifford.clifford-algebra

• gauge-theory.yang-mills.actionopen unit 03.07.05 →

  Yang-Mills action

  Defines $\mathrm{YM}(A)=\frac{1}{2}\Vert F_A{\Vert }_{L^2}^2$, proves gauge invariance, derives $d_A^{\ast }{F}_A=0$, relates four-dimensional self-duality to Yang-Mills, and separates metric-dependent action from Chern-Weil topological classes.

  requires: bundle.principal-bundle, diffgeo.connection.connection, diffgeo.connection.curvature, lie-algebra.lie-algebra, topology.de-rham-cohomology, char-classes.chern-weil.homomorphism

• physics.cft.basicsopen unit 03.10.02 →

  CFT basics

  Introduces two-dimensional conformal symmetry, primary fields, stress tensor, Witt and Virasoro algebras, central charge, OPEs, radial quantization, and the state-operator correspondence.

  requires: functional-analysis.hilbert-space, lie-algebra.central-extension, lie-algebra.infinite-dimensional-representations, lie-algebra.virasoro

• index-theory.atiyah-singer.index-theoremopen unit 03.09.10 →

  Atiyah-Singer index theorem

  States ${\mathrm{ind}}_a(D)={\mathrm{ind}}_t([\sigma (D)])$, explains symbol K-theory and Bott-periodic topological index, specializes to the spin Dirac formula $⟨\hat{A}(TM)\mathrm{ch}(E),[M]⟩$, and connects heat-kernel local index theory to Chern-Weil forms.

  requires: functional-analysis.fredholm.operators, spin-geometry.dirac.dirac-operator, k-theory.bott.periodicity, char-classes.pontryagin-chern.definitions, char-classes.chern-weil.homomorphism, diffgeo.elliptic-operators

• spin-geometry.spin-groupopen unit 03.09.03 →

  Spin group $\mathrm{Spin}(n)$

  Connected double cover of $\mathrm{SO}(n)$ for $n\ge 3$. Even part of the Pin group. Realizes universal cover for $n=3$. Spin(4) is SU(2) × SU(2). Spin(6) is SU(4).

  requires: linalg.bilinear-form, spin-geometry.clifford.clifford-algebra, lie-groups.connected-double-cover

• diffgeo.operator.symbolopen unit 03.09.07 →

  Symbol of a differential operator

  Principal symbols of linear differential operators, order filtration, lower-order quotient, composition rule, cotangent-bundle interpretation, and Dirac symbol.

  requires: linalg.vector-space, diffgeo.smooth-manifold, bundle.vector-bundle

• diffgeo.elliptic-operatorsopen unit 03.09.09 →

  Elliptic operators on a manifold

  Elliptic differential operators, invertible principal symbols off the zero section, Laplacian and Dirac examples, parametrices, elliptic regularity, Fredholmness, and symbol K-theory class.

  requires: diffgeo.operator.symbol, functional-analysis.unbounded-self-adjoint, bundle.vector-bundle, diffgeo.smooth-manifold

• char-classes.invariant-polynomial.adjoint-invariantopen unit 03.06.05 →

  Invariant polynomial on a Lie algebra

  Symmetric multilinear $G$-invariant functions on a Lie algebra; the algebra $S^{\bullet }({\mathfrak{g}}^{\ast }{)}^G$. Group/Lie-algebra invariance equivalence for connected $G$. Generators for matrix Lie algebras: $\mathrm{tr}(A^k)$, elementary symmetric polynomials of eigenvalues, Pfaffian for $\mathfrak{so}$. Chevalley's restriction theorem. The input to Chern-Weil theory.

  requires: lie-algebra.lie-algebra, linalg.vector-space, linalg.bilinear-form

• lie-algebra.lie-algebraopen unit 03.04.01 →

  Lie algebra

  Lie algebras as vector spaces with a bilinear, antisymmetric, Jacobi-respecting bracket; classical examples ${\mathfrak{gl}}_n$, ${\mathfrak{sl}}_n$, ${\mathfrak{so}}_n$, ${\mathfrak{u}}_n$; the adjoint representation; ideals and homomorphisms; the commutator-on-an-associative-algebra construction. The Killing form, simple/semisimple/solvable structure, Cartan classification, and the exponential map / BCH formula are at Master tier.

  requires: linalg.vector-space

• lie-algebra.central-extensionopen unit 03.11.01 →

  Central extension of a Lie algebra

  Central extensions of Lie algebras, one-dimensional extensions by 2-cocycles, coboundary equivalence, relation to projective representations, and Virasoro central charge.

  requires: lie-algebra.lie-algebra

• lie-algebra.infinite-dimensional-representationsopen unit 03.11.02 →

  Infinite-dimensional Lie algebra representations

  Lie algebra representations on infinite-dimensional vector spaces, modules, highest-weight representations, central elements acting by scalars under irreducibility hypotheses, and the representation-theoretic bridge from central extensions to CFT.

  requires: lie-algebra.lie-algebra, lie-algebra.central-extension, functional-analysis.hilbert-space

• lie-algebra.virasoroopen unit 03.11.03 →

  Virasoro algebra

  Witt algebra of Laurent vector fields, Virasoro central extension, Gelfand-Fuchs cocycle, central charge, highest-weight modules, and the CFT stress-tensor mode algebra.

  requires: lie-algebra.central-extension, lie-algebra.infinite-dimensional-representations

• symplectic-geometry.symplectic-vector-spaceopen unit 05.01.01 →

  Symplectic vector space

  A vector space with a nondegenerate skew form. Central theorem: standard symplectic basis theorem. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 01.01.03, 01.01.15

• symplectic-geometry.symplectic-manifoldopen unit 05.01.02 →

  Symplectic manifold

  A smooth manifold with a closed nondegenerate 2-form. Central theorem: symplectic volume form theorem. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 03.02.01, 03.04.02, 03.04.04, 05.01.01

• symplectic-geometry.symplectic-groupopen unit 05.01.03 →

  Symplectic group

  The linear transformations preserving a symplectic form. Central theorem: matrix criterion for the symplectic group. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 03.03.01, 03.03.03, 05.01.01

• symplectic-geometry.darboux-theoremopen unit 05.01.04 →

  Darboux's theorem

  The local normal form for every symplectic form. Central theorem: Darboux local coordinate theorem. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.01.02, 03.04.04

• symplectic-geometry.moser-trickopen unit 05.01.05 →

  Moser's trick

  Path-method proof technique. Given ${\omega }_t$ a path of cohomologous symplectic forms with ${\dot{\omega }}_t=d{\alpha }_t$, define $X_t$ by ${\iota }_{X_t}{\omega }_t=-{\alpha }_t$; the time-1 flow ${\psi }_1$ satisfies ${\psi }_1^{\ast }{\omega }_1={\omega }_0$. Originator-anchor for Darboux's theorem (modern proof), Weinstein neighbourhood theorem, equivariant Darboux, and smooth structure of regular symplectic reduction.

  requires: symplectic-geometry.symplectic-manifold, symplectic-geometry.darboux-theorem

• symplectic-geometry.weinstein-neighbourhoodopen unit 05.05.02 →

  Weinstein Lagrangian neighbourhood theorem

  Closed Lagrangian $L\subset (M,\omega )$ has a tubular neighbourhood symplectomorphic to a neighbourhood of the zero section in $T^{\ast }L$ with the canonical form $-d\lambda$. Three-step proof: Lagrangian splitting + normal bundle = $T^{\ast }L$; tubular diffeomorphism via exponential map; Moser's trick on path of forms vanishing on $L$. Equivariant version. Generating-function bridge to symplectomorphisms. Foundational for Floer-theoretic comparison and Arnold-Givental conjectures. Originator: Weinstein 1971 Symplectic manifolds and their Lagrangian submanifolds (Adv. Math. 6).

  requires: symplectic-geometry.lagrangian-submanifold, symplectic-geometry.symplectic-manifold, symplectic-geometry.moser-trick

• symplectic-geometry.hamiltonian-vector-fieldopen unit 05.02.01 →

  Hamiltonian vector field

  The vector field determined by a function and the symplectic form. Central theorem: Hamiltonian flow preserves the symplectic form. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.01.02, 03.04.04

• symplectic-geometry.poisson-bracketopen unit 05.02.02 →

  Poisson bracket and Poisson manifold

  A lie bracket on functions encoding hamiltonian dynamics. Central theorem: Poisson bracket satisfies Jacobi identity. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.02.01, 03.04.01

• symplectic-geometry.integrable-systemopen unit 05.02.03 →

  Integrable system

  Many commuting conserved quantities on a symplectic manifold. Central theorem: commuting Hamiltonians preserve common level sets. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.02.01, 05.02.02

• symplectic-geometry.action-angle-coordinatesopen unit 05.02.04 →

  Action-angle coordinates

  Canonical coordinates near compact invariant tori. Central theorem: Liouville-Arnold local normal form. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.02.03, 05.01.04

• symplectic-geometry.cotangent-bundleopen unit 05.02.05 →

  Cotangent bundle as canonical symplectic manifold

  The natural symplectic form on a cotangent bundle. Central theorem: canonical one-form gives the cotangent symplectic form. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 03.02.01, 03.04.02, 03.04.04, 05.01.02

• symplectic-geometry.coadjoint-orbitopen unit 05.03.01 →

  Coadjoint orbit

  An orbit in the dual of a lie algebra with a natural symplectic form. Central theorem: Kirillov-Kostant-Souriau form is symplectic. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 03.03.01, 03.04.01, 05.02.02

• symplectic-geometry.moment-mapopen unit 05.04.01 →

  Moment map

  A map whose components generate an infinitesimal group action. Central theorem: moment map components are Hamiltonians for fundamental fields. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 03.03.02, 05.02.01, 05.02.02

• symplectic-geometry.symplectic-reductionopen unit 05.04.02 →

  Marsden-Weinstein symplectic reduction

  A quotient construction producing a smaller symplectic manifold. Central theorem: regular reduction theorem. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.04.01, 03.03.02, 05.01.02

• symplectic-geometry.lagrangian-submanifoldopen unit 05.05.01 →

  Lagrangian submanifold

  A half-dimensional submanifold on which the symplectic form vanishes. Central theorem: graph of a closed one-form is Lagrangian. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.01.02, 03.02.01

• symplectic-geometry.almost-complex-structureopen unit 05.06.01 →

  Almost-complex structure on a symplectic manifold

  An endomorphism squaring to minus identity compatible with the symplectic form. Central theorem: compatible almost-complex structures exist. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.01.02, 03.05.02

• symplectic-geometry.pseudoholomorphic-curveopen unit 05.06.02 →

  Pseudoholomorphic curve

  A surface whose tangent map intertwines complex structures. Central theorem: energy-area identity for pseudoholomorphic curves. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.06.01, 03.02.01

• complex-geometry.newlander-nirenbergopen unit 05.06.03 →

  Newlander-Nirenberg integrability theorem

  An almost-complex structure $J$ on $M$ comes from a complex-manifold structure if and only if its Nijenhuis tensor $N_J(X,Y)=[JX,JY]-J[JX,Y]-J[X,JY]-[X,Y]$ vanishes. Equivalent formulations: involutivity of $T^{1,0}M$ under the Lie bracket; ${\bar{\partial }}^2=0$ on ${\Lambda }^{\ast }$; local existence of $n$ smooth $\mathbb{C}$-valued functions $z_1,\dots ,z_n$ with $\bar{\partial }{z}_i=0$ and $d{z}_1\wedge \cdots \wedge d{z}_n\ne 0$. Originator: Newlander-Nirenberg 1957 (smooth case via Hörmander-style $L^2$-estimates for $\bar{\partial }$); predecessors Korn 1914 / Lichtenstein 1916 (real-dimension 2), Cartan-Kähler 1934 (real-analytic case via Frobenius), Eckmann-Frölicher 1951 (formal version); modern simplifications Malgrange 1969, Webster 1989. Real-dimension cases: dim 2 automatic; dim 4 generically obstructed; $S^4$ admits no almost-complex structure (Wu / Massey); $S^6$ integrability is the famous open problem on the $G_2$-invariant structure. Cross-strand bridge between symplectic almost-complex geometry and complex geometry; prerequisite for Kodaira-Spencer deformation theory of complex structures.

  requires: symplectic-geometry.almost-complex-structure, symplectic-geometry.pseudoholomorphic-curve

• symplectic-geometry.non-squeezingopen unit 05.07.01 →

  Gromov non-squeezing theorem

  The rigidity theorem forbidding symplectic squeezing of a ball into a thin cylinder. Central theorem: Gromov non-squeezing theorem. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.01.02, 05.06.02

• symplectic-geometry.symplectic-capacityopen unit 05.07.02 →

  Symplectic capacity

  A numerical invariant measuring symplectic size. Central theorem: capacity monotonicity under symplectic embeddings. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.07.01, 05.01.02

• symplectic-geometry.arnold-conjectureopen unit 05.08.01 →

  Arnold conjecture and Floer homology setup

  Fixed points of hamiltonian diffeomorphisms counted through floer theory. Central theorem: Arnold fixed-point lower bound statement. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.02.01, 05.05.01, 05.06.02, 05.07.02

• symplectic-geometry.floer-homologyopen unit 05.08.02 →

  Floer homology

  A homology theory generated by hamiltonian orbits or lagrangian intersections. Central theorem: boundary squares to zero under compactness and gluing. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.08.01, 05.06.02

• symplectic-geometry.maslov-indexopen unit 05.08.03 →

  Maslov index

  An integer measuring winding of lagrangian subspaces. Central theorem: Maslov index is homotopy invariant with fixed endpoints. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.05.01, 05.01.01

• symplectic-geometry.conley-zehnder-indexopen unit 05.08.04 →

  Conley-Zehnder index

  An index grading nondegenerate hamiltonian periodic orbits. Central theorem: Conley-Zehnder index changes by Maslov index under loop composition. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.01.03, 05.08.03, 05.02.01

• stat-mech.partition-functionopen unit 08.01.01 →

  Partition function (statistical mechanics)

  The weighted sum of all allowed states in a statistical system. Central theorem: thermodynamic derivatives of log partition function. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 00.02.05

• stat-mech.ising-modelopen unit 08.01.02 →

  Ising model

  A lattice model whose spins take two values and interact with neighbors. Central theorem: one-dimensional Ising transfer-matrix solution. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.01

• stat-mech.boltzmann-distributionopen unit 08.01.03 →

  Boltzmann distribution and canonical ensemble

  The probability rule assigning lower weight to higher energy at fixed temperature. Central theorem: canonical distribution maximizes entropy with fixed mean energy. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.01

• stat-mech.free-energyopen unit 08.01.04 →

  Free energy

  The thermodynamic potential obtained from the logarithm of the partition function. Central theorem: free energy generates canonical thermodynamics. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.01, 08.01.03

• stat-mech.mean-fieldopen unit 08.02.01 →

  Mean-field theory and Curie-Weiss model

  An approximation replacing many neighbors by an average field. Central theorem: Curie-Weiss self-consistency equation. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.02, 08.01.04

• stat-mech.spontaneous-symmetry-breakingopen unit 08.02.02 →

  Spontaneous symmetry breaking

  The selection of asymmetric equilibrium states from symmetric equations. Central theorem: mean-field double-well criterion for broken symmetry. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.02.01, 01.02.01

• stat-mech.mermin-wagneropen unit 08.02.03 →

  Mermin-Wagner theorem

  A low-dimensional obstruction to breaking continuous symmetries at positive temperature. Central theorem: absence of continuous-symmetry long-range order in two dimensions. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.02.02, 02.01.05

• stat-mech.onsager-solutionopen unit 08.03.01 →

  Onsager solution of the 2D Ising model (transfer matrix)

  The exact solution locating the two-dimensional ising critical point. Central theorem: Onsager critical temperature formula. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.02, 08.03.02

• stat-mech.transfer-matrixopen unit 08.03.02 →

  Transfer matrix

  A linear operator that advances a lattice model one slice at a time. Central theorem: largest eigenvalue controls thermodynamic free energy. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 01.01.03, 02.11.01

• stat-mech.real-space-rgopen unit 08.04.01 →

  Renormalisation group (real-space block decimation)

  A scale-changing transformation on statistical systems. Central theorem: fixed points organize long-distance behavior. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.02, 08.01.04

• stat-mech.wilson-fisheropen unit 08.04.02 →

  Wilson-Fisher fixed point and universality

  The non-gaussian fixed point governing many critical phenomena below four dimensions. Central theorem: epsilon-expansion fixed point to first order. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.04.01, 08.06.01, 08.04.03

• stat-mech.beta-functionopen unit 08.04.03 →

  Beta function (renormalisation group)

  The vector field describing how couplings change with scale. Central theorem: fixed points are zeros of the beta function. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.04.01

• stat-mech.block-spin-decimationopen unit 08.04.04 →

  Block-spin decimation

  A concrete coarse-graining map replacing blocks of spins by effective spins. Central theorem: decimation induces effective couplings. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.04.01

• stat-mech.critical-exponentsopen unit 08.05.01 →

  Critical exponents and scaling laws

  Numbers measuring power-law behavior near a phase transition. Central theorem: Rushbrooke scaling relation. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.02.01, 08.04.02, 08.01.04

• stat-mech.correlation-functionsopen unit 08.05.02 →

  Correlation functions (statistical mechanics)

  Expectations of products of observables at separated points. Central theorem: connected correlations detect fluctuations. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.01, 08.05.01

• stat-mech.gaussian-fieldopen unit 08.06.01 →

  Gaussian field theory and free boson

  A field theory whose action is quadratic and whose correlations are determined by a green kernel. Central theorem: Gaussian Wick factorization. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.05.02, 02.11.08

• stat-mech.conformal-criticalityopen unit 08.06.02 →

  Conformal symmetry at criticality

  Scale symmetry enhanced by angle-preserving transformations at critical points. Central theorem: two-dimensional primary-field two-point form. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.05.01, 08.06.01, 03.10.02

• stat-mech.path-integralopen unit 08.07.01 →

  Path integral formulation of statistical mechanics

  A continuum weighted sum over field configurations. Central theorem: saddle-point expansion around a stationary configuration. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.01, 08.06.01, 03.04.08

• stat-mech.wilson-lattice-gaugeopen unit 08.08.01 →

  Wilson's lattice gauge theory

  A lattice regularization of gauge fields using group elements on links. Central theorem: Wilson plaquette action approximates Yang-Mills action. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 03.03.01, 03.05.07, 08.08.02

• stat-mech.wilson-actionopen unit 08.08.02 →

  Wilson action

  The plaquette action measuring lattice curvature in gauge theory. Central theorem: small-plaquette expansion recovers curvature squared. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 03.07.05, 03.03.01

• stat-mech.effective-field-theoryopen unit 08.08.03 →

  Effective field theory

  A scale-dependent description retaining operators relevant at the chosen resolution. Central theorem: irrelevant operators are suppressed at long distance. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.04.01, 08.07.01

• stat-mech.wick-rotationopen unit 08.09.01 →

  Quantum-classical correspondence (Wick rotation)

  The relation between quantum time evolution and statistical weights after imaginary-time continuation. Central theorem: thermal trace as imaginary-time path integral. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.07.01, 02.11.08

• stat-mech.bosonic-fock-space-second-quantisationopen unit 08.10.01 →

  Bosonic Fock space and second quantisation

  The bosonic Fock space ${\mathcal{F}}_s(\mathcal{H})={⨁}_{n\ge 0}{S}^n\mathcal{H}$ is the symmetric tensor algebra of a one-particle Hilbert space, with creation and annihilation operators $a^{\ast }(f),a(g)$ satisfying $[a(f),a^{\ast }(g)]=⟨f,g⟩$ on the dense finite-particle subspace. Central theorem: the CCR algebra plus the Cook 1953 construction realises every free bosonic quantum theory as a Fock-space representation. Stat-mech-side framing: the grand-canonical free-Bose-gas partition function $\Xi ={\prod }_k(1-e^{-\beta ({ϵ}_k-\mu )}{)}^{-1}$ is the thermal trace over ${\mathcal{F}}_s(\mathcal{H})$, and the Klein-Gordon free field's mode expansion $\varphi (x)=\int dk[a(\mathbf{k})e^{-ikx}+a^{\ast }(\mathbf{k})e^{ikx}]/(2{\omega }_k{)}^{1/2}$ is the operator-valued tempered distribution living on ${\mathcal{F}}_s(L^2(X_m,d{\lambda }_m))$. Companion to the QM-side quantum-mechanics.bosonic-fock-space-second-quantisation (12.13.01), which emphasises the functional-analytic / Stone-von-Neumann angle; this stat-mech entry emphasises the operational / partition-function / mode-expansion angle. Gaussian-Fock correspondence (Glimm-Jaffe Ch. 6) identifies operator-side vacuum expectations with measure-side Gaussian moments via Wick rotation, the bridge between this unit and the Euclidean field-theory strand.

  requires: 01.01.03, 02.11.08, 02.11.03, 03.01.04, 08.01.01, 08.06.01, 08.07.01, 08.09.01

• stat-mech.phi-4-dyson-seriesopen unit 08.10.03 →

  φ⁴ theory and the Dyson series

  The interaction Hamiltonian of φ⁴ theory generates a time-ordered exponential whose perturbative expansion is the Dyson series. Central theorem: Wick's theorem reduces the vacuum expectation of a time-ordered product of free fields to a sum over pairings of Feynman propagators. First non-trivial application: tree-level 2→2 scattering amplitude $\mathcal{M}=-i\lambda$. One-loop bubble diagram exhibits the logarithmic UV divergence cured by coupling renormalisation, yielding the one-loop beta function $\beta (\lambda )=3{\lambda }^2/(16{\pi }^2)$. Stat-side framing: the Dyson expansion is the perturbative series of the Euclidean path-integral generating functional around the Gaussian free-field measure; the Wick-rotated theory is the field-theoretic continuum limit of the critical Ising model in the Wilsonian sense. Triviality in $d=4$ (Aizenman-Duminil-Copin 2021) is the modern closure of the four-dimensional φ⁴ chapter.

  requires: stat-mech.path-integral, stat-mech.wick-rotation, quantum-mechanics.bosonic-fock-space-second-quantisation

• complex-analysis.meromorphic-functionopen unit 06.01.05 →

  Meromorphic function

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.01

• complex-analysis.cauchy-integral-formulaopen unit 06.01.02 →

  Cauchy integral formula

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.01

• complex-analysis.residue-theoremopen unit 06.01.03 →

  Residue theorem

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.02, 06.01.05

• complex-analysis.analytic-continuationopen unit 06.01.04 →

  Analytic continuation

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.01

• complex-analysis.riemann-mapping-theoremopen unit 06.01.06 →

  Riemann mapping theorem

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.01, 06.01.02

• complex-analysis.riemann-sphereopen unit 06.01.07 →

  Riemann sphere

  Distinctive Ahlfors item from plans/fasttrack/lars-ahlfors-complex-analysis.md §3 audit. Three-tier unit covering one-point compactification, the two-chart holomorphic atlas, biholomorphism with ${\mathbb{P}}^1(\mathbb{C})$, Möbius automorphism group ${\mathrm{PSL}}_2(\mathbb{C})$, function field $\mathbb{C}(z)$, Riemann-Roch in genus 0, and Grothendieck's splitting of vector bundles on ${\mathbb{P}}^1$.

  requires: complex-analysis.holomorphic-function

• complex-analysis.mobius-transformationsopen unit 06.01.08 →

  Möbius (linear-fractional) transformations

  Distinctive Ahlfors item 7 from plans/fasttrack/lars-ahlfors-complex-analysis.md §3 audit. Three-tier unit covering the group ${\mathrm{PSL}}_2(\mathbb{C})=\mathrm{Aut}({\mathbb{C}}_{\infty })$, three-point uniqueness via cross-ratio, classification by trace squared (parabolic / elliptic / hyperbolic / loxodromic), circle-and-line preservation, and the bridge to hyperbolic geometry on $\mathbb{H}$ via ${\mathrm{PSL}}_2(\mathbb{R})$ and the modular group ${\mathrm{PSL}}_2(\mathbb{Z})$. Foundational for Schwarz-Pick, Schwarz-Christoffel, and the modular function $\lambda /j$.

  requires: complex-analysis.riemann-sphere

• complex-analysis.cauchy-riemannopen unit 06.01.10 →

  Cauchy-Riemann equations and harmonic conjugate

  Distinctive Ahlfors item 9 from plans/fasttrack/lars-ahlfors-complex-analysis.md §3 audit. Three-tier unit covering the differential characterisation of holomorphicity: real Jacobian as complex multiplication, equivalence of complex differentiability with the CR system $u_x=v_y$, $u_y=-v_x$, harmonic conjugate existence on simply-connected domains, Wirtinger $\partial /\bar{\partial }$ formalism, Hartogs's separately-holomorphic theorem, and the connection to elliptic-regularity for general elliptic PDE.

  requires: complex-analysis.riemann-sphere

• complex-analysis.harmonic-functionsopen unit 06.01.11 →

  Harmonic functions on the plane

  Distinctive Ahlfors item 10 from plans/fasttrack/lars-ahlfors-complex-analysis.md §3 audit. Three-tier unit covering the planar Laplace equation $u_{xx}+u_{yy}=0$, the connection $f=u+iv$ holomorphic $\Rightarrow u,v$ harmonic and the converse on simply-connected domains via harmonic conjugate, the mean-value property, the maximum principle, the Poisson integral as the disc Dirichlet solution, the Liouville theorem for bounded plane harmonic functions, and the elliptic-regularity perspective that frames harmonic-function theory as the prototypical scalar elliptic PDE on ${\mathbb{R}}^2$. Foundational for Schwarz reflection, Schwarz lemma, Perron's method on general domains, Hardy-space theory, and the higher-dimensional generalisation $\Delta u=\sum u_{x_i{x}_i}=0$ on ${\mathbb{R}}^n$.

  requires: complex-analysis.cauchy-riemann

• complex-analysis.max-modulus-schwarzopen unit 06.01.12 →

  Maximum modulus + Schwarz lemma

  Distinctive Ahlfors item 11 from plans/fasttrack/lars-ahlfors-complex-analysis.md §3 audit. Three-tier unit covering the maximum modulus principle on connected open sets (non-constant holomorphic $f$ cannot attain an interior modulus maximum), the Schwarz lemma on the unit disc ($f:\mathbb{D}\to \mathbb{D}$ with $f(0)=0$ satisfies $|f(z)|\le |z|$ and $|f^{\prime }(0)|\le 1$, with rotation as the only equality case), the disc automorphism group $\mathrm{Aut}(\mathbb{D})\cong \mathrm{PSU}(1,1)$, the Schwarz-Pick hyperbolic-metric contraction, Schwarz-Ahlfors-Pick on negatively-curved metrics, Phragmén-Lindelöf on unbounded sectors, Hadamard three-circles convexity, and Cartan's lemma in several complex variables. Foundational for the Riemann mapping theorem's uniqueness statement, Koenigs linearisation in complex dynamics, the modular-function bridge to Picard's theorems, and the Kobayashi-hyperbolic framework on complex manifolds.

  requires: complex-analysis.cauchy-riemann, complex-analysis.harmonic-functions

• complex-analysis.argument-principleopen unit 06.01.13 →

  Argument principle and Rouché's theorem

  Distinctive Ahlfors item from plans/fasttrack/lars-ahlfors-complex-analysis.md §3 audit (P1, Ch. 4 §5). Three-tier unit covering the argument principle ($(2\pi i{)}^{-1}{\oint }_{\gamma }{f}^{\prime }/fdz=N-P$ for meromorphic $f$ with no zeros or poles on $\gamma$, equivalently the winding number of $f\circ \gamma$ around the origin), the generalised argument principle with a holomorphic weight $g$, Rouché's theorem in both the asymmetric form $|g|<|f|$ on $\partial D$ and the symmetric form $|f+g|<|f|+|g|$ on $\partial D$, the open mapping theorem (non-constant holomorphic functions are open maps, with the local $m$-to-$1$ refinement), Hurwitz's theorem on uniform limits of non-vanishing holomorphic functions, and the fundamental theorem of algebra proved cleanly via Rouché on a large circle. Foundational for the local-degree theory of branched coverings ($\mathrm{06.02.02}$) and the Riemann-Hurwitz formula, the existence half of the Riemann mapping theorem ($\mathrm{06.01.06}$, via Hurwitz applied to Montel-extremal sequences), Jensen's formula and Nevanlinna value-distribution theory, and the argument-principle-as-index-theorem perspective that bridges to the Atiyah-Singer index theorem on higher-dimensional elliptic operators.

  requires: complex-analysis.cauchy-riemann

• complex-analysis.branch-point-ramificationopen unit 06.02.01 →

  Branch point and ramification

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.04, 06.03.01

• complex-analysis.branched-coveringsopen unit 06.02.02 →

  Branched coverings of Riemann surfaces

  Forster-distinctive item from plans/fasttrack/forster-riemann-surfaces.md priority 2 (item 9). Three-tier unit organising branched covers as a category, with Riemann's existence theorem as the master equivalence.

  requires: 04.04.02, 06.05.02

• alg-geom.riemann-existence-theoremopen unit 06.02.03 →

  Riemann's existence theorem for algebraic curves

  The converse direction to complex-analysis.branched-coverings (06.02.02): every compact Riemann surface is biholomorphic to the analytification of a smooth projective algebraic curve. Three proof routes (Riemann-Roch + very-ample embedding; function-field generation with $P(f,g)=0$; branched-cover monodromy + GAGA). Master tier covers GAGA, the function-field-of-curves equivalence of categories, Belyi's theorem, and Chow's theorem as the higher-dimensional analogue.

  requires: 06.02.02, 06.04.04

• complex-analysis.genus-riemann-surfaceopen unit 06.03.02 →

  Genus of a Riemann surface

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.03.01

• complex-analysis.uniformization-theoremopen unit 06.03.03 →

  Uniformization theorem

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.03.01, 06.01.06

• complex-analysis.divisor-riemann-surfaceopen unit 06.05.01 →

  Divisor on a Riemann surface

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.05, 06.03.01

• complex-analysis.holomorphic-line-bundleopen unit 06.05.02 →

  Holomorphic line bundle on a Riemann surface

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.05.01, 06.03.01, 03.05.02

• complex-analysis.riemann-hurwitz-formulaopen unit 06.05.03 →

  Riemann-Hurwitz formula

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.02.01, 06.03.02

• complex-analysis.holomorphic-one-formopen unit 06.06.01 →

  Holomorphic 1-form / abelian differential

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.03.01, 03.04.02

• complex-analysis.period-matrixopen unit 06.06.02 →

  Period matrix

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.06.01, 06.03.02

• complex-analysis.jacobian-varietyopen unit 06.06.03 →

  Jacobian variety

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.06.02, 06.06.04

• complex-analysis.abel-jacobi-mapopen unit 06.06.04 →

  Abel-Jacobi map

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.05.01, 06.06.01

• complex-analysis.theta-functionopen unit 06.06.05 →

  Theta function

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.06.02, 06.06.03

• complex-analysis.holomorphic-several-variablesopen unit 06.07.01 →

  Holomorphic functions of several variables

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.01

• complex-analysis.hartogs-phenomenonopen unit 06.07.02 →

  Hartogs phenomenon

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.07.01

• rep-theory.lie-algebra-representationopen unit 07.06.01 →

  Lie algebra representation

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 03.04.01, 01.01.03, 07.01.01

• rep-theory.universal-enveloping-algebraopen unit 07.06.02 →

  Universal enveloping algebra

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 03.04.01, 03.01.02, 07.06.01

• rep-theory.root-systemopen unit 07.06.03 →

  Root system

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 03.04.01, 01.01.15, 07.04.01

• rep-theory.weyl-groupopen unit 07.06.04 →

  Weyl group

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 07.06.03

• rep-theory.dynkin-diagramopen unit 07.06.05 →

  Dynkin diagram

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 07.06.03, 07.06.04

• rep-theory.verma-moduleopen unit 07.06.06 →

  Verma module

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 07.06.01, 07.06.02, 07.03.01

• rep-theory.weyl-character-formulaopen unit 07.06.07 →

  Weyl character formula

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 07.03.01, 07.06.03, 07.06.04

• rep-theory.weyl-dimension-formulaopen unit 07.06.08 →

  Weyl dimension formula

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 07.06.07

• rep-theory.borel-weil-theoremopen unit 07.06.09 →

  Borel-Weil theorem

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 07.06.07, 03.05.02

• rep-theory.compact-lie-group-representationopen unit 07.07.01 →

  Compact Lie group representation

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 03.03.01, 07.01.01

• rep-theory.peter-weyl-theoremopen unit 07.07.02 →

  Peter-Weyl theorem

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 07.07.01, 02.11.08, 07.01.01

• rep-theory.haar-measureopen unit 07.07.03 →

  Haar measure

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 03.03.01, 03.04.03

• rep-theory.characteropen unit 07.01.03 →

  Character of a representation

  Class function ${\chi }_{\rho }(g)=\mathrm{tr}(\rho (g))$. Originated by Frobenius via Dedekind's group-determinant problem. Powers orthogonality, dimension formula, decomposition algorithms. Generalises to compact Lie groups via integration against Haar measure.

  requires: rep-theory.group-representation

• rep-theory.character-orthogonalityopen unit 07.01.04 →

  Character orthogonality

  Row and column orthogonality. Gives multiplicity formula $n_V=⟨\chi ,{\chi }_V⟩$ and isomorphism criterion (representations isomorphic iff characters agree). Frobenius's first orthogonality relation; Schur's 1905 derivation via Schur's lemma is the modern textbook approach.

  requires: rep-theory.character, rep-theory.schur-lemma

• rep-theory.regular-representationopen unit 07.01.05 →

  Regular representation

  $\mathbb{C}G$ as a $G$-representation. Contains every irreducible $V$ with multiplicity $\dim V$, giving $|G|={\sum }_V(\dim V{)}^2$. Character ${\chi }_{\mathrm{reg}}(e)=|G|$, ${\chi }_{\mathrm{reg}}(g)=0$ for $g\ne e$. Foundational computational object.

  requires: rep-theory.group-representation

• rep-theory.tensor-product-of-representationsopen unit 07.01.06 →

  Tensor product of representations

  $({\rho }_1\otimes {\rho }_2)(g)={\rho }_1(g)\otimes {\rho }_2(g)$ on $V_1\otimes V_2$. Character ${\chi }_{V_1\otimes V_2}={\chi }_{V_1}\cdot {\chi }_{V_2}$. Decomposition into irreducibles (Clebsch-Gordan). Foundation of Schur-Weyl duality.

  requires: rep-theory.group-representation, linalg.tensor-product

• rep-theory.induced-representationopen unit 07.01.07 →

  Induced representation

  ${\mathrm{Ind}}_H^{G}W=\mathbb{C}G{\otimes }_{\mathbb{C}H}W$. Character formula ${\chi }_{\mathrm{Ind}W}(g)=\frac{1}{|H|}{\sum }_{x\in G:x^{-1}gx\in H}{\chi }_W(x^{-1}gx)$. Foundation of Mackey theory and ${\mathrm{GL}}_n({\mathbb{Q}}_p)$ representations.

  requires: rep-theory.group-representation, groups.group

• rep-theory.frobenius-reciprocityopen unit 07.01.08 →

  Frobenius reciprocity

  Adjunction ${\mathrm{Hom}}_G({\mathrm{Ind}}_H^{G}W,V)\cong {\mathrm{Hom}}_H(W,{\mathrm{Res}}_H^{G}V)$. Prototype of all categorical adjunctions, predating category theory by 50+ years. Powers Mackey theory, Brauer's theorem on induced characters, Langlands correspondences.

  requires: rep-theory.induced-representation, rep-theory.character

• rep-theory.maschke-theoremopen unit 07.02.01 →

  Maschke's theorem

  Complete reducibility of finite-group representations over fields where $|G|$ is invertible. Averaging-projection proof. Failure in characteristic dividing $|G|$ launches modular representation theory (Brauer 1935).

  requires: rep-theory.group-representation

• spin-geometry.heat-kernel-indexopen unit 03.09.20 →

  Heat-kernel proof of the Atiyah-Singer index theorem

  The McKean-Singer formula $\mathrm{ind}(D)=\mathrm{Str}(e^{-t{D}^2})$ holds for every $t>0$. Large-$t$ limit returns the analytic index from the harmonic projection; small-$t$ asymptotic expansion of the heat kernel produces the local index density $\hat{A}({\nabla }^{TM})\mathrm{ch}({\nabla }^E)$ (spin Dirac case). Getzler 1986 rescaling reduces the small-$t$ computation to the harmonic-oscillator heat kernel of Mehler. Alvarez-Gaumé 1983 reproduces the result by supersymmetric path-integral evaluation. The proof package supplies the local form, not just the index integer.

  requires: index-theory.atiyah-singer.index-theorem, diffgeo.elliptic-operators, spin-geometry.dirac.dirac-operator, char-classes.pontryagin-chern.definitions

• spin-geometry.family-equivariant-indexopen unit 03.09.21 →

  Family, equivariant, and Lefschetz fixed-point index theorems

  Family index of an elliptic family $D=\{D_b{\}}_{b\in B}$ is a class $\mathrm{ind}(D)\in K^0(B)$ whose Chern character equals the integral of $\hat{A}\cdot \mathrm{ch}(\sigma )$ over the fibre. Equivariant index of a $G$-equivariant elliptic operator lives in the representation ring $R(G)$. Lefschetz fixed-point formula expresses the equivariant index via local contributions at the fixed-point set $M^g$. The three refinements share a heat-kernel proof: insert the family / group action into the supertrace, invoke localisation.

  requires: index-theory.atiyah-singer.index-theorem, spin-geometry.heat-kernel-index, k-theory.bott.periodicity, bundle.principal-bundle.connection

• spin-geometry.pseudodifferentialopen unit 03.09.22 →

  Sobolev spaces, pseudodifferential operators, and elliptic parametrices

  Sobolev spaces $H^s$ measure regularity by integrability of derivatives. Embedding theorem $H^s\hookrightarrow C^k$ for $s>k+n/2$. Rellich-Kondrachov compactness $H^s\hookrightarrow H^{s^{\prime }}$ for $s>s^{\prime }$ on bounded domains. Pseudodifferential operators of order $m$ are quantisations of symbols $a(x,\xi )\in S^m$. Elliptic symbols admit a parametrix $Q\in {\Psi }^{-m}$ with $QP-I,PQ-I\in {\Psi }^{-\infty }$ smoothing. On a closed manifold smoothing operators are compact between Sobolev spaces; Atkinson then yields Fredholmness of every elliptic operator. Notation: Thom class $\tau \in H^{\ast }$ (cohomology), $\Lambda \in K^{\ast }$ (K-theory) per LM.

  requires: functional-analysis.compact-operators, diffgeo.operator.symbol, diffgeo.elliptic-operators

• spin-geometry.dirac-bundleopen unit 03.09.14 →

  Generalised Dirac bundles and the Bochner-Weitzenböck identity

  A Dirac bundle is a Hermitian vector bundle equipped with a fibrewise Clifford action of $T^{\ast }M$ and a metric-compatible connection whose Levi-Civita-Leibniz rule holds. The Dirac operator is $D=c\circ {\nabla }^{\mathfrak{E}}$. The universal Bochner-Weitzenböck identity reads $D^2=({\nabla }^{\mathfrak{E}}{)}^{\ast }{\nabla }^{\mathfrak{E}}+{\mathcal{R}}^{\mathfrak{E}}$ with ${\mathcal{R}}^{\mathfrak{E}}=\frac{1}{2}{\sum }_{j,k}c(e^j)c(e^k)R^{\mathfrak{E}}(e_j,e_k)$. Specialisations: spinor bundle ($\mathcal{R}=\frac{1}{4}\mathrm{Scal}$, Lichnerowicz 1963), de Rham bundle ($\mathcal{R}$ = Ricci on 1-forms, Bochner 1946; Weitzenböck curvature operator on $k$-forms), twisted spinor bundle ($\mathcal{R}=\frac{1}{4}\mathrm{Scal}+F^E$, Atiyah-Singer twist).

  requires: spin-geometry.spinor-bundle, spin-geometry.dirac.dirac-operator, bundle.principal-bundle.connection, diffgeo.de-rham

• spin-geometry.clk-diracopen unit 03.09.15 →

  Cl_k-linear Dirac operators and the KO-valued index

  A ${\mathrm{Cl}}_k$-linear Dirac bundle carries a graded right ${\mathrm{Cl}}_k$-action commuting with the Clifford action of $T^{\ast }M$. Its Clifford-index $\hat{\alpha }(D)$ lives in ${\hat{\mathfrak{M}}}_k=M_k/i^{\ast }{M}_{k+1}$, the ABS module quotient, which equals $K{O}_k$ via the Atiyah-Bott-Shapiro isomorphism. The Cl_k-linear AS theorem (Atiyah-Singer Index IV) computes the topological side as the KO-pushforward of the symbol class. The $\alpha$-invariant of Hitchin is the special case $\hat{\alpha }(D_{\mathfrak{S}})$ on a spin manifold, the foundational psc obstruction in dimensions where the integer Dirac index vanishes. Notation disambiguation: ${\mathrm{Cl}}_k$ here is the standard real Clifford algebra ${\mathrm{Cl}}_{0,k}$ (positive-definite in the sign convention $v^2=-|v{|}^2$), distinct from the ${\mathrm{Cl}}_{r,s}$ chessboard family of 03.09.11.

  requires: spin-geometry.dirac-bundle, spin-geometry.clifford.clifford-algebra, index-theory.atiyah-singer.index-theorem, k-theory.bott.periodicity

• spin-geometry.exercise-pack-ch1open unit 03.09.E1 →

  Clifford and spin algebra exercise pack (Lawson-Michelsohn Ch. I supplement)

  Twelve exercises covering Clifford chessboard low-rank computations, Pin/Spin extensions, automorphism inner-products, Atiyah-Bott-Shapiro module classifications, K-theory orientations, and exceptional Lie group descriptions. Cross-cuts the existing units 03.09.02, 03.09.03, 03.09.11, and (in low-dim and chessboard exercises) 03.09.13. Difficulty distribution: 4 easy / 5 medium / 3 hard. Each exercise carries a hint and full answer in <details> blocks. Exercise-pack-only unit type — slimmed frontmatter with tiers_present: [intermediate], no Lean infrastructure.

  requires: spin-geometry.clifford.clifford-algebra, spin-geometry.spin-group, spin-geometry.clifford-chessboard

• spin-geometry.calibrated-geometriesopen unit 03.09.19 →

  Calibrated geometries — Special Lagrangian, associative, coassociative, Cayley

  A calibration of degree $p$ on a Riemannian manifold is a closed $p$-form of comass at most 1 globally. Submanifolds calibrated by $\phi$ — those whose oriented unit tangent $p$-vector lies in the contact set $\{\xi :\phi (\xi )=1\}$ — are volume-minimisers in their homology class by the Harvey-Lawson fundamental theorem (one chain of inequalities: Stokes plus comass bound). The four named calibrated geometries are: Special Lagrangian on a Calabi-Yau $n$-fold ($\mathrm{Re}\Omega$, holonomy $\mathrm{SU}(n)$, contact set $\mathrm{SU}(n)/\mathrm{SO}(n)$); associative on a $G_2$ 7-fold ($\varphi$, contact set $G_2/\mathrm{SO}(4)$); coassociative on a $G_2$ 7-fold ($\ast \varphi$); Cayley on a Spin(7) 8-fold ($\Phi$). Each calibrating form is the spinor square of a parallel spinor preserved by the special holonomy. McLean 1998 computed the moduli-space dimensions: $b_1$ for SL, $b_{+}^2$ for coassociative, normal-Dirac kernel for associative and Cayley.

  requires: spin-geometry.triality, spin-geometry.spinor-bundle, spin-geometry.structure.spin-structure

• spin-geometry.psc-obstructionopen unit 03.09.16 →

  Positive scalar curvature obstruction theory

  The classical psc obstruction chain spans 1963–1992 in four links. Lichnerowicz 1963: $D^2={\nabla }^{\ast }\nabla +\frac{1}{4}\mathrm{Scal}$ on the spinor bundle, with positive scalar curvature forcing $\ker D=0$ and hence $\hat{A}(M)[M]=0$. Hitchin 1974: the Cl_n-linear refinement, with the α-invariant $\alpha (M)\in K{O}_n$ vanishing under psc; the new content is the $\mathbb{Z}/2$ obstructions in dimensions $n\equiv 1,2\mathrm{mod}8$, detecting exotic spheres without psc. Gromov-Lawson 1980/83: the surgery theorem (psc preserved under codim-$\ge 3$ surgery) and the enlargeable manifold theorem (manifolds admitting arbitrarily small Lipschitz maps to $S^n$ from finite covers admit no psc, ruling out $T^n$). Stolz 1992: in the simply-connected case, $\alpha (M)=0$ is equivalent to psc-existence in dimension $\ge 5$. Lawson is co-originator of Gromov-Lawson. Notation: the α-invariant symbol is $\alpha (M)$ (notation decision #24, pinned in this unit).

  requires: spin-geometry.dirac-bundle, spin-geometry.clk-dirac, spin-geometry.spinor-bundle, spin-geometry.structure.spin-structure

• spin-geometry.witten-positive-massopen unit 03.09.17 →

  Witten positive-mass theorem via spinors

  Witten 1981 reproved the Schoen-Yau positive-mass theorem in three pages by introducing a harmonic spinor with prescribed asymptotic value ${\psi }_{\infty }$, applying the Lichnerowicz formula on the asymptotically flat spin 3-manifold, and identifying the integration-by-parts boundary term at infinity with $-4\pi m_{\mathrm{ADM}}|{\psi }_{\infty }{|}^2$. Result: $4\pi m_{\mathrm{ADM}}={\int }_M|\nabla \psi {|}^2+\frac{1}{4}{\int }_M\mathrm{Scal}|\psi {|}^2\ge 0$ under non-negative scalar curvature; equality forces a parallel spinor and hence flat ${\mathbb{R}}^3$. The Parker-Taubes 1982 reformulation made the existence of the Witten spinor rigorous via weighted Sobolev spaces. The Dirac-Witten operator $\hat{D}=D-\frac{1}{2}{K}^{ij}c(e_i){\nabla }_{e_j}$ extends the argument to the spacetime case (positive-energy theorem with dominant energy condition).

  requires: spin-geometry.spinor-bundle, spin-geometry.structure.spin-structure, spin-geometry.dirac-bundle

• spin-geometry.berger-holonomyopen unit 03.09.18 →

  Berger holonomy classification and parallel spinors

  Berger 1955 classified the restricted holonomy of simply-connected, irreducible, non-symmetric Riemannian manifolds: $\mathrm{SO}(n)$ (generic), $\mathrm{U}(n)$ (Kähler), $\mathrm{SU}(n)$ (Calabi-Yau, Ricci-flat), $\mathrm{Sp}(n)$ (hyperkähler), $\mathrm{Sp}(n)\mathrm{Sp}(1)$ (quaternionic Kähler, Einstein), $G_2$ (dim 7, Ricci-flat), $\mathrm{Spin}(7)$ (dim 8, Ricci-flat). The proof is a representation-theoretic case analysis: which irreducible subalgebras of $\mathfrak{so}(n)$ admit non-zero curvature tensors satisfying the first Bianchi identity? Wang 1989 established the bijection between Berger's special holonomies $\{\mathrm{SU}(n),\mathrm{Sp}(n),G_2,\mathrm{Spin}(7)\}$ and the existence of a non-zero parallel spinor, with parallel-spinor counts 2, $n+1$, 1, 1 respectively. Bryant 1987 constructed local exceptional-holonomy metrics; Joyce 1996/2000 constructed compact $G_2$ and $\mathrm{Spin}(7)$ manifolds. The Wang bijection is the structural foundation of the Harvey-Lawson calibrated-geometry framework: every parallel spinor produces a calibration via spinor squaring.

  requires: spin-geometry.spinor-bundle, spin-geometry.structure.spin-structure, spin-geometry.spin-group

• spin-geometry.exercise-pack-ch4open unit 03.09.E2 →

  Chapter IV applications exercise pack (Lawson-Michelsohn Ch. IV supplement)

  Fourteen exercises covering Chapter IV applications of spin geometry. Distribution: 4 easy / 7 medium / 3 hard. Group I (4 exercises): psc obstruction — Lichnerowicz on flat torus, Â-genus of K3, α-invariant on $\mathbb{H}{P}^2$, enlargeable propagation under product. Group II (3 exercises): Witten positive-mass theorem — ADM mass of Schwarzschild, Witten boundary identity, equality case via parallel-spinor rigidity. Group III (3 exercises): Berger holonomy — list size, parallel-spinor count on K3, Calabi-Yau Ricci-flatness. Group IV (4 exercises): calibrated geometries — comass of volume form, Wirtinger inequality, Special Lagrangian lines, $G_2$ associative 3-planes via the cross product. Cross-cuts the four Batch-2 units 03.09.16–03.09.19. Each exercise carries a hint and full answer in <details> blocks. Exercise-pack-only unit type — slimmed frontmatter with tiers_present: [intermediate], no Lean infrastructure.

  requires: spin-geometry.psc-obstruction, spin-geometry.witten-positive-mass, spin-geometry.berger-holonomy, spin-geometry.calibrated-geometries

• diffgeo.de-rham.kunnethopen unit 03.04.12 →

  Künneth formula for de Rham cohomology — two proofs

  $H_{\mathrm{dR}}^{\ast }(M\times F)\cong H_{\mathrm{dR}}^{\ast }(M)\otimes H_{\mathrm{dR}}^{\ast }(F)$ for finite-type $M$, proved twice. First proof: Mayer-Vietoris induction over a good cover, with the cross-product map ${\pi }_M^{\ast }\alpha \wedge {\pi }_F^{\ast }\beta$ as the natural transformation, base case the Poincaré lemma on ${\mathbb{R}}^n\times F$, inductive step via the five lemma on the MV ladder. Second proof: tic-tac-toe ascent on the Čech-de Rham double complex of the product cover $\mathcal{U}\times \mathcal{V}$, factoring as Čech-of-constant-coefficients tensored. The dual-proof presentation is Bott-Tu's pedagogical signature — one theorem, told twice; the second telling is shorter because the first installed the machinery. Failure mode: infinite-type manifolds; cross-product is injective but the tensor-product splitting fails. Consequences: $H^{\ast }(T^n)={\Lambda }^{\ast }[{\theta }_1,\dots ,{\theta }_n]$ exterior algebra of dimension $2^n$; multiplicativity of Euler characteristic $\chi (M\times F)=\chi (M)\chi (F)$; Poincaré-polynomial multiplication $P_{M\times F}(t)=P_M(t)P_F(t)$.

  requires: diffgeo.de-rham.mayer-vietoris, diffgeo.de-rham.good-cover-induction, diffgeo.de-rham.cech-de-rham-double-complex, topology.de-rham-cohomology

• alg-top.singular-cohomologyopen unit 03.04.13 →

  Singular cohomology and the de Rham theorem (with $\mathbb{Z}$ coefficients)

  Singular complex $C_{\ast }(X)={⨁}_n\mathbb{Z}⟨\sigma :{\Delta }^n\to X⟩$ with face boundary $\partial$; singular cohomology $H_{\mathrm{sing}}^{\ast }(X;A)=H^{\ast }(\mathrm{Hom}(C_{\ast },A))$. Key constructions: cone construction — every chain on a contractible space is a boundary, hence acyclic; small chains — for an open cover, every chain is chain-equivalent to a sum of chains supported in single opens, giving a singular MV long exact sequence. De Rham theorem $H_{\mathrm{dR}}^k(M)\cong H_{\mathrm{sing}}^k(M;\mathbb{R})$ via three routes: (i) MV induction over a good cover comparing the de Rham and singular MV functors via the integration pairing; (ii) Čech-de Rham double complex collapses to both Čech and de Rham, identifying both with ${\check{H}}^{\ast }(\mathcal{U};\underline{\mathbb{R}})$; (iii) sheaf-cohomology via Leray on the constant sheaf $\underline{\mathbb{R}}$ with the Poincaré-lemma fine resolution. The integer-coefficient version requires the singular complex (de Rham only sees $\mathbb{R}$ coefficients). Eilenberg-Steenrod axioms 1952 axiomatised the construction; de Rham 1931 proved the analytic equivalence directly.

  requires: diffgeo.de-rham.mayer-vietoris, topology.de-rham-cohomology, diffgeo.de-rham.cech-de-rham-double-complex, alg-geom.sheaf-cohomology

• alg-geom.cohomology.local-system-monodromyopen unit 04.03.02 →

  Local systems, monodromy, and twisted cohomology

  A local system on $X$ is a locally constant sheaf $\mathcal{L}$ of $A$-modules — concretely, a sheaf such that every point has a neighborhood where $\mathcal{L}$ is the constant sheaf with stalk a fixed $A$-module $V$. On a connected, locally simply-connected $X$, the category of local systems is equivalent to the category of ${\pi }_1(X,x_0)$-representations: parallel transport along loops gives the monodromy representation $\rho :{\pi }_1(X)\to \mathrm{Aut}(V)$. Examples: constant sheaf (trivial monodromy); orientation local system ${\mathrm{or}}_M$ on a non-orientable manifold (monodromy $\mathbb{Z}/2$ via the orientation double cover); Möbius local system on $S^1$. Twisted cohomology $H^{\ast }(X;\mathcal{L})$ is the cohomology with local-system coefficients; for orientation-twisted coefficients on a non-orientable manifold, Poincaré duality recovers via $H^k(M;\underline{\mathbb{R}})\cong H_c^{n-k}(M;{\mathrm{or}}_M)$ (the twisted Poincaré duality of Bott-Tu §7). Cohomology $H^{\ast }(X;\mathcal{L})$ on $K(\pi ,1)$-spaces equals group cohomology $H^{\ast }(\pi ;V)$ with $V$ a $\pi$-module via $\rho$.

  requires: alg-geom.sheaf-cohomology, topology.cover.double-cover, topology.covering-space, topology.homotopy

• diffgeo.de-rham.exercise-packs.chapter-iopen unit 03.04.E1 →

  Mayer-Vietoris and degree-theory exercise pack (Bott-Tu Ch. I supplement)

  Fourteen exercises covering Chapter I of Bott-Tu — Mayer-Vietoris computation, degree theory, Hopf invariant, sphere cohomology by induction, ${\mathbb{R}}^n$-minus-$k$-points, the punctured torus, Stokes-on-manifolds-with-boundary applications. Distribution: 4 easy / 6 medium / 4 hard. Each exercise carries a hint and full solution in <details> blocks. Cross-cuts N1 (MV), N2 (good cover), N4 (Thom and degree), and the deepening D1 of 03.04.06. Exercise-pack-only unit type — slimmed frontmatter with tiers_present: [intermediate], no Lean infrastructure.

  requires: diffgeo.de-rham.mayer-vietoris, diffgeo.de-rham.good-cover-induction, diffgeo.de-rham.thom-cv-cohomology, topology.de-rham-cohomology

• diffgeo.de-rham.mayer-vietorisopen unit 03.04.07 →

  Mayer-Vietoris sequence for de Rham cohomology

  For a smooth manifold $M=U\cup V$, the short exact sequence $0\to {\Omega }^{\ast }(M)\to {\Omega }^{\ast }(U)\oplus {\Omega }^{\ast }(V)\to {\Omega }^{\ast }(U\cap V)\to 0$ produces the long exact Mayer-Vietoris sequence in de Rham cohomology. Surjectivity is the partition-of-unity argument; the connecting homomorphism $\delta :H^q(U\cap V)\to H^{q+1}(M)$ is the canonical zig-zag of homological algebra. Compactly-supported variant has reversed maps; both extend to arbitrary good covers via the generalised Mayer-Vietoris sequence (the row exactness of the Čech-de Rham double complex). The canonical inductive computation of $H^{\ast }(S^n)$ runs through this machinery — Bott-Tu's pedagogical heart. Closes Bott-Tu §1 → §2 within-chapter sequencing.

  requires: topology.de-rham-cohomology, diffgeo.exterior-derivative, diffgeo.differential-forms, diffgeo.smooth-manifold

• diffgeo.de-rham.good-cover-inductionopen unit 03.04.10 →

  Good covers, finite-dimensionality of de Rham cohomology, and the Mayer-Vietoris induction

  A good cover of a smooth manifold is an open cover $\mathcal{U}$ such that every finite intersection $U_{{\alpha }_0\cdots {\alpha }_p}$ is diffeomorphic to a Euclidean ball (in particular contractible). Existence: choose a Riemannian metric and use the fact that geodesically convex normal-coordinate neighborhoods exist around every point and have geodesically convex finite intersections. Finite good cover exists on every compact manifold; countable good cover on every paracompact manifold. The Mayer-Vietoris induction over a finite good cover proves: $H_{\mathrm{dR}}^{\ast }(M)$ is finite-dimensional in each degree on a compact manifold, vanishes above $\dim M$, and admits explicit inductive computation. Foundation for both the Čech-de Rham double-complex theorem and the Künneth / Poincaré-duality dual proofs.

  requires: diffgeo.de-rham.mayer-vietoris, topology.de-rham-cohomology, diffgeo.smooth-manifold

• diffgeo.de-rham.cech-de-rham-double-complexopen unit 03.04.11 →

  Čech-de Rham double complex and the tic-tac-toe principle

  Bigraded complex $K^{p,q}(\mathcal{U})={\prod }_{{\alpha }_0<\cdots <{\alpha }_p}{\Omega }^q(U_{{\alpha }_0\cdots {\alpha }_p})$ on a cover $\mathcal{U}$, with Čech differential $\delta$ horizontal (alternating-sum restriction), de Rham differential $d$ vertical (componentwise exterior derivative), and total differential $D=d+(-1{)}^p\delta$ on bidegree $(p,q)$. Total complex ${\mathrm{Tot}}^n(K)={⨁}_{p+q=n}{K}^{p,q}$. On a good cover, both row-collapse (yielding $H_{\mathrm{dR}}^{\ast }(M)$) and column-collapse (yielding ${\check{H}}^{\ast }(\mathcal{U};\underline{\mathbb{R}})$) compute the same total cohomology — Weil's 1952 proof of the de Rham theorem. Tic-tac-toe ascent is the diagonal-staircase algorithm that makes the equivalence explicit. Generalised Mayer-Vietoris is the row exactness; tic-tac-toe Künneth and Poincaré duality follow from the bigraded structure. Prototype of the spectral sequence of a filtered complex (Bott-Tu §14). Notation $K^{p,q}$, ${\mathrm{Tot}}^n$, $\delta$ vs $d$, $D=d+(-1{)}^p\delta$ — Pass 4 §3.4 decisions #12, #13, #27, #28.

  requires: diffgeo.de-rham.good-cover-induction, topology.de-rham-cohomology, diffgeo.de-rham.mayer-vietoris, alg-geom.sheaf-cohomology

• diffgeo.de-rham.thom-cv-cohomologyopen unit 03.04.09 →

  Compactly-supported cohomology, integration along the fiber, and the de Rham Thom isomorphism

  ${\Omega }_c^{\ast }(M)$ compactly supported forms; $H_c^{\ast }(M)$ compactly supported cohomology. For a vector bundle $\pi :E\to M$, ${\Omega }_{cv}^{\ast }(E)$ is the compactly-vertical complex (forms whose support meets every fiber in a compact set; Bott-Tu coinage), and $H_{cv}^{\ast }(E)$ its cohomology. Integration along the fiber ${\pi }_{\ast }:{\Omega }_{cv}^{p+r}(E)\to {\Omega }^p(M)$ for a rank-$r$ oriented bundle commutes with $d$ and induces the Thom isomorphism $H^p(M)\cong H_{cv}^{p+r}(E)$, with inverse $\omega \mapsto {\pi }^{\ast }\omega \wedge \Phi$ for the Thom class $\Phi \in H_{cv}^r(E)$ characterised by ${\int }_{\mathrm{fiber}}\Phi =1$. Global angular form $\psi$ on the unit-sphere bundle of an oriented rank-$r$ bundle, with Bott-Tu sign convention $d\psi =-{\pi }^{\ast }e(E)$. Provides the de Rham model of the Euler class as the obstruction to a global section, and dual proof (via Čech-de Rham of [03.04.11]) of the Thom isomorphism. Notation ${\Omega }_{cv}^{\ast }$, $H_{cv}^{\ast }$, $\Phi$, $\psi$ — Pass 4 §3.4 decisions #4, #8, #9, #21, #22.

  requires: diffgeo.de-rham.mayer-vietoris, topology.de-rham-cohomology, bundle.vector-bundle, diffgeo.stokes-theorem

• homotopy.spectral-sequence.filtered-complexopen unit 03.13.01 →

  Spectral sequences — exact couples, filtered complexes, double complexes

  A spectral sequence is a tower of bigraded pages $\{E_r^{p,q},d_r{\}}_{r\ge 1}$ with $d_r$ of bidegree $(r,1-r)$ (notation decision #15), $H(E_r,d_r)=E_{r+1}$, and limiting page $E_{\infty }$ recovering an associated graded of a filtration on a target $H^{\ast }$. The convergence symbol $\Rightarrow$ (decision #30) denotes this abutment. Three equivalent presentations: Massey 1952 exact couples (the algebraic structure that drives the page-advance), filtered cochain complexes (the geometric source — bounded filtrations give the cleanest convergence), and double complexes with two anticommuting differentials $d^{\prime },d^{\prime \prime }$ producing two spectral sequences ${}_{I}E$ and ${}_{II}E$ (decision #29) both abutting to $H^{\ast }(\mathrm{Tot})$. The Čech-de Rham double complex of [03.04.11] is the concrete prototype: its two collapsing spectral sequences prove the de Rham theorem on a good cover. Multiplicative structure: a filtered DGA gives each page the structure of a bigraded ring, $d_r$ a derivation, with $E_{\infty }$ the associated graded of the cup product. Edge homomorphisms and transgression supply the partial maps used in low-degree calculations. Master section channels Leray 1946 directly: invented at Oflag XVII-A, the prisoner-of-war camp where Leray hid his fluid-mechanics expertise behind the topological work that became sheaf theory and spectral sequences.

  requires: topology.de-rham-cohomology, diffgeo.de-rham.cech-de-rham-double-complex

• homotopy.spectral-sequence.serreopen unit 03.13.02 →

  Leray-Serre spectral sequence and the Gysin sequence

  For a fibration $F\to E\stackrel{\pi }{\to }B$ with $B$ simply connected (or a local system describing monodromy on ${\pi }_1(B)$), the Leray-Serre spectral sequence has $E_2^{p,q}=H^p(B;H^q(F))$ and converges multiplicatively to $H^{p+q}(E)$. Three canonical computations following Bott-Tu: (i) the Hopf fibration $S^1\to S^3\to S^2$ — non-trivial $d_2$ kills classes in degrees 1 and 2, giving $H^{\ast }(S^3)$ with classes only in 0 and 3; (ii) the trivial $S^1\to T^2\to S^1$ where the spectral sequence collapses at $E_2$ and recovers the Künneth product; (iii) the unit-circle bundle of $\mathcal{O}(1)\to \mathbb{C}{P}^k$ giving the cohomology of $S^{2k+1}$ via Gysin. The Gysin sequence is the long exact sequence

  requires: homotopy.spectral-sequence.filtered-complex, diffgeo.de-rham.mayer-vietoris, bundle.vector-bundle

• homotopy.spectral-sequence.leray-hirschopen unit 03.13.03 →

  Leray-Hirsch theorem and the splitting principle for vector bundles

  Leray-Hirsch theorem. For a fibre bundle $F\to E\stackrel{\pi }{\to }B$ with $H^{\ast }(F)$ free over the coefficient ring and finitely generated, if there exist global classes $e_1,\dots ,e_k\in H^{\ast }(E)$ whose restrictions to each fibre form a basis of $H^{\ast }(F)$, then $H^{\ast }(E)$ is a free $H^{\ast }(B)$-module on $\{e_i\}$:

  requires: homotopy.spectral-sequence.serre, char-classes.pontryagin-chern.definitions, bundle.vector-bundle

• homotopy.spectral-sequence.exercise-packopen unit 03.13.E1 →

  Spectral-sequence computation exercise pack (Bott-Tu Ch. III supplement)

  Eighteen exercises covering general spectral-sequence machinery, Leray-Serre and Gysin computations, Leray-Hirsch and the splitting principle, and Eilenberg-Moore on the path-loop fibration. Distribution: 5 easy / 9 medium / 4 hard. Group I (5 easy): bidegree of $d_r$, convergence symbol semantics, exact-couple from a SES, two-filtration $E_2$ identification, collapse-at-$E_2$ from a row/column hypothesis. Group II (4 medium): Hopf fibration $S^3\to S^2$ Serre SS, $S^1\to S^3\to S^2$ Gysin, $H^{\ast }(\mathbb{C}{P}^n)$ via Borel construction, Borel computation of $H^{\ast }(BU(n))$. Group III (5 medium): Leray-Hirsch on the projectivization $\mathbb{P}(E)\to M$, splitting principle for Pontryagin classes, $H^{\ast }(\mathrm{Gr}(2,4))$, Whitney via splitting, $H^{\ast }(\Omega S^n)$ via Serre. Group IV (4 hard): ${\pi }_4(S^3)=\mathbb{Z}/2$ via Postnikov truncation of $K(\mathbb{Z},3)$, Eilenberg-Moore on the path-loop fibration, transgression in a specific bundle, multiplicative structure. Cross-cuts the three Batch units 03.13.01–03. Each exercise carries a hint and full answer in <details> blocks. Exercise-pack-only unit type — slimmed frontmatter with tiers_present: [intermediate], no Lean infrastructure.

  requires: homotopy.spectral-sequence.filtered-complex, homotopy.spectral-sequence.serre, homotopy.spectral-sequence.leray-hirsch

• homotopy.rational.sullivan-minimal-modelsopen unit 03.12.06 →

  Sullivan minimal models and rational homotopy theory

  For a simply-connected space $X$ of finite rational type, the Sullivan minimal model is a quasi-isomorphism $\phi :(\Lambda V,d)\stackrel{\sim }{\to }{A}_{PL}(X)$ where $\Lambda V$ is a free graded-commutative algebra on a graded $\mathbb{Q}$-vector space $V$, the differential $d$ satisfies $dV\subseteq {\Lambda }^{\ge 2}V$ (minimality), and $A_{PL}$ is the piecewise-polynomial de Rham functor sending $X$ to compatible polynomial forms over each simplex. Sullivan's main theorem (1977): $V^n\cong {\mathrm{Hom}}_{\mathbb{Z}}({\pi }_n(X),\mathbb{Q})$, and the differential encodes Whitehead products and higher Massey products. Existence and uniqueness via the lifting lemma for minimal Sullivan algebras. Worked examples: $S^{2k}$ → $\Lambda (x,y)$ with $\mathrm{deg}x=2k$, $\mathrm{deg}y=4k-1$, $dy=x^2$; $S^{2k+1}$ → $\Lambda (x)$ with $\mathrm{deg}x=2k+1$, $dx=0$; $\mathbb{C}{P}^n$ → $\Lambda (x,y)$ with $\mathrm{deg}x=2$, $\mathrm{deg}y=2n+1$, $dy=x^{n+1}$. Halperin's algorithm computes the minimal model of a fibration $F\to E\to B$ as a perturbed tensor product $M_B\otimes \Lambda V_F$ with transgression encoded by the perturbation $\tau$. Formality theorem (Deligne-Griffiths-Morgan-Sullivan 1975): simply-connected compact Kähler manifolds have minimal models determined by their cohomology rings via the $d{d}^c$-lemma. Master section channels Sullivan 1977 Publ. IHÉS 47 directly: rational homotopy theory as the differential-form calculus on rational invariants, parallel to Quillen's DG-Lie-algebraic side, with Bott-Tu §19 the canonical pedagogical exposition.

  requires: topology.eilenberg-maclane, topology.de-rham-cohomology, homotopy.spectral-sequence.serre

• homotopy.universal-bundle-borelopen unit 03.08.05 →

  Universal bundle, $H^{\ast }(BU(k))$, and the Borel presentation of flag-manifold cohomology

  The universal complex rank-$k$ bundle ${\gamma }_k\to BU(k)$ is realised in the Grassmannian model $BU(k)=G_k({\mathbb{C}}^{\infty })=\underrightarrow{\lim }{G}_k({\mathbb{C}}^n)$ as the colimit ${\gamma }_k=\underrightarrow{\lim }{\gamma }_k^n$ of the tautological bundles ${\gamma }_k^n\to G_k({\mathbb{C}}^n)$ (notation decision #19). Steenrod's classification theorem identifies $[X,BU(k)]$ with rank-$k$ complex vector bundles on $X$ up to isomorphism, the bijection sending $[f]$ to $f^{\ast }{\gamma }_k$. The cohomology ring $H^{\ast }(BU(k);\mathbb{Z})=\mathbb{Z}[c_1,\dots ,c_k]$ is the polynomial algebra on the universal Chern classes, $\mathrm{deg}{c}_i=2i$; computed via the Leray-Serre spectral sequence of the fibration $\mathbb{C}{P}^{\infty }\to BU(k-1)\to BU(k)$ (or directly from the Schubert cell decomposition of the Grassmannian). Borel presentation (1953): for a compact Lie group $G$ with maximal torus $T$ of rank $r$ and Weyl group $W=N_G(T)/T$,

  requires: k-theory.classifying-spaces, homotopy.spectral-sequence.serre, char-classes.pontryagin-chern.definitions, homotopy.spectral-sequence.leray-hirsch

• homotopy.whitehead-tower-rational-hurewiczopen unit 03.12.07 →

  Whitehead tower, rational Hurewicz theorem, and Serre's finiteness

  The Whitehead tower of a connected space $X$ is a sequence $\cdots \to X⟨3⟩\to X⟨2⟩\to X⟨1⟩\to X$ of fibrations such that $X⟨n⟩$ is $n$-connected and the fibre $X⟨n⟩\to X⟨n-1⟩$ is the Eilenberg-MacLane space $K({\pi }_n(X),n-1)$. Dual to the Postnikov tower (which truncates from above), the Whitehead tower truncates from below: at each stage one kills the lowest non-vanishing homotopy group. The basepoint-loop space $\Omega X$ is the fibre of $PX\to X$ where $PX$ is the based path space (notation decision #32). Hurewicz theorem (1935-36): for a path-connected space $X$ with ${\pi }_i(X)=0$ for $i<n$, the Hurewicz map ${\pi }_n(X)\to H_n(X;\mathbb{Z})$ is an isomorphism (and surjective with abelianisation kernel for $n=1$). Rational Hurewicz: for simply-connected $X$, if ${\pi }_i(X)\otimes \mathbb{Q}=0$ for $1\le i\le n-1$, then ${\pi }_n(X)\otimes \mathbb{Q}\stackrel{\sim }{\to }{H}_n(X;\mathbb{Q})$ is an isomorphism, and ${\pi }_i(X)\otimes \mathbb{Q}\stackrel{\sim }{\to }{H}_i(X;\mathbb{Q})$ is an isomorphism for $1\le i\le 2n-2$. Hopf invariant $H(f)\in \mathbb{Z}$ (notation decision #34) of a map $f:S^{2n-1}\to S^n$: the integer counted by the linking number of two preimage spheres of generic regular values. Adams 1960 proved $H(f)=\pm 1$ only when $n\in \{1,2,4,8\}$. Serre's finiteness theorem (1953): for $k\ne n$ and except ${\pi }_{4k-1}(S^{2k})$ which has a $\mathbb{Z}$-summand from the Hopf invariant, the homotopy groups ${\pi }_k(S^n)$ are finite. Computations via Whitehead tower: ${\pi }_4(S^3)=\mathbb{Z}/2$ via $S^3\to K(\mathbb{Z},3)$ killing ${\pi }_3$; ${\pi }_5(S^3)=\mathbb{Z}/2$ via further truncation. Master section channels J.H.C. Whitehead 1953 + Hurewicz 1935-36 + Serre 1953 directly.

  requires: topology.eilenberg-maclane, homotopy.rational.sullivan-minimal-models, homotopy.spectral-sequence.serre

• homotopy.rational.exercise-packopen unit 03.12.E1 →

  Rational homotopy and Sullivan minimal-model exercise pack (Bott-Tu Ch. III §19 supplement)

  Eight exercises covering the Sullivan minimal-model machinery and rational homotopy. Distribution: 2 easy / 4 medium / 2 hard. Easy: minimal model of $S^{2k}$ (closed case), minimal model of $\mathbb{C}{P}^n$. Medium: minimal model of $\Omega S^n$ via Bott-Samelson; rational Hurewicz for a simply-connected space; minimal model of a fibration via Halperin's algorithm; minimal model of a Lie group as exterior algebra on primitive generators. Hard: Sullivan's solution to Serre's question ${\pi }_{\ast }(S^n)\otimes \mathbb{Q}$; formality of compact Kähler manifolds via the $d{d}^c$-lemma. Cross-cuts N12 and N14. Each exercise carries a hint and full answer in <details> blocks. Exercise-pack-only unit type — slimmed frontmatter with tiers_present: [intermediate], no Lean infrastructure.

  requires: homotopy.rational.sullivan-minimal-models, homotopy.whitehead-tower-rational-hurewicz, topology.de-rham-cohomology

• alg-geom.cech-schemesopen unit 04.03.03 →

  Čech cohomology of sheaves on schemes

  Čech cochain complex $C^p(\mathcal{U},\mathcal{F})={\prod }_{i_0<\cdots <i_p}\mathcal{F}(U_{i_0\cdots i_p})$ with alternating-sum-of-restriction differential $\delta :C^p\to C^{p+1}$; Čech cohomology ${\check{H}}^p(\mathcal{U},\mathcal{F})=\ker {\delta }^p/\mathrm{im}{\delta }^{p-1}$; refinement colimit ${\check{H}}^p(X,\mathcal{F})={\mathrm{colim}}_{\mathcal{U}}{\check{H}}^p(\mathcal{U},\mathcal{F})$. Cartan's comparison theorem: for separated scheme $X$, affine open cover $\mathcal{U}$, and quasi-coherent $\mathcal{F}$, the canonical map ${\check{H}}^{\ast }(\mathcal{U},\mathcal{F})\to H^{\ast }(X,\mathcal{F})$ is an isomorphism in all degrees. Foundation for the cohomology computation on ${\mathbb{P}}_k^n$ via the standard cover $\{D_{+}(x_i)\}$ — Hartshorne III.5. Affine vanishing (Serre): $H^p(\mathrm{Spec}A,\tilde{M})=0$ for $p>0$. Two-set covers reproduce Mayer-Vietoris. Čech-derived spectral sequence $E_2^{p,q}={\check{H}}^p(\mathcal{U},{\mathcal{H}}^q(\mathcal{F}))\Rightarrow H^{p+q}(X,\mathcal{F})$ degenerates when higher derived presheaves vanish on intersections. Originator: Čech 1932 (combinatorial topology); modern scheme version Serre 1955 FAC.

  requires: alg-geom.scheme, alg-geom.affine-scheme, alg-geom.sheaf-cohomology

• alg-geom.cohomology-projectiveopen unit 04.03.04 →

  Cohomology of line bundles on projective space

  Theorem (Hartshorne III.5.1, Serre 1955): for $X={\mathbb{P}}_k^n$ and $\mathcal{O}(d)$ the standard twisting sheaf, $H^0({\mathbb{P}}^n,\mathcal{O}(d))=S_d$ the degree-$d$ piece of $S=k[x_0,\dots ,x_n]$, of dimension $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ for $d\ge 0$ and zero otherwise; $H^n({\mathbb{P}}^n,\mathcal{O}(d))$ has basis Laurent monomials $x_0^{a_0}\cdots x_n^{a_n}$ with all $a_i<0$ and $\sum a_i=d$, of dimension $\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right)$ for $d\le -n-1$ and zero otherwise; $H^p({\mathbb{P}}^n,\mathcal{O}(d))=0$ for $0<p<n$. Proof via Čech on the standard cover $\{D_{+}(x_i){\}}_{i=0}^n$: localised polynomial rings $S_{x_{i_0}\cdots x_{i_p}}$, alternating-sum differential, monomial bookkeeping with negative-support decomposition. Serre duality on ${\mathbb{P}}^n$: $H^n({\mathbb{P}}^n,\mathcal{O}(d))\cong H^0({\mathbb{P}}^n,\mathcal{O}(-d-n-1){)}^{\ast }$ via cup-product pairing, with dual basis matching $a_i+b_i=-1$; canonical sheaf ${\omega }_{{\mathbb{P}}^n}=\mathcal{O}(-n-1)$ read off from this. Euler-Poincaré characteristic: $\chi ({\mathbb{P}}^n,\mathcal{O}(d))=\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ as a polynomial identity in $d$. Hilbert polynomial of ${\mathcal{O}}_{{\mathbb{P}}^n}$ of degree $n$, leading coefficient $1/n!$, constant term $1$. Bott vanishing: $H^p({\mathbb{P}}^n,{\Omega }^q(d))=0$ for $0<p<n$, $d>0$ via Euler exact sequence and the cohomology table. Riemann-Roch on ${\mathbb{P}}^n$ via Chern character and Todd class recovers $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ from $\int e^{dH}\mathrm{td}({\mathbb{P}}^n)$. Originator: Serre 1955 FAC; standard pedagogical reference Hartshorne III.5.

  requires: alg-geom.cech-schemes, alg-geom.projective-scheme

• alg-geom.serre-vanishing-finitenessopen unit 04.03.05 →

  Serre's vanishing and finiteness theorems

  Theorem (Serre 1955; Hartshorne III.5.2). Setup: $A$ a noetherian ring, $X$ a noetherian projective scheme over $A$ with very ample ${\mathcal{O}}_X(1)$, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-module. (1) Finiteness: $H^i(X,\mathcal{F})$ is a finitely generated $A$-module for every $i\ge 0$, and $H^i(X,\mathcal{F})=0$ for $i>\dim X$ (Grothendieck vanishing). (2) Vanishing: there exists $n_0=n_0(\mathcal{F})$ such that $H^i(X,\mathcal{F}(n))=0$ for every $n\ge n_0$ and every $i>0$. Proof outline (3 ingredients). (i) Reduce to $X={\mathbb{P}}_A^N$ via the closed immersion $i:X\hookrightarrow {\mathbb{P}}_A^N$: $i_{\ast }$ is exact with $R^p{i}_{\ast }=0$ for $p\ge 1$ (closed immersion is affine), so the Leray spectral sequence collapses to $H^p(X,\mathcal{F})\cong H^p({\mathbb{P}}_A^N,i_{\ast }\mathcal{F})$. (ii) Compute $H^p({\mathbb{P}}_A^N,\mathcal{O}(d))$ explicitly via Čech on the standard cover [04.03.04]: $H^0=A[x_0,\dots ,x_N{]}_d$ free of rank $\left(\genfrac{}{}{0ex}{}{N+d}{N}\right)$ for $d\ge 0$, $H^N=$ free $A$-module of rank $\left(\genfrac{}{}{0ex}{}{-d-1}{N}\right)$ for $d\le -N-1$, zero in middle and high degrees. (iii) Reduce general coherent $\mathcal{F}$ to line bundles via Serre's theorem A: every coherent $\mathcal{F}$ on ${\mathbb{P}}_A^N$ admits a surjection $\mathcal{O}(-d{)}^{\oplus k}\twoheadrightarrow \mathcal{F}$, kernel $\mathcal{K}$ coherent, long exact sequence in cohomology, descending induction on $i$ with base case $i>N$ (Grothendieck vanishing). Cohomological characterisation of ampleness (Hartshorne III.5.3): $\mathcal{L}$ ample iff for every coherent $\mathcal{F}$ there is $n_0(\mathcal{F})$ with $H^i(X,\mathcal{F}\otimes {\mathcal{L}}^n)=0$ for $i>0$ and $n\ge n_0$. Castelnuovo-Mumford regularity: $\mathcal{F}$ is $m$-regular iff $H^i({\mathbb{P}}_A^N,\mathcal{F}(m-i))=0$ for $i\ge 1$; the regularity index $\mathrm{reg}(\mathcal{F})$ is the smallest such $m$, finite, monotone ($m$-regular implies $m^{\prime }$-regular for $m^{\prime }\ge m$), and provides the explicit threshold $n_0=\mathrm{reg}(\mathcal{F})$ for Serre vanishing; $\mathcal{F}(\mathrm{reg}(\mathcal{F}))$ is globally generated and the multiplication map $H^0(\mathcal{F}(m))\otimes H^0(\mathcal{O}(d))\to H^0(\mathcal{F}(m+d))$ is surjective (Mumford 1966). Generalisations: Grothendieck's relative finiteness/vanishing for proper morphisms (EGA III §2); Serre duality as the dual statement; Kodaira vanishing $H^i(X,{\omega }_X\otimes \mathcal{L})=0$ for ample $\mathcal{L}$ on smooth projective $X/\mathbb{C}$ (one-twist refinement, characteristic zero); Kawamata-Viehweg vanishing for nef-and-big $\mathbb{Q}$-divisors; Serre's affineness criterion ($X$ affine iff $H^i(X,\mathcal{F})=0$ for every coherent $\mathcal{F}$ and $i\ge 1$). Worked example on ${\mathbb{P}}^2$ with ideal sheaf ${\mathcal{I}}_C$ of a smooth plane curve $C$ of degree $e$ uses the SES $0\to \mathcal{O}(d-e)\to \mathcal{O}(d)\to {\mathcal{O}}_C(d)\to 0$ to compute the explicit threshold. Originator: Serre 1955 FAC; modernised in Hartshorne III.5.2-3 (1977) and EGA III (1961-63).

  requires: alg-geom.cech-schemes, alg-geom.cohomology-projective

• alg-geom.cech-cohomology-line-bundlesopen unit 06.04.02 →

  Čech cohomology of holomorphic line bundles

  For a compact Riemann surface $X$, a holomorphic line bundle $L\to X$ with transition functions $g_{ij}$ on a cover $\mathcal{U}=\{U_i\}$, and the sheaf $\mathcal{O}(L)$ of holomorphic sections, the Čech cochain complex $C^p(\mathcal{U},\mathcal{O}(L))={\prod }_{i_0<\cdots <i_p}\Gamma (U_{i_0\cdots i_p},L)$ with alternating-sum-of-restriction differential gives the Čech cohomology ${\check{H}}^p(X,L)={\mathrm{colim}}_{\mathcal{U}}{\check{H}}^p(\mathcal{U},L)$. Real dimension two forces $H^p(X,L)=0$ for $p\ge 2$, leaving only $H^0$ (global sections) and $H^1$. Dolbeault comparison: the fine resolution $0\to \mathcal{O}(L)\to C^{\infty }(L)\stackrel{\bar{\partial }}{\to }{C}^{\infty }(L\otimes \overline{T^{\ast }X})\to 0$ identifies ${\check{H}}^p(X,L)\cong H_{\bar{\partial }}^{0,p}(X,L)$ canonically. Worked computation on ${\mathbb{P}}^1$ with $L=\mathcal{O}(d)$ on the standard two-set cover $\{U_0,U_1\}$ with transition $z\mapsto z^d$ on ${\mathbb{C}}^{\ast }$: $\dim H^0(\mathcal{O}(d))=d+1$ for $d\ge 0$ and zero for $d\le -1$; by Serre duality $\dim H^1(\mathcal{O}(d))=-d-1$ for $d\le -2$ and zero otherwise. Mittag-Leffler problem (Forster §26): the obstruction to globalising prescribed principal parts on $X$ lives in $H^1(X,L)$, recovering Cousin I (additive principal parts) and Cousin II (multiplicative divisors) as cohomological-obstruction problems. Computational tool: short exact sequences $0\to \mathcal{O}(-D)\to \mathcal{O}\to {\mathcal{O}}_D\to 0$ for a divisor $D$ produce long exact sequences relating $H^{\ast }(X,\mathcal{O}(-D))$ to $H^{\ast }(X,\mathcal{O})$ and the local data of $D$, the inductive backbone of every line-bundle cohomology computation on a curve. Originator: Čech 1932 (combinatorial topology) + Behnke-Stein 1949 (Math. Ann. 120) for holomorphic-bundle data on Riemann surfaces; Serre 1955 FAC for the algebraic-side counterpart on schemes.

  requires: complex-analysis.holomorphic-line-bundle, complex-analysis.riemann-roch-compact-rs

• complex-analysis.stein-riemann-surfacesopen unit 06.09.01 →

  Stein Riemann surfaces

  A non-compact Riemann surface $X$ is Stein when it is holomorphically convex (for every compact $K\subset X$ the holomorphic hull $\hat{K}=\{z:|f(z)|\le {\sup }_K|f|,\forall f\in \mathcal{O}(X)\}$ is compact) and holomorphically separable (for every $p\ne q$ in $X$ there is $f\in \mathcal{O}(X)$ with $f(p)\ne f(q)$). Equivalent characterisations on a non-compact Riemann surface: existence of a strictly subharmonic exhaustion $\varphi :X\to \mathbb{R}$; vanishing $H^q(X,\mathcal{F})=0$ for $q\ge 1$ and every coherent analytic sheaf $\mathcal{F}$ (Theorem B); solvability of the $\bar{\partial }$-equation $\bar{\partial }f=g$ with no compatibility condition; solvability of Cousin I and II problems. Behnke-Stein theorem (1949): every non-compact Riemann surface is Stein — the trivial classification on a curve, replaced by an inequivalent rigidity condition only in higher complex dimension. Cohomological consequences: $H^1(X,\mathcal{O})=0$ (Mittag-Leffler always solvable), $H^1(X,{\mathcal{O}}^{\ast })=0$ for the simply-connected part (every line bundle on a non-compact RS is holomorphically trivial), classical Runge approximation extends to all of $X$ when $X\setminus K$ has no relatively compact components. Examples = the entire class of non-compact Riemann surfaces: $\mathbb{C}$, ${\mathbb{C}}^{\ast }$, $\mathbb{C}\setminus \mathbb{N}$, the upper half-plane $\mathbb{H}$, the open unit disc $\mathbb{D}$, every annulus, the universal cover ${\mathbb{H}}^2$ of a genus-$g\ge 2$ surface, every compact Riemann surface minus a finite point set. Higher-dimensional Stein theory: same axioms produce the Cartan-Serre theorems A and B in arbitrary complex dimension, with Hörmander's $L^2$-method and Grauert's Oka principle as analytic engines. Bridge to symplectic topology: every Stein manifold carries a canonical Weinstein structure compatible with the cotangent symplectic form (Cieliebak-Eliashberg 2012). Bridge to algebraic geometry via Serre's GAGA on the projective side and the affine-versus-Stein dictionary on the open side. Originators: Behnke-Stein 1949 (RS case), Stein 1951 (general Stein-manifold definition), Cartan 1951–53 (Theorems A and B).

  requires: complex-analysis.holomorphic-line-bundle, alg-geom.cech-cohomology-line-bundles

• complex-analysis.theorems-a-and-b-stein-rsopen unit 06.09.02 →

  Theorems A and B for Stein Riemann surfaces

  Theorem A: on a Stein Riemann surface, every coherent analytic sheaf is generated by its global sections. Theorem B: on a Stein Riemann surface, $H^q(X,\mathcal{F})=0$ for every coherent analytic $\mathcal{F}$ and every $q\ge 1$. Forster's 1-d proof: exhaustion by relatively compact Runge opens combined with Schwartz finiteness and a limit-passage argument. Forces Cousin I, Cousin II, and Mittag-Leffler on $X$ from a single vanishing identity. Originators: Cartan 1951–53; Cartan-Serre 1953.

  requires: complex-analysis.stein-riemann-surfaces, alg-geom.cech-cohomology-line-bundles

• complex-analysis.behnke-steinopen unit 06.09.03 →

  Behnke-Stein theorem

  Behnke-Stein theorem (1949). Every connected non-compact Riemann surface is Stein. Cornerstone classification of non-compact Riemann surfaces; combined with the uniformisation theorem on the compact side gives a complete picture of complex one-folds. Equivalent reformulations: (1) every non-compact RS admits a strictly subharmonic exhaustion $\varphi :X\to \mathbb{R}$; (2) for every coherent analytic sheaf $\mathcal{F}$ on a non-compact $X$, $H^q(X,\mathcal{F})=0$ for $q\ge 1$ (Theorem B); (3) $H^1(X,\mathcal{O})=0$, i.e. Mittag-Leffler holds globally; (4) $\mathrm{Pic}(X)=0$, every line bundle holomorphically trivial. Proof outline: construct a Runge exhaustion $K_1⋐K_2⋐\cdots$ with $\bigcup K_n=X$; build a strictly subharmonic exhaustion via a limit-passage Runge approximation; solve $\bar{\partial }f=g$ globally via Hörmander $L^2$ on each pseudoconvex piece + Mittag-Leffler / Fréchet limit passage; deduce cohomological vanishing and holomorphic separability + holomorphic convexity from the exhaustion. Why dim 1 is special. Every domain in $\mathbb{C}$ is automatically holomorphically convex (Runge's theorem 1885 generalises). In higher dim, this fails — ${\mathbb{C}}^2\setminus \{0\}$ is not Stein. The notion of "non-compact + open" implies Stein in dim 1; in higher dim, pseudoconvexity is required (Levi's theorem). The dim-1 proof is therefore much simpler than the general Stein-manifold theory. Higher-dim picture. Stein $\iff$ holomorphically convex $\iff$ pseudoconvex (Bergman 1934 / Oka 1942 / Bremermann 1954 / Norguet 1954) — the Levi problem. In dim $n\ge 2$, not every non-compact connected complex $n$-fold is Stein; Behnke-Stein is the dim-1 "easy" case. Examples (all Stein on the dim-1 side): $\mathbb{C}$, ${\mathbb{C}}^{\ast }$, $\mathbb{C}\setminus K$ for $K$ finite, open Riemann surfaces of any genus, disc, half-plane, annulus, compact RS minus a finite point set, universal cover $\widetilde{{\Sigma }_g}$ of higher-genus ${\Sigma }_g$ (the disc). Connection to Riemann's mapping theorem: every simply connected proper open $\Omega \subset \mathbb{C}$ biholomorphic to the unit disc — special case of Behnke-Stein for simply connected non-compact RS, biholomorphic to $\mathbb{C}$ or the disc by uniformisation. Connection to uniformisation: compact RS (3 types — ${\mathbb{P}}^1$, elliptic, hyperbolic); non-compact (universal cover $\mathbb{C}$ or disc); Behnke-Stein gives the structure of the quotient by deck transformations. Originators. Heinrich Behnke + Karl Stein 1949 Entwicklung analytischer Funktionen auf Riemannschen Flächen (Math. Ann. 120). Karl Stein 1951 abstracted to Stein manifolds in higher dim. Levi problem in higher dim solved by Bremermann + Norguet + Oka in 1954.

  requires: complex-analysis.stein-riemann-surfaces, complex-analysis.theorems-a-and-b-stein-rs

• complex-analysis.cousin-i-additiveopen unit 06.09.04 →

  Cousin I (additive)

  Cousin I problem (additive). Given an open cover $\{U_i\}$ of a non-compact Riemann surface $X$ and meromorphic functions $f_i\in \mathcal{M}(U_i)$ with $f_i-f_j\in \mathcal{O}(U_i\cap U_j)$ on every overlap, find a global meromorphic $f\in \mathcal{M}(X)$ with $f-f_i\in \mathcal{O}(U_i)$ for every $i$. The differences $g_{ij}=f_i-f_j$ form a Čech 1-cocycle in the structure sheaf ${\mathcal{O}}_X$, and solvability is exactly the cohomology class $[\{g_{ij}\}]\in H^1(X,{\mathcal{O}}_X)$ vanishing. By Behnke-Stein 1949 + Cartan-Serre Theorem B, $H^1(X,{\mathcal{O}}_X)=0$ on every non-compact Riemann surface, so Cousin I is unconditionally solvable. Sheaf-theoretic formulation. Use the short exact sequence $0\to {\mathcal{O}}_X\to {\mathcal{M}}_X\to {\mathcal{P}}_X\to 0$ where ${\mathcal{M}}_X$ is the meromorphic-function sheaf and ${\mathcal{P}}_X={\mathcal{M}}_X/{\mathcal{O}}_X$ is the principal-parts sheaf with stalks the Laurent-tail rings. The connecting homomorphism $\delta :\Gamma (X,{\mathcal{P}}_X)\to H^1(X,{\mathcal{O}}_X)$ is the Cousin I obstruction map; the datum is solvable when $\delta (\sigma )=0$. Classical Mittag-Leffler. The 1884 planar case on $\mathbb{C}$: for $a_n\to \infty$ and prescribed Laurent tails $P_n(1/(z-a_n))$, the convergence-factor series ${\sum }_n(P_n(1/(z-a_n))-q_n(z))$ with $q_n$ a partial Taylor expansion is the explicit Čech coboundary trivialisation. Closed form for poles at integers with residue 1: $\pi \cot (\pi z)$. Compact case. On a compact RS of genus $g$, $H^1(X,{\mathcal{O}}_X)\cong {\mathbb{C}}^g$ (Hodge / Serre duality on a curve), so a generic Cousin I datum is not solvable; the obstruction dimension is the genus. Higher-dimensional picture. Cousin posed the problem in 1895 on bidiscs in ${\mathbb{C}}^n$ and solved the polydisc case by iterated Cauchy integrals. Oka 1937 (J. Sci. Hiroshima Univ.) settled Cousin I on every domain of holomorphy in ${\mathbb{C}}^n$. Cartan-Serre 1953 (Cartan séminaire, CRAS 237) extended to abstract Stein manifolds via Theorem B. Hörmander 1965 supplied the modern $L^2$-PDE engine. Failure on non-Stein manifolds. ${\mathbb{C}}^2\setminus \{0\}$: Hartogs extension forces every holomorphic function to extend to all of ${\mathbb{C}}^2$, breaking the Stein hypothesis and producing a non-zero $H^1(X,{\mathcal{O}}_X)$. The dichotomy Cousin-I-solvable-vs-not coincides with Stein-vs-not in arbitrary complex dimension. Cousin II (multiplicative). Replace differences with ratios $f_i/f_j\in {\mathcal{O}}^{\ast }(U_i\cap U_j)$; obstruction in $H^1(X,{\mathcal{O}}_X^{\ast })=\mathrm{Pic}(X)$. On a non-compact RS, both $H^1(X,{\mathcal{O}}_X)$ and $H^2(X,\mathbb{Z})$ vanish, so Cousin II is also unconditionally solvable; on Stein manifolds in higher dimension, Cousin II can fail when $H^2(X,\mathbb{Z})\ne 0$. Originators. Pierre Cousin 1895 (Acta Math. 19) — thesis problem in ${\mathbb{C}}^n$; Mittag-Leffler 1884 (Acta Math. 4) — planar special case; Oka 1937 — domains of holomorphy in ${\mathbb{C}}^n$; Cartan-Serre 1953 — Stein manifolds in arbitrary dimension; Behnke-Stein 1949 — RS case via the dimension-one Stein theorem.

  requires: complex-analysis.theorems-a-and-b-stein-rs, complex-analysis.behnke-stein

• complex-analysis.cousin-ii-multiplicativeopen unit 06.09.05 →

  Cousin II (multiplicative)

  Cousin II problem (multiplicative). Given an open cover $\{U_i\}$ of a Riemann surface $X$ and non-zero meromorphic $f_i\in {\mathcal{M}}^{\ast }(U_i)$ with $f_i/f_j\in {\mathcal{O}}^{\ast }(U_i\cap U_j)$ (holomorphic and non-vanishing) on every overlap, find a global $f\in {\mathcal{M}}^{\ast }(X)$ with $f/f_i\in {\mathcal{O}}^{\ast }(U_i)$ for every $i$. The ratios $g_{ij}=f_i/f_j$ form a multiplicative Čech 1-cocycle in ${\mathcal{O}}_X^{\ast }$, and solvability is exactly the cohomology class $[\{g_{ij}\}]\in H^1(X,{\mathcal{O}}_X^{\ast })=\mathrm{Pic}(X)$ vanishing — equivalently, the holomorphic line bundle determined by the cocycle is holomorphically trivial. Exponential sheaf sequence. $0\to 2\pi i\mathbb{Z}\to {\mathcal{O}}_X\stackrel{\exp }{\to }{\mathcal{O}}_X^{\ast }\to 0$ produces the long exact sequence segment $H^1(X,{\mathcal{O}}_X)\to \mathrm{Pic}(X)\stackrel{c_1}{\to }{H}^2(X,\mathbb{Z})\to H^2(X,{\mathcal{O}}_X)$ where $c_1$ is the first Chern class. Non-compact RS case. On a connected non-compact Riemann surface, $H^1(X,{\mathcal{O}}_X)=0$ (Behnke-Stein + Theorem B) and $H^2(X,\mathbb{Z})=0$ (top-degree integer cohomology of a connected non-compact 2-manifold), so $\mathrm{Pic}(X)=0$ and Cousin II is unconditionally solvable. The two-fold vanishing chain — analytic + topological — distinguishes Cousin II from Cousin I, which only needs the analytic part. Sheaf-theoretic formulation. Short exact sequence $0\to {\mathcal{O}}_X^{\ast }\to {\mathcal{M}}_X^{\ast }\to {\mathcal{D}}_X\to 0$ where ${\mathcal{D}}_X={\mathcal{M}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$ is the divisor sheaf; connecting map $\delta :\Gamma (X,{\mathcal{D}}_X)\to \mathrm{Pic}(X)$ is the Cousin II obstruction. Classical Weierstrass product. The 1876 planar case on $\mathbb{C}$: for $a_n\to \infty$ and multiplicities $m_n$, the product ${\prod }_n{E}_{p_n}(z/a_n{)}^{m_n}$ with elementary factors $E_p(z)=(1-z)\exp (z+z^2/2+\cdots +z^p/p)$ is the explicit multiplicative Čech coboundary trivialisation. Closed form for simple zeros at every integer: $\sin (\pi z)$. Compact case. On a compact RS of genus $g$, $\mathrm{Pic}(X)$ sits in $0\to {\mathrm{Pic}}^0(X)\to \mathrm{Pic}(X)\stackrel{\mathrm{deg}}{\to }\mathbb{Z}\to 0$ with ${\mathrm{Pic}}^0(X)$ the Jacobian, a $g$-dim complex torus. Generic Cousin II data are not solvable; obstruction is the line bundle's degree (topological) plus its class in ${\mathrm{Pic}}^0$ (analytic). Higher-dimensional picture. Cousin posed Cousin II in 1895 on polydiscs in ${\mathbb{C}}^n$ and solved them directly (contractible). Oka 1939 (J. Sci. Hiroshima Univ. Ser. A 9) gave the first counterexample on a domain in ${\mathbb{C}}^2$ with non-trivial $H^2(\mathbb{Z})$ and formulated the Oka principle: on a Stein manifold, every continuous Cousin II datum has a holomorphic solution iff the topological obstruction vanishes. Grauert 1958 (Math. Ann. 135) extended to vector bundles of arbitrary rank — the Oka-Grauert principle: holomorphic and topological classifications of complex vector bundles on a Stein manifold agree as bijections. Cousin I vs Cousin II. Cousin I obstruction in $H^1(\mathcal{O})$, purely analytic; Cousin II obstruction in $\mathrm{Pic}$, sandwiched between analytic and topological via the exponential sequence. In dim 1 non-compact, both vanish; in dim ≥ 2 Stein, Cousin I always solvable but Cousin II conditional on topology. Originators. Pierre Cousin 1895 (Acta Math. 19) — thesis problem in ${\mathbb{C}}^n$; Weierstrass 1876 — planar special case (entire functions with prescribed zeros); Oka 1939 — counterexample + Oka principle; Grauert 1958 — Oka-Grauert principle for vector bundles; Cartan-Serre 1953 — cohomological reformulation via the exponential sequence on Stein manifolds; Behnke-Stein 1949 — RS case via dimension-one Stein theorem.

  requires: complex-analysis.cousin-i-additive, complex-analysis.theorems-a-and-b-stein-rs

• alg-geom.sheaf-cohomology-surveyopen unit 06.04.07 →

  Survey of sheaf cohomology on Riemann surfaces

  Synoptic survey gathering the four pictures of sheaf cohomology on a Riemann surface and the comparison theorems that make them equivalent. (1) Čech: ${\check{H}}^{\ast }(X,\mathcal{F})$ via the alternating-sum-of-restriction complex on open covers and the colimit over refinements (computational; dovetails with the line-bundle case in 06.04.02). (2) Dolbeault / harmonic: $H^q(X,\mathcal{F})\cong {\mathcal{H}}^q(X,\mathcal{F})$ via the harmonic kernel of the Hodge-Laplace, with the Bochner-Kodaira-Nakano refinement (analytic; the framework of 06.04.05). (3) Derived-functor: $H^{\ast }(X,\mathcal{F})=R^{\ast }\Gamma (X,-)$ in the abelian category of sheaves of ${\mathcal{O}}_X$-modules (Grothendieck 1957); most flexible, least computational. (4) Singular / topological for constant coefficient sheaves: $H^{\ast }(X,\mathbb{Z})$ via singular cochains on the underlying topological surface, matched with $H_{dR}^{\ast }(X,\mathbb{C})$ via de Rham. Comparison theorems. Cartan-Leray: Čech equals derived-functor on a paracompact Hausdorff space. Dolbeault: fine resolution by smooth $(0,q)$-forms identifies derived-functor cohomology with $\bar{\partial }$-cohomology, with harmonic representatives via the Hodge theorem on compact Kähler. Hodge: $H^k(X,\mathbb{C})={⨁}_{p+q=k}{H}^{p,q}(X)$ with $H^{p,q}$ the harmonic $(p,q)$-forms; on a Riemann surface this is the genus identification $H^1(X,\mathbb{C})=H^{1,0}(X)\oplus H^{0,1}(X)={\mathbb{C}}^g\oplus \overline{{\mathbb{C}}^g}$. Serre duality (Serre 1955 Un théorème de dualité): $H^q(X,\mathcal{F})\cong H^{n-q}(X,{\mathcal{F}}^{\vee }\otimes {\omega }_X{)}^{\ast }$ for coherent $\mathcal{F}$ on a smooth proper $n$-dimensional scheme; the curve case ($n=1$) is 06.04.04. GAGA (Serre 1956 Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier 6): on a smooth projective variety over $\mathbb{C}$, algebraic and analytic coherent cohomology agree, identifying the analytic Čech / Dolbeault picture with the algebraic Hartshorne III picture. Computational toolkit. Long exact sequence in cohomology from a short exact sequence of sheaves; Mayer-Vietoris from a two-set cover; twisting / shift via $0\to \mathcal{O}(L-p)\to \mathcal{O}(L)\to {\mathbb{C}}_p\to 0$; Schwartz finiteness $\dim H^q(X,\mathcal{F})<\infty$ for coherent $\mathcal{F}$ on compact $X$; Riemann-Roch $\chi (L)=\mathrm{deg}L+1-g$ for line bundles on a curve; Serre vanishing $H^1(X,L)=0$ for $\mathrm{deg}L\ge 2g-1$. Standard cohomology table (compact RS of genus $g$): $h^0({\mathcal{O}}_X)=1$, $h^1({\mathcal{O}}_X)=g$; $h^0({\omega }_X)=g$, $h^1({\omega }_X)=1$; for $\mathcal{O}(D)$ with $\mathrm{deg}D>2g-2$, $h^0=\mathrm{deg}D-g+1$ and $h^1=0$ (non-special range); for $\mathrm{deg}D<0$, $h^0=0$ and $h^1=h^0({\omega }_X\otimes \mathcal{O}(-D))$; for $0\le \mathrm{deg}D\le 2g-2$, both sides require Riemann-Roch + Serre + speciality bookkeeping. Higher-dimensional generalisation. Cartan-Serre Theorems A and B on Stein manifolds; Kodaira vanishing on compact Kähler with positive line bundles; Grothendieck cohomology on noetherian schemes via injective resolutions; the four-pictures-agree mantra extends with Hodge replaced by mixed Hodge (Deligne) on singular varieties and de Rham replaced by crystalline / étale on positive-characteristic schemes. Failure modes. Non-Kähler compact complex manifolds (Hopf surface): Hodge symmetry $h^{p,q}=h^{q,p}$ fails, so harmonic-vs-Dolbeault identification is incomplete. Non-paracompact spaces: Čech-vs-derived-functor can differ (counterexamples in Godement). Non-coherent sheaves on a Riemann surface: $H^{\ast }$ may be infinite-dimensional, and harmonic theory does not directly apply. Originators: Čech 1932 (combinatorial cohomology); Cartan 1953-55 (sheaf-theoretic refinement, séminaire); Serre 1955 FAC (algebraic-side coherent cohomology); Hodge 1941 (harmonic representatives); Dolbeault 1953 (Dolbeault cohomology); Hörmander 1965 ($L^2$ existence); Grothendieck 1957 Tôhoku (derived-functor cohomology in an abelian category).

  requires: alg-geom.cech-cohomology-line-bundles, complex-analysis.dbar-hilbert-pde, alg-geom.serre-duality-curves

• complex-analysis.dbar-hilbert-pdeopen unit 06.04.05 →

  Hilbert-space PDE for $\bar{\partial }$

  Setup: compact Riemann surface $X$ with Hermitian metric $\omega$ and Hermitian holomorphic line bundle $(L,h)$. The $L^2$ space of $L$-valued $(0,q)$-forms is the completion of ${\Omega }^{0,q}(X,L)$ under $⟨s,t⟩={\int }_X⟨s,t{⟩}_{h}d{V}_{\omega }$. The Cauchy-Riemann operator $\bar{\partial }:L_{(0,q)}^2(X,L)\to L_{(0,q+1)}^2(X,L)$ is a closed densely-defined unbounded operator with Hilbert adjoint ${\bar{\partial }}^{\ast }$. The Hodge-Laplace ${\Delta }_{\bar{\partial }}=\bar{\partial }{\bar{\partial }}^{\ast }+{\bar{\partial }}^{\ast }\bar{\partial }$ is self-adjoint, elliptic of order two, non-negative. Hodge-Dolbeault theorem on a compact Riemann surface: $L_{(0,q)}^2(X,L)={\mathcal{H}}^q(X,L)\oplus \overline{\mathrm{im}\bar{\partial }}\oplus \overline{\mathrm{im}{\bar{\partial }}^{\ast }}$ orthogonally; ${\mathcal{H}}^q(X,L)=\ker {\Delta }_{\bar{\partial }}$ is finite-dimensional and canonically isomorphic to $H^q(X,L)$ via the Dolbeault fine resolution. Solvability: $\bar{\partial }f=g$ for $g\in L_{(0,1)}^2(X,L)$ closed has a solution iff $g\perp {\mathcal{H}}^1(X,L)$; the canonical solution is $f={\bar{\partial }}^{\ast }Gg$, the Bergman / Green-operator output. Bochner-Kodaira-Nakano formula ${\Delta }_{\bar{\partial }}={\Delta }_{\partial }+[i\Theta (L,h),\Lambda ]$ on $L$-valued forms gives the curvature lower bound $⟨{\Delta }_{\bar{\partial }}\alpha ,\alpha ⟩\ge {\int }_X(i\Theta /\omega )|\alpha {|}^{2}dV$ on $(0,1)$-forms; positive curvature forces ${\mathcal{H}}^1=0$, recovering Kodaira vanishing $H^1(X,L)=0$ for $\mathrm{deg}L>2g-2$. Hörmander's $L^2$ existence theorem: on a complete Kähler manifold with curvature $i\Theta (L,h)\ge c\omega$, every $\bar{\partial }$-closed $g\in L_{(0,q)}^2$ admits a solution with $\Vert f{\Vert }^2\le c^{-1}\Vert g{\Vert }^2$ — foundation of complex analysis on Stein manifolds. Schwartz finiteness theorem $\dim H^q(X,\mathcal{F})<\infty$ for coherent $\mathcal{F}$ on compact $X$ via $\bar{\partial }$-Hodge theory + ellipticity of ${\Delta }_{\bar{\partial }}$. Spectral discreteness on compact $X$: eigenvalues $0={\lambda }_0<{\lambda }_1\le {\lambda }_2\le \cdots \to \infty$ with finite multiplicities; the Bergman kernel of $\ker \bar{\partial }$ encodes geometric data (Tian 1990, Zelditch 1998). Worked examples: ${\mathbb{P}}^1$ with $\mathcal{O}(d)$ (Kodaira positive for $d\ge 1$ gives $H^1=0$); elliptic curve $\mathbb{C}/\Lambda$ trivial bundle (harmonic $(0,1)$-form $d\bar{z}$ generates $H^1(X,\mathcal{O})\cong \mathbb{C}$); higher genus with $\mathrm{deg}L\ge 2g-1$ gives $H^1=0$. Originators: Hörmander 1965 (Acta Math. 113); Andreotti-Vesentini 1965 (compact Kähler $L^2$); Hodge 1941 (harmonic representatives); Kodaira 1953 (vanishing).

  requires: alg-geom.cech-cohomology-line-bundles, functional-analysis.bounded-operators

• analysis.real-number-axiomsopen unit 02.02.01 →

  Real-number axioms (ordered field)

  The standard axiomatic characterisation of $\mathbb{R}$ as a complete ordered field. Thirteen axioms in three families: nine field axioms (closure, associativity, commutativity of $+,\cdot$; identity elements $0\ne 1$; additive inverse; multiplicative inverse for nonzero; distributivity), three order axioms (trichotomy, transitivity, compatibility of $<$ with $+$ and $\cdot$), and one completeness axiom (every non-empty bounded-above subset has a least upper bound). Apostol's Axiom 11 is the standard textbook framing. *Categoricity:** any two complete ordered fields are uniquely order-isomorphic, so $\mathbb{R}$ is uniquely characterised up to isomorphism. Archimedean property as theorem from completeness: for every $x\in \mathbb{R}$ some $n\in \mathbb{N}$ has $n>x$; proof by contradiction via $\sup \mathbb{N}$. Equivalent reformulations of (C): Cauchy completeness plus Archimedean, nested-interval property, monotone convergence theorem, Bolzano-Weierstrass. Independence of axioms demonstrated by witnessing structures: $\mathbb{Z}$ violates (F6), $\mathbb{C}$ admits no compatible order, $\mathbb{Q}$ violates (C). Constructive vs classical completeness (Bishop-Bridges 1967) — Cauchy completeness is intuitionistically valid; LUB completeness needs LEM. Non-Archimedean extensions (Robinson 1966) — hyperreals $^\mathbb{R}$relaxArchimedean,restoringinfinitesimals.\ast \ast Tarski1948\ast \ast —thefirst-ordertheoryofrealclosedfieldsisdecidablebyquantifierelimination,contrastingwiththeundecidabilityof$\mathbb{N}$. Foundational for [02.02.02] sup/inf, [02.03.01] sequence convergence, [02.04.04] FTC, and the entire single-variable analysis strand. Originators: Hilbert 1899 (Grundlagen der Geometrie) for the axiomatisation; Apostol 1967 for the canonical pedagogical presentation; Dedekind 1872 / Cantor 1872 for the construction-from-rationals alternative; Bourbaki 1940- for the modern formalisation; Tarski 1948 for the elementary theory's decision procedure.

  requires: set-theory.function

• analysis.cauchy-bolzano-weierstrassopen unit 02.03.02 →

  Cauchy sequences and Bolzano-Weierstrass

  Two convergence criteria for sequences in $\mathbb{R}$ that together encode the metric form of completeness. Cauchy sequence: $(a_n)$ is Cauchy iff for every $\epsilon >0$ some $N$ has $|a_m-a_n|<\epsilon$ whenever $m,n\ge N$ — terms eventually cluster tightly against one another rather than against a named limit. Bolzano-Weierstrass theorem: every bounded sequence in $\mathbb{R}$ has a convergent subsequence, proved by repeated bisection of a bounding interval and the nested-interval consequence of completeness. Cauchy criterion theorem: a real sequence converges iff it is Cauchy, proved by combining the triangle inequality (convergent implies Cauchy) with boundedness plus Bolzano-Weierstrass plus the Cauchy condition (Cauchy implies convergent). The two theorems package the metric form of the LUB axiom: completeness in the sense of "every Cauchy sequence converges" is equivalent over the Archimedean ordered field axioms to the least-upper-bound completeness of [02.02.01]. Worked example: $a_n=(-1{)}^n/n$ is bounded between $-1$ and $1$; the even-index subsequence converges to $0$; the full sequence converges to $0$; Bolzano-Weierstrass guarantees the subsequence's existence from boundedness alone. Master scope: metric-space completeness (every Cauchy sequence converges); the Cauchy completion functor (every metric space embeds densely into a complete one, generalising the construction of $\mathbb{R}$ from $\mathbb{Q}$); sequential compactness (every sequence has a convergent subsequence, equivalent to compactness on metric spaces but not in general); Heine-Borel in ${\mathbb{R}}^n$ (closed and bounded equals compact equals sequentially compact); failures in infinite dimensions — the closed unit ball of ${\ell }^2$ is bounded but the standard-basis sequence $(e_n)$ has no convergent subsequence, so Bolzano-Weierstrass fails; Banach's contraction principle uses Cauchy completeness as input. Originators: Bolzano 1817 Rein analytischer Beweis for the original intermediate-value-style clustering argument; Weierstrass 1860s Berlin lectures (published in his Werke 1894-1895) for the bisection proof now standard; Cauchy 1821 Cours d'analyse for the Cauchy criterion as a working definition (without a proof of completeness); modern unified treatment in Apostol Calculus Vol. 1 Ch. 10 and Rudin Principles of Mathematical Analysis Ch. 3. Foundational for [02.01.05] metric-space completeness, [02.03.03] series-convergence tests via the Cauchy criterion for partial sums, [02.04.02] the extreme-value theorem and the IVT via compactness, [02.05.01] multi-variable limits using Cauchy and sequential compactness, and [02.11.04] Banach-space completeness.

  requires: analysis.real-number-axioms

• analysis.multi-variable-limit-continuityopen unit 02.05.01 →

  Multi-variable limit and continuity

  Limit and continuity for $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$. Open ball $B_r(a)=\{x:\Vert x-a\Vert <r\}$ from the Euclidean norm. Definition: ${\lim }_{x\to a}f(x)=L$ iff for every $\epsilon >0$ there exists $\delta >0$ with $0<\Vert x-a\Vert <\delta \Rightarrow \Vert f(x)-L\Vert <\epsilon$; $f$ is continuous at $a$ iff ${\lim }_{x\to a}f(x)=f(a)$; continuous iff continuous at every point of its domain. Worked example contrast: $f(x,y)=x^2+y^2$ is continuous at the origin, but $g(x,y)=xy/(x^2+y^2)$ has no limit at the origin — the approach $y=x$ gives $1/2$ while $y=0$ gives $0$, so directional limits disagree. Sequential characterisation theorem: $f$ is continuous at $a$ iff for every sequence $x_k\to a$, the image sequence $f(x_k)\to f(a)$. The forward direction picks $\delta$ from $\epsilon$ and $N$ from $\delta$; the contrapositive of the converse builds a witness sequence with $\Vert x_k-a\Vert <1/k$ but $\Vert f(x_k)-f(a)\Vert \ge {\epsilon }_0$. Bridge to the general-topology characterisation (preimage of open is open, [02.01.02]); to uniform continuity and the multivariable Heine-Cantor theorem on compact domains; to the differential structure built on top of continuity in [02.05.02]; to first-countability — sequential continuity equals topological continuity exactly when the domain has countable neighbourhood bases. The path-independence requirement is the central insight: in ${\mathbb{R}}^n$ for $n\ge 2$, "$x$ approaches $a$" allows infinitely many directions, so a candidate limit must agree along all of them. Master scope: multivariable Heine-Cantor (continuous on compact equals uniformly continuous); composition continuity via preimage axioms; Tietze extension on normal spaces; Banach fixed-point theorem on complete metric spaces; first-countability and the gap to general topological continuity. Originators: Cauchy 1821 (Cours d'analyse) for single-variable $\epsilon$-$\delta$; Riemann's lectures of the 1850s for the multivariable case in pedagogical practice; Heine 1872 and Cantor 1872 (independently) for uniform continuity on compact sets; Fréchet's 1906 thesis introduced the metric / abstract-space framework that subsumes both single- and multi-variable limits. Foundational for [02.05.02] partial derivative and the differential, [02.05.03] chain rule, [02.05.04] inverse and implicit function theorems, and [02.05.05] Taylor and extrema.

  requires: analysis.real-number-axioms

• analysis.multivariable-chain-ruleopen unit 02.05.03 →

  Chain rule for multi-variable functions

  The composition rule for differentiation between Euclidean spaces. Setup: $g:U\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^m$ differentiable at $a$ with derivative $Dg(a):{\mathbb{R}}^n\to {\mathbb{R}}^m$ (the Jacobian) and $f:V\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^p$ differentiable at $b=g(a)$ with derivative $Df(b)$. Statement: $f\circ g$ is differentiable at $a$ and $D(f\circ g)(a)=Df(g(a))\circ Dg(a)$; in Jacobian-matrix form, $J_{f\circ g}(a)=J_f(g(a))\cdot J_g(a)$, an ordinary $p\times m$ by $m\times n$ matrix product yielding a $p\times n$ matrix. Worked example: $g(t)=(\cos t,\sin t)$ and $f(x,y)=x^2+y^2$ give $(f\circ g)(t)=1$; direct derivative is $0$; chain-rule gives $(2\cos t,2\sin t)\cdot (-\sin t,\cos t{)}^T=0$. Proof structure (linear-approximation form): define remainders ${\rho }_g(h)=g(a+h)-g(a)-Dg(a)h$ and ${\rho }_f(k)=f(b+k)-f(b)-Df(b)k$, both $o(\Vert h\Vert )$ and $o(\Vert k\Vert )$ respectively; set $k(h)=Dg(a)h+{\rho }_g(h)$; then $(f\circ g)(a+h)=f(g(a))+Df(b)\cdot Dg(a)h+Df(b){\rho }_g(h)+{\rho }_f(k(h))$; bound $\Vert Df(b){\rho }_g(h)\Vert \le \Vert Df(b){\Vert }_{\mathrm{op}}\Vert {\rho }_g(h)\Vert$ and $\Vert k(h)\Vert \le (\Vert Dg(a){\Vert }_{\mathrm{op}}+1)\Vert h\Vert$ on a small ball; conclude the two remainder terms are each $o(\Vert h\Vert )$. Bridge to the implicit and inverse function theorems via $D{g}^{-1}(b)=Dg(a{)}^{-1}$ (chain rule applied to $g^{-1}\circ g=\mathrm{id}$); to the differential structure of smooth manifolds where the chain rule is functoriality of the tangent functor; to the de Rham complex where pullback of forms satisfies $(\psi \circ \varphi {)}^{\ast }={\varphi }^{\ast }\circ {\psi }^{\ast }$; to the change-of-variables formula in multi-variable integration with $|\det Dg|$ correction factor. Master scope: Banach-space chain rule (same proof, Fréchet derivative in normed spaces); pushforward on tangent vectors with $(\psi \circ \varphi {)}_{\ast }={\psi }_{\ast }\circ {\varphi }_{\ast }$ (functoriality); pullback on differential forms with $(\psi \circ \varphi {)}^{\ast }={\varphi }^{\ast }\circ {\psi }^{\ast }$ (contravariance); Faà di Bruno formula for the $n$-th derivative of a composition with sum over set partitions; Itô formula in stochastic calculus with the half-Hessian quadratic-variation correction term; categorical view of differentiation as a functor from smooth manifolds to vector bundles. Originators: Leibniz 1684 (Nova methodus pro maximis et minimis, Acta Eruditorum) — originator of the single-variable rule $dy/dx\cdot dx/dt$; Cauchy and Lagrange (early nineteenth century) for rigorous proofs; Cartan c. 1900 for the intrinsic differential / coordinate-free form; Apostol Vol. 2 1969 for the canonical pedagogical multi-variable presentation; Faà di Bruno 1855 for the combinatorial higher-order formula; Itô 1944 for the stochastic chain rule. Foundational for [02.05.04] implicit and inverse function theorems, [02.05.05] Taylor and extrema (multi-variable Taylor uses the chain rule iteratively), [02.06.*] ODE systems (variational equations via the chain rule), [03.02.01] smooth manifolds (chain rule is the structural reason coordinate changes patch together), [03.04.04] exterior derivative (pullback functoriality is the chain rule), and the change-of-variables formula in multi-variable integration.

  requires: analysis.multi-variable-limit-continuity

• analysis.implicit-inverse-function-theoremsopen unit 02.05.04 →

  Implicit and inverse function theorems

  The two foundational local-invertibility theorems for multi-variable maps. Inverse function theorem: if $f:U\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ is $C^k$ on open $U$ and $Df(a)$ is invertible at $a\in U$, then there are open neighbourhoods $V\ni a$ and $W\ni f(a)$ with $f{|}_V:V\to W$ a $C^k$ bijection with $C^k$ inverse, and $D(f^{-1})(f(a))=(Df(a){)}^{-1}$. Implicit function theorem: if $g:U\subseteq {\mathbb{R}}^{n+m}\to {\mathbb{R}}^m$ is $C^k$ with $g(a,b)=0$ and $D_{2}g(a,b)$ (the Jacobian in the last $m$ variables) is invertible, then there is a neighbourhood $V\ni a$ and a $C^k$ map $h:V\to {\mathbb{R}}^m$ with $h(a)=b$ and $g(x,h(x))=0$ for $x\in V$, with derivative $Dh(a)=-D_{2}g(a,b{)}^{-1}{D}_{1}g(a,b)$. Worked example (inverse): $f(x,y)=(x^2-y^2,2xy)$ at $(1,0)$: $Df=\mathrm{diag}(2,2)$, determinant $4$, invertible, so local inverse exists (globally $f$ is the complex squaring map, 2-to-1). Worked example (implicit): $g(x,y)=x^2+y^2-1$ at $(1/\sqrt{2},1/\sqrt{2})$: $g_y=2y=\sqrt{2}\ne 0$, so the unit circle is locally a graph $y=\sqrt{1-x^2}$. Proof structure (inverse implies implicit): define $F(x,y)=(x,g(x,y))$; then $DF(a,b)$ is block lower-triangular with diagonal blocks $I_n$ and $D_{2}g(a,b)$, hence invertible iff $D_{2}g(a,b)$ is invertible; apply inverse function theorem to $F$ and read off $h$ from the second component of the inverse. Proof structure (inverse function theorem): define $T_y(x)=x-Df(a{)}^{-1}(f(x)-y)$; this is a contraction on a small closed ball for $y$ near $f(a)$, with contraction constant $1/2$ from continuity of $Df$; Banach contraction principle gives a unique fixed point — the inverse value $f^{-1}(y)$. Bridge to the manifold structure on level sets (a regular level set of a $C^k$ submersion is a $C^k$ submanifold — the regular-value theorem of [03.02.01]); to the constant-rank theorem (a $C^k$ map of constant rank is locally equivalent to a linear projection $(x_1,\dots ,x_n)\mapsto (x_1,\dots ,x_r,0,\dots ,0)$ in suitable coordinates); to the Banach-space inverse function theorem with the same contraction-mapping proof under a topological-isomorphism hypothesis; to the holomorphic inverse function theorem powering the local theory of Riemann surfaces and the definition of étale morphisms in algebraic geometry; to the real-analytic inverse function theorem via Cauchy-Kovalevskaya majorant series; to the Nash-Moser hard implicit function theorem on tame Fréchet spaces with loss-of-derivatives compensation, used in KAM theory and the Riemannian embedding problem. Master scope: Banach-space form with topological-isomorphism hypothesis; failure of inverse on Banach spaces when $Df(a)$ is algebraically bijective but lacks continuous inverse (witness $T(x_n)=(x_n/n)$ on $c_0$); holomorphic form and étale morphisms; real-analytic form; constant rank theorem subsuming both inverse and implicit theorems; Nash-Moser smoothed Newton iteration on tame Fréchet spaces; applications to KAM, isometric Riemannian embedding, and nonlinear PDE. Originators: Newton's Method of Fluxions (1671, published 1736) for single-variable series-form forerunner; Lagrange Théorie des fonctions analytiques 1797 for multi-variable series form; Cauchy 1831 Turin lectures for the analytic majorant-series form; Dini 1877 Lezioni di analisi infinitesimale for the modern ${\mathbb{R}}^n$ form under continuous-partials hypothesis; Goursat 1903 Bulletin de la Société Mathématique de France 31 for the contraction-mapping proof under $C^1$ hypothesis; Banach 1922 Fundamenta Mathematicae 3 for the contraction-mapping principle as an abstract tool; Dieudonné 1960 Foundations of Modern Analysis for the Banach-space textbook presentation; Apostol 1969 Vol. 2 Ch. 13 for the canonical pedagogical presentation; Nash 1956 Annals of Mathematics 63 for the smoothed Newton iteration in the Riemannian embedding problem; Moser 1966 Annali Scuola Norm. Sup. Pisa 20 for the Nash-Moser refinement and KAM applications; Hamilton 1982 Bulletin of the AMS 7 for the tame-Fréchet-space framework. Foundational for [02.05.05] Taylor and extrema (Lagrange multiplier rule is a direct corollary), [03.02.01] smooth manifolds (regular-value theorem makes level sets into submanifolds), [03.02.02] the constant-rank theorem and submanifold structure, [04.] étale morphisms in algebraic geometry, [05.09.] KAM theory via the Nash-Moser hard IFT, [06.*] Riemann surfaces via the holomorphic IFT, and the Picard-Lindelöf existence theorem for ODEs which shares the Banach contraction engine.

  requires: analysis.multivariable-chain-rule

• analysis.multivariable-taylor-extremaopen unit 02.05.05 →

  Taylor's theorem and extrema in several variables

  Multi-variable Taylor expansion plus the second-derivative test for extrema. Taylor's theorem (multi-variable, Lagrange form): for $f:U\to \mathbb{R}$ of class $C^{k+1}$ on a convex open $U\subseteq {\mathbb{R}}^n$, $a\in U$, $a+h\in U$, there exists $c$ on the segment from $a$ to $a+h$ with $f(a+h)={\sum }_{|\alpha |\le k}{D}^{\alpha }f(a)h^{\alpha }/\alpha !+{\sum }_{|\alpha |=k+1}{D}^{\alpha }f(c)h^{\alpha }/\alpha !$, using multi-index notation $\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{N}}^n$, $|\alpha |=\sum {\alpha }_i$, $\alpha !={\alpha }_1!\cdots {\alpha }_n!$, $h^{\alpha }=h_1^{{\alpha }_1}\cdots h_n^{{\alpha }_n}$, $D^{\alpha }={\partial }^{|\alpha |}/(\partial x_1^{{\alpha }_1}\cdots \partial x_n^{{\alpha }_n})$. The second-order specialisation reads $f(a+h)=f(a)+\nabla f(a)\cdot h+\frac{1}{2}{h}^T{H}_f(a)h+o(\Vert h{\Vert }^2)$ where $H_f(a)=({\partial }^{2}f/\partial x_i\partial x_j(a))$ is the Hessian matrix, symmetric by Clairaut-Schwarz for $C^2$ functions. Second-derivative test theorem: for a $C^2$ function $f$ with $\nabla f(a)=0$ and $H=H_f(a)$ the Hessian: (1) $H$ positive definite ⇒ strict local min; (2) $H$ negative definite ⇒ strict local max; (3) $H$ indefinite (both positive and negative eigenvalues) ⇒ saddle; (4) $H$ semidefinite with a zero eigenvalue ⇒ inconclusive. Worked examples: $f(x,y)=x^2+y^2$, critical point $(0,0)$, Hessian $\mathrm{diag}(2,2)$, both eigenvalues positive, local minimum; $g(x,y)=x^2-y^2$, critical point $(0,0)$, Hessian $\mathrm{diag}(2,-2)$, mixed signs, saddle. Proof structure (case (1)): Taylor expansion at $a$ with $\nabla f(a)=0$ gives $f(a+h)-f(a)=\frac{1}{2}{h}^T{H}_f(c(h))h$ for $c(h)$ on the segment; by continuity of $H_f$, $h^T{H}_f(c(h))h\ge \frac{1}{2}{\lambda }_{\min }(H)\Vert h{\Vert }^2$ for small $\Vert h\Vert$, with ${\lambda }_{\min }>0$ the smallest eigenvalue of $H$; the Rayleigh-quotient bound dominates the remainder, giving strict positivity. Counterexamples for case (4): $f(x,y)=x^4+y^4$ has zero Hessian at origin and is a strict local min; $g(x,y)=x^4-y^4$ has zero Hessian at origin and is a saddle; $h(x,y)=x^2-y^3$ has positive semidefinite Hessian (one zero eigenvalue) but the origin is not a local min. Bridge to Morse theory — a Morse function has only non-degenerate critical points, each labelled by an index (number of negative Hessian eigenvalues), and the topology of sublevel sets changes by attaching a cell of dimension equal to the index as $c$ passes a critical value; foundation for the h-cobordism theorem and Floer homology. To Lagrange multipliers via the implicit function theorem [02.05.04]: constrained extrema are critical points of the Lagrangian $L(x,\lambda )=f(x)-\lambda g(x)$, and the bordered-Hessian classifies them. To catastrophe theory (Thom 1972): generic degenerate critical points in $r$-parameter families with $r\le 4$ have one of seven normal forms (fold, cusp, swallowtail, butterfly, three umbilics). To Laplace's method: $\int e^{-Nf(x)}dx\sim e^{-Nf(a^{\ast })}(2\pi /N{)}^{n/2}/\sqrt{\det H_f(a^{\ast })}$ as $N\to \infty$ for a non-degenerate minimum $a^{\ast }$. Master scope: integral form of the Taylor remainder; Morse lemma (local quadratic normal form at non-degenerate critical points via parametric inverse function theorem); Lagrange multiplier rule with bordered-Hessian classification; Thom's elementary catastrophes; Laplace's method and the saddle-point method in complex analysis; multi-jet classification of degenerate critical points. Originators: Taylor 1715 for the single-variable formula; Lagrange 1797 for the remainder form; Cauchy 1821 Cours d'analyse for the first rigorous $\epsilon$-$\delta$ proof; Hesse 1857 for the Hessian determinant in algebraic geometry; Morse 1925 for Morse theory; Milnor 1963 Morse Theory for the modern textbook presentation; Smale 1961 for the h-cobordism theorem application; Thom 1972 for catastrophe theory; Mather 1968–1971 for the rigorous proofs; Apostol 1969 Vol. 2 Ch. 9 for the canonical undergraduate pedagogical presentation. Foundational for Morse theory and the topology of manifolds (pending unit in 03.12.), Lagrange multipliers in constrained optimisation, the saddle-point method in complex analysis (pending unit in 06.), the stationary-phase asymptotic in oscillatory-integral theory, and partition-function expansions in statistical mechanics.

  requires: analysis.implicit-inverse-function-theorems

• classical-mechanics.kinematicsopen unit 09.01.01 →

  Kinematics — position, velocity, acceleration

  (to be filled during production)

  requires: analysis.multivariable-chain-rule, set-theory.function, ode.phase-space-vector-field

• classical-mechanics.newtons-lawsno unit yet

  Newton's laws of motion

  Three laws (inertia, F=ma, action-reaction) stated formally as ODE initial-value problems on phase space. Intermediate tier proves uniqueness via Picard-Lindelof. Master tier develops four substantive strands: (1) conservative forces, the work-energy theorem, and the virial theorem; (2) the central-force two-body problem reduced to effective one-body form, Kepler orbits and the Laplace-Runge-Lenz vector; (3) rigid body dynamics with the inertia tensor, Euler's equations, and intermediate-axis instability; (4) the geometric reformulation on Riemannian manifolds with Galilean invariance and the bridge to Lagrangian/Hamiltonian mechanics. Connects to conservation laws [09.01.03], action principle [09.02.01], Noether's theorem [09.03.01], Hamilton's equations [09.04.02].

  requires: linalg.vector-space, ode.phase-space-vector-field

• classical-mechanics.conservation-lawsopen unit 09.01.03 →

  Conservation laws — energy, momentum, angular momentum

  (to be filled during production)

  requires: classical-mechanics.newtons-laws, linalg.vector-space, ode.phase-space-vector-field, ode.first-integrals

• classical-mechanics.action-principleno unit yet

  The action principle and variational calculus

  (to be filled during production)

  requires: classical-mechanics.newtons-laws, classical-mechanics.conservation-laws, analysis.multivariable-chain-rule, ode.phase-space-vector-field

• classical-mechanics.euler-lagrange-equationsno unit yet

  Euler-Lagrange equations

  (to be filled during production)

  requires: classical-mechanics.newtons-laws, ode.phase-space-vector-field

• classical-mechanics.noethers-theoremno unit yet

  Noether's theorem — symmetries and conservation laws

  (to be filled during production)

  requires: classical-mechanics.euler-lagrange-equations, classical-mechanics.conservation-laws, linalg.vector-space, ode.first-integrals

• modern-geometry.quantum-free-particle-e3open unit 03.14.01 →

  Quantum free particle as a representation of E(3)

  (to be filled during production)

  requires: (none)

• classical-mechanics.legendre-transformopen unit 05.00.03 →

  Legendre transform — from Lagrangian to Hamiltonian

  (to be filled during production)

  requires: classical-mechanics.euler-lagrange-equations, classical-mechanics.conservation-laws, analysis.multivariable-chain-rule

• classical-mechanics.hamiltons-equationsno unit yet

  Hamilton's equations

  (to be filled during production)

  requires: classical-mechanics.legendre-transform, symplectic-geometry.symplectic-manifold, symplectic-geometry.hamiltonian-vector-field, symplectic-geometry.poisson-bracket, symplectic-geometry.cotangent-bundle

• modern-geometry.complex-structures-squeezedopen unit 03.14.02 →

  Complex structures and quantization: squeezed states

  (to be filled during production)

  requires: (none)

• classical-mechanics.canonical-transformationsno unit yet

  Canonical transformations

  (to be filled during production)

  requires: classical-mechanics.hamiltons-equations, linalg.vector-space, analysis.multivariable-chain-rule

• classical-mechanics.hamilton-jacobiopen unit 05.05.04 →

  Hamilton-Jacobi equation

  (to be filled during production)

  requires: classical-mechanics.hamiltons-equations, classical-mechanics.legendre-transform, ode.phase-space-vector-field

• classical-mechanics.action-angle-variablesno unit yet

  Action-angle variables

  (to be filled during production)

  requires: classical-mechanics.hamilton-jacobi, classical-mechanics.hamiltons-equations

• classical-mechanics.kam-theorem-chaosno unit yet

  KAM theorem and chaos

  (to be filled during production)

  requires: classical-mechanics.hamiltons-equations, ode.poincare-bendixson, ode.limit-cycle-lienard

• electromagnetism.coulomb-gaussno unit yet

  Coulomb's law and Gauss's law

  (to be filled during production)

  requires: classical-mechanics.newtons-laws, linalg.vector-space, analysis.multivariable-chain-rule

• electromagnetism.laplace-boundary-value-problemsno unit yet

  Laplace equation and boundary value problems

  (to be filled during production)

  requires: analysis.multivariable-chain-rule, linalg.vector-space

• electromagnetism.biot-savart-ampereno unit yet

  Biot-Savart law and Ampere's law

  (to be filled during production)

  requires: classical-mechanics.newtons-laws, linalg.vector-space

• electromagnetism.faradays-law-inductionno unit yet

  Faraday's law and electromagnetic induction

  (to be filled during production)

  requires: electromagnetism.biot-savart-ampere, classical-mechanics.newtons-laws

• electromagnetism.maxwell-equations-differential-formno unit yet

  Maxwell's equations in differential form

  (to be filled during production)

  requires: diffgeo.exterior-derivative, diffgeo.stokes-theorem

• electromagnetism.em-waves-wave-equationno unit yet

  EM waves and the wave equation

  (to be filled during production)

  requires: electromagnetism.faradays-law-induction, classical-mechanics.kinematics

• electromagnetism.special-relativity-lorentzno unit yet

  Special relativity — postulates and Lorentz transformations

  (to be filled during production)

  requires: classical-mechanics.kinematics, linalg.vector-space, analysis.multivariable-chain-rule

• electromagnetism.relativistic-dynamicsno unit yet

  Relativistic kinematics and dynamics

  (to be filled during production)

  requires: classical-mechanics.newtons-laws, classical-mechanics.conservation-laws

• electromagnetism.covariant-em-faraday-tensorno unit yet

  Covariant electrodynamics — Faraday tensor

  (to be filled during production)

  requires: electromagnetism.relativistic-dynamics, electromagnetism.maxwell-equations-differential-form

• electromagnetism.radiation-larmorno unit yet

  Radiation from accelerating charges — Larmor formula

  (to be filled during production)

  requires: electromagnetism.covariant-em-faraday-tensor, classical-mechanics.newtons-laws

• statistical-mechanics.thermodynamic-lawsno unit yet

  First and second laws of thermodynamics

  (to be filled during production)

  requires: classical-mechanics.noethers-theorem, analysis.multivariable-chain-rule

• statistical-mechanics.thermodynamic-potentialsno unit yet

  Thermodynamic potentials and Legendre transforms

  (to be filled during production)

  requires: classical-mechanics.legendre-transform, analysis.multivariable-chain-rule

• statistical-mechanics.maxwell-boltzmann-kinetic-theoryno unit yet

  Maxwell-Boltzmann distribution from kinetic theory

  (to be filled during production)

  requires: classical-mechanics.conservation-laws, classical-mechanics.kinematics, ode.phase-space-vector-field

• statistical-mechanics.microcanonical-ensembleno unit yet

  Microcanonical ensemble

  (to be filled during production)

  requires: statistical-mechanics.maxwell-boltzmann-kinetic-theory, classical-mechanics.hamiltons-equations, ode.phase-space-vector-field

• statistical-mechanics.canonical-ensembleno unit yet

  Canonical ensemble and partition function

  (to be filled during production)

  requires: stat-mech.boltzmann-distribution, stat-mech.free-energy, analysis.multivariable-taylor-extrema

• statistical-mechanics.bose-einstein-distributionno unit yet

  Bose-Einstein distribution

  (to be filled during production)

  requires: statistical-mechanics.microcanonical-ensemble, statistical-mechanics.maxwell-boltzmann-kinetic-theory

• statistical-mechanics.fermi-dirac-electron-gasno unit yet

  Fermi-Dirac distribution and electron gas

  (to be filled during production)

  requires: statistical-mechanics.bose-einstein-distribution, quantum-mechanics.stern-gerlach-spin-half

• statistical-mechanics.ising-phase-transitionsno unit yet

  Ising model and phase transitions

  (to be filled during production)

  requires: statistical-mechanics.bose-einstein-distribution, statistical-mechanics.fermi-dirac-electron-gas

• statistical-mechanics.renormalization-group-critical-phenomenano unit yet

  Critical phenomena and renormalization group

  (to be filled during production)

  requires: statistical-mechanics.bose-einstein-distribution, statistical-mechanics.fermi-dirac-electron-gas

• quantum-mechanics.wave-particle-dualityopen unit 12.01.01 →

  Wave-particle duality and the double-slit

  (to be filled during production)

  requires: classical-mechanics.kinematics, classical-mechanics.newtons-laws

• quantum-mechanics.stern-gerlach-spin-halfno unit yet

  Stern-Gerlach and spin-1/2

  (to be filled during production)

  requires: linear-algebra.linear-transformation-rank-nullity, linear-algebra.eigenvalue-eigenvector, spin-geometry.clifford.clifford-algebra, rep-theory.lie-algebra-representation, rep-theory.compact-lie-group-representation

• quantum-mechanics.hilbert-space-formalismno unit yet

  Hilbert-space formalism of quantum mechanics

  (to be filled during production)

  requires: linear-algebra.eigenvalue-eigenvector, functional-analysis.banach-space, functional-analysis.normed-vector-space, functional-analysis.inner-product-space, functional-analysis.hilbert-space, functional-analysis.bounded-operators, functional-analysis.unbounded-self-adjoint, quantum-mechanics.stern-gerlach-spin-half

• quantum-mechanics.hilbert-spaces-kets-brasno unit yet

  Hilbert spaces, kets, and bras

  (to be filled during production)

  requires: quantum-mechanics.stern-gerlach-spin-half, linalg.vector-space

• quantum-mechanics.operators-observables-hermiticityno unit yet

  Operators, observables, and Hermiticity

  (to be filled during production)

  requires: quantum-mechanics.wave-particle-duality, quantum-mechanics.stern-gerlach-spin-half, linalg.vector-space

• quantum-mechanics.bosonic-fock-space-second-quantisationopen unit 12.13.01 →

  Bosonic Fock space and second quantisation

  (to be filled during production)

  requires: linear-algebra.eigenvalue-eigenvector, functional-analysis.banach-space, functional-analysis.inner-product-space, functional-analysis.hilbert-space, functional-analysis.bounded-operators, functional-analysis.unbounded-self-adjoint, rep-theory.lie-algebra-representation, quantum-mechanics.stern-gerlach-spin-half

• quantum-mechanics.schrodinger-heisenberg-picturesno unit yet

  Schrödinger and Heisenberg pictures

  (to be filled during production)

  requires: quantum-mechanics.hilbert-space-formalism, quantum-mechanics.wave-particle-duality

• quantum-mechanics.fermionic-fock-space-pauli-anticommutatorsopen unit 12.13.02 →

  Fermionic Fock space, Pauli exclusion, and anticommutators

  (to be filled during production)

  requires: linear-algebra.eigenvalue-eigenvector, functional-analysis.banach-space, functional-analysis.inner-product-space, functional-analysis.hilbert-space, functional-analysis.bounded-operators, spin-geometry.clifford.clifford-algebra, rep-theory.lie-algebra-representation, quantum-mechanics.stern-gerlach-spin-half, quantum-mechanics.schrodinger-heisenberg-pictures

• quantum-mechanics.particle-in-a-boxopen unit 12.04.01 →

  Particle in a box

  (to be filled during production)

  requires: quantum-mechanics.hilbert-space-formalism, quantum-mechanics.operators-observables-hermiticity

• quantum-mechanics.quantum-harmonic-oscillatorno unit yet

  Quantum harmonic oscillator

  (to be filled during production)

  requires: quantum-mechanics.operators-observables-hermiticity, quantum-mechanics.hilbert-space-formalism

• quantum-mechanics.angular-momentum-su2no unit yet

  Hydrogen atom bound states

  (to be filled during production)

  requires: quantum-mechanics.particle-in-a-box, quantum-mechanics.operators-observables-hermiticity

• quantum-mechanics.time-independent-perturbation-theoryno unit yet

  Time-independent perturbation theory

  (to be filled during production)

  requires: quantum-mechanics.particle-in-a-box, quantum-mechanics.operators-observables-hermiticity

• quantum-mechanics.path-integral-formulationno unit yet

  Path integral formulation of quantum mechanics

  (to be filled during production)

  requires: quantum-mechanics.quantum-harmonic-oscillator, classical-mechanics.action-principle

• quantum-mechanics.dirac-equation-relativistic-spinno unit yet

  Dirac equation and relativistic spin

  (to be filled during production)

  requires: quantum-mechanics.schrodinger-heisenberg-pictures, electromagnetism.special-relativity-lorentz

• general-relativity.equivalence-principleno unit yet

  The equivalence principle

  (to be filled during production)

  requires: classical-mechanics.newtons-laws

• general-relativity.tensors-smooth-manifoldsopen unit 13.02.01 →

  Tensors on smooth manifolds

  (to be filled during production)

  requires: algebra.tensor-product, electromagnetism.special-relativity-lorentz

• general-relativity.geodesics-parallel-transportopen unit 13.02.02 →

  Geodesics and parallel transport

  (to be filled during production)

  requires: algebra.tensor-product, classical-mechanics.action-principle

• general-relativity.riemann-curvature-tensorno unit yet

  Riemann curvature tensor

  (to be filled during production)

  requires: general-relativity.tensors-smooth-manifolds

• general-relativity.einstein-field-equationsno unit yet

  Einstein field equations

  (to be filled during production)

  requires: general-relativity.tensors-smooth-manifolds, classical-mechanics.newtons-laws

• general-relativity.schwarzschild-solutionno unit yet

  Schwarzschild solution

  (to be filled during production)

  requires: diffgeo.differential-forms, diffgeo.exterior-derivative, diffgeo.connection.vector-bundle-connection, bundle.connection.curvature

• general-relativity.orbits-schwarzschildno unit yet

  Orbits in Schwarzschild geometry

  (to be filled during production)

  requires: general-relativity.geodesics-parallel-transport, classical-mechanics.newtons-laws

• general-relativity.linearized-gr-gravitational-wavesno unit yet

  Linearized GR and gravitational waves

  (to be filled during production)

  requires: general-relativity.riemann-curvature-tensor, electromagnetism.em-waves-wave-equation

• general-relativity.flrw-cosmology-friedmannopen unit 13.08.01 →

  FLRW cosmology and Friedmann equations

  (to be filled during production)

  requires: general-relativity.tensors-smooth-manifolds, electromagnetism.special-relativity-lorentz

• chemistry.atomic-structure-electron-configurationsopen unit 14.01.01 →

  Atomic structure and electron configurations

  (to be filled during production)

  requires: quantum-mechanics.angular-momentum-su2

• chemistry.lewis-structures-vsepropen unit 14.02.01 →

  Lewis structures and VSEPR

  (to be filled during production)

  requires: (none)

• chemistry.hybridization-valence-bondno unit yet

  Hybridization and valence bond theory

  (to be filled during production)

  requires: chemistry.atomic-structure-electron-configurations

• chemistry.stoichiometry-gas-lawsopen unit 14.03.01 →

  Stoichiometry and gas laws

  Stoichiometry (mole ratios, limiting reagent, theoretical yield) and the ideal gas law PV = nRT with Dalton's law of partial pressures. Intermediate: limiting-reagent theorem with extent-of-reaction proof, van der Waals corrections. Master: kinetic theory derivation of PV = nRT from molecular collisions, Maxwell-Boltzmann distribution and characteristic speeds, equipartition theorem, virial expansion with statistical-mechanical B(T) expression and Mayer cluster-integral proof, cubic equations of state (Redlich-Kwong, Peng-Robinson), Graham's law of effusion, mean free path, Joule-Thomson coefficient and inversion temperature, principle of corresponding states, generalised compressibility charts. Originators: Richter, Dalton, Avogadro, Boyle, Charles, Clausius, Maxwell, Boltzmann, van der Waals, Andrews, Graham, Joule-Thomson, Kamerlingh Onnes. Cycle 4 Track B deepening from 3512w to 8004w.

  requires: (none)

• chemistry.hydrogen-atom-quantum-chemistryno unit yet

  Hydrogen atom quantum chemistry

  (to be filled during production)

  requires: chemistry.atomic-structure-electron-configurations

• physical-chemistry.mo-theory-homonuclear-diatomicsno unit yet

  Molecular orbital theory for homonuclear diatomics

  (to be filled during production)

  requires: linear-algebra.eigenvalue-eigenvector, rep-theory.group-representation, rep-theory.character

• chemistry.chemical-thermodynamics-equilibriumopen unit 14.06.01 →

  Chemical thermodynamics: free energies and equilibrium

  (to be filled during production)

  requires: chemistry.stoichiometry-gas-laws

• chemistry.chemical-kinetics-rate-lawsopen unit 14.08.01 →

  Chemical kinetics: rate laws and the Arrhenius equation

  (to be filled during production)

  requires: chemistry.stoichiometry-gas-laws

• chemistry.acid-base-bronsted-lewis-pkaopen unit 14.10.01 →

  Acid-base chemistry: Bronsted-Lowry, Lewis, and pKa

  (to be filled during production)

  requires: chemistry.atomic-structure-electron-configurations

• chemistry.electrochemistry-nernst-cellsopen unit 14.11.01 →

  Electrochemistry: the Nernst equation and electrochemical cells

  (to be filled during production)

  requires: chemistry.acid-base-bronsted-lewis-pka

• chemistry.structure-stereochemistryopen unit 15.01.01 →

  Structure of organic molecules — stereochemistry

  (to be filled during production)

  requires: (none)

• chemistry.functional-groups-nomenclatureopen unit 15.02.01 →

  Functional groups and nomenclature

  (to be filled during production)

  requires: (none)

• chemistry.acids-bases-organicopen unit 15.03.01 →

  Acids and bases in organic chemistry

  Deepened Cycle-A Track-B unit. Master tier covers four substantive H2 sections: Hammett equation with sigma+/sigma- and Yukawa-Tsuno extensions; acid-base catalysis with Brønsted law and enzymatic serine-protease examples; quantitative pKa prediction via Taft sigma-I/sigma-R separation and DFT thermodynamic cycles; borderline acid-base phenomena including carbon acids (pKa > 20), kinetic vs thermodynamic acidity, KIE on proton transfer, and superacids (Hammett H_0 function). Key originators: Brønsted 1923, Hammett 1937, Taft 1952, Olah 1972. Bridge paragraph connects pKa-comparison to leaving-group ability in [15.04.02] and enolate formation in [15.07.01].

  requires: chemistry.functional-groups-nomenclature

• organic-chemistry.sn1-vs-sn2-mechanismno unit yet

  SN1 vs SN2 substitution mechanisms

  (to be filled during production)

  requires: chemistry.chemical-kinetics-rate-laws

• chemistry.electrophilic-addition-alkenesopen unit 15.05.01 →

  Electrophilic addition to alkenes

  (to be filled during production)

  requires: chemistry.acid-base-bronsted-lewis-pka

• chemistry.aromatic-chemistry-eas-huckelno unit yet

  Aromatic chemistry — EAS, Huckel

  (to be filled during production)

  requires: chemistry.structure-stereochemistry

• chemistry.carbonyl-nucleophilic-additionopen unit 15.07.01 →

  Carbonyl chemistry — nucleophilic addition

  (to be filled during production)

  requires: chemistry.functional-groups-nomenclature

• chemistry.retrosynthetic-analysisopen unit 15.10.01 →

  Retrosynthetic analysis

  (to be filled during production)

  requires: (none)

• chemistry.nmr-spectroscopy-organicopen unit 15.11.01 →

  NMR spectroscopy of organic molecules

  (to be filled during production)

  requires: organic-chemistry.sn1-vs-sn2-mechanism

• chemistry.amino-acids-protein-chemistryno unit yet

  Amino acids and protein chemistry

  (to be filled during production)

  requires: (none)

• chemistry.nucleic-acid-chemistryno unit yet

  Nucleic acid chemistry

  (to be filled during production)

  requires: (none)

• chemistry.enzyme-mechanismno unit yet

  Enzyme mechanism

  (to be filled during production)

  requires: chemistry.chemical-kinetics-rate-laws

• chemistry.periodic-trends-quantifiedno unit yet

  Periodic trends quantified

  (to be filled during production)

  requires: chemistry.hydrogen-atom-quantum-chemistry

• chemistry.symmetry-group-theory-chemistryopen unit 16.02.01 →

  Symmetry and group theory in chemistry

  Point groups, character tables, and irreducible representations applied to molecular symmetry. Covers SALC construction via projection operators, vibrational-mode decomposition, spectroscopic selection rules (Laporte, mutual exclusion), the Jahn-Teller theorem, crystallographic restriction, correlation tables, and factor-group analysis. Proves the decomposition formula for reducible representations, the crystallographic restriction theorem, and the direct-product identity for the totally symmetric irrep. Originator chain: Hessel 1830, Bethe 1929, Wigner 1931, Jahn-Teller 1937, Cotton 1963. Shipped at ~8100w with lean_status: none.

  requires: chemistry.periodic-trends-quantified

• chemistry.crystal-field-theory-fundamentalsopen unit 16.03.01 →

  Crystal field theory fundamentals

  (to be filled during production)

  requires: chemistry.symmetry-group-theory-chemistry

• inorganic-chemistry.crystal-field-octahedralno unit yet

  Crystal field splitting in octahedral complexes

  (to be filled during production)

  requires: rep-theory.character-orthogonality, quantum-mechanics.stern-gerlach-spin-half, chemistry.hydrogen-atom-quantum-chemistry

• chemistry.coordination-chemistryopen unit 16.04.01 →

  Coordination chemistry

  (to be filled during production)

  requires: (none)

• chemistry.organometallic-16-18-electronopen unit 16.05.01 →

  Organometallic chemistry

  (to be filled during production)

  requires: chemistry.coordination-chemistry

• chemistry.bioinorganic-metalloenzymesopen unit 16.06.01 →

  Bioinorganic chemistry

  Metalloenzymes and metalloproteins across the periodic table. Covers haemoglobin cooperative binding (Hill equation, MWC allosteric model, Perutz stereochemical mechanism, Bohr effect, 2,3-BPG), iron-sulphur clusters and Marcus electron-transfer theory, photosynthetic water oxidation (Kok S-state cycle, Mn4CaO5 structure), zinc enzymes (carbonic anhydrase, carboxypeptidase, alcohol dehydrogenase, Lewis acid catalysis), metal ion homeostasis (metallochaperones, IRP/IRE system, Irving-Williams selectivity), vitamin B12 organometallic biochemistry (adenosylcobalamin radical rearrangements, methylcobalamin methyl transfer), and cytochrome P450 hydroxylation (Compound I, oxygen-rebound mechanism). Key theorem: Hill equation proof for cooperative binding. Prerequisites: coordination chemistry [16.04.01]. Deepened in Cycle 4 Track B to 8000+ words with 7 substantive Master H2 sections.

  requires: chemistry.coordination-chemistry, chemistry.enzyme-mechanism

• chemistry.solid-state-chemistryopen unit 16.07.01 →

  Solid-state chemistry

  Covers crystal structures (NaCl, CsCl, ZnS, diamond, CaF2) and the 14 Bravais lattices; radius-ratio rules for structure prediction; Madelung constants and Born-Lande lattice energies; the Born-Haber cycle; Kapustinskii approximation; Bloch theorem and band theory (nearly-free electron model, tight-binding model, reciprocal lattice, Brillouin zone); classification of metals, semiconductors, and insulators by band gap; carrier concentration, effective mass, and density of states; semiconductor doping (n-type, p-type); optical properties and direct vs indirect gaps; point defects (vacancies, interstitials, Frenkel, Schottky) and defect thermodynamics; ionic conductivity and fast ion conductors; non-stoichiometry; colour centres (F-centres); covalent network solids (diamond, graphite, graphene, silicates); metastable polymorphs and phase transitions. Key theorems: packing efficiency of NaCl structure (proved), Bloch theorem, Born-Lande equation. Originator chain: Bragg 1913 (X-ray diffraction), Madelung 1909 (Coulomb sum), Born-Lande 1918 (lattice energy), Bravais 1850 (lattice classification), Bloch 1928 (band theory), Wilson 1931 (semiconductor classification). Status: deepened to 8001w in Cycle 4 Track B with 5 substantive Master H2 sections.

  requires: (none)

• symplectic.souriau-gibbs-stateopen unit 05.03.02 →

  Souriau Gibbs state on a symplectic G-space

  (to be filled during production)

  requires: (none)

• biology.biomolecules-cells-overviewopen unit 17.01.01 →

  Biomolecules in cells

  (to be filled during production)

  requires: chemistry.lewis-structures-vsepr

• biology.cell-membranes-structureopen unit 17.02.01 →

  Cell membranes: structure

  (to be filled during production)

  requires: (none)

• biology.membrane-transportopen unit 17.02.02 →

  Membrane transport

  (to be filled during production)

  requires: chemistry.electrochemistry-nernst-cells

• biology.cellular-organization-organellesno unit yet

  Cellular organization: organelles

  (to be filled during production)

  requires: (none)

• biology.cytoskeleton-contractileno unit yet

  Cytoskeleton and contractile proteins

  (to be filled during production)

  requires: biology.skeletal-muscle-physiology

• biology.cellular-respiration-glycolysis-cacopen unit 17.04.01 →

  Cellular respiration: glycolysis and CAC

  (to be filled during production)

  requires: chemistry.chemical-thermodynamics-equilibrium

• biology.oxidative-phosphorylationno unit yet

  Oxidative phosphorylation and ATP synthesis

  (to be filled during production)

  requires: chemistry.electrochemistry-nernst-cells

• biology.photosynthesisno unit yet

  Photosynthesis: light and dark reactions

  (to be filled during production)

  requires: (none)

• biology.dna-replicationno unit yet

  DNA replication

  (to be filled during production)

  requires: (none)

• biology.transcriptionno unit yet

  Transcription

  (to be filled during production)

  requires: (none)

• biology.translationno unit yet

  Translation

  (to be filled during production)

  requires: (none)

• biology.mutation-repairno unit yet

  Mutation and repair

  (to be filled during production)

  requires: (none)

• biology.cell-signaling-gpcrsno unit yet

  Cell signaling: receptors and GPCRs

  (to be filled during production)

  requires: biology.transcription

• biology.rtk-mapk-cascadeopen unit 17.07.02 →

  Receptor tyrosine kinases and the MAPK signaling cascade

  Receptor tyrosine kinases, ligand-induced dimerization and trans-autophosphorylation, GRB2-SOS adapter recruitment via SH2 phospho-tyrosine recognition, Ras GTP loading, and the three-tier Raf-MEK-ERK MAPK cascade. Quantitative content: Hill-function kinetics at each tier, Huang-Ferrell composition theorem for cascade ultrasensitivity, Goldbeter-Koshland zero-order ultrasensitivity, distributive multisite phosphorylation. Master tier covers four named sub-sections: stacked-Hill mathematics of ultrasensitivity; bistability via positive feedback (saddle-node bifurcations, oscillations from delayed negative feedback per Kholodenko); spatial dynamics including scaffold proteins, KSR, and reaction-diffusion gradients of active ERK; and cross-system connections to oncology via RTK/Ras/Raf mutations and clinical inhibitors (EGFR-TKIs, vemurafenib, trametinib, sotorasib).

  requires: biology.cell-signaling-gpcrs

• biology.cell-cycle-mitosisopen unit 17.08.01 →

  Cell cycle and mitosis

  (to be filled during production)

  requires: biology.cell-signaling-gpcrs

• biology.resting-membrane-potentialopen unit 17.09.01 →

  Resting membrane potential and ion channels

  (to be filled during production)

  requires: chemistry.electrochemistry-nernst-cells

• cellular-neuroscience.action-potential-ionic-basisno unit yet

  The action potential — ionic basis

  (to be filled during production)

  requires: ode.limit-cycle-lienard

• biology.innate-immunityopen unit 17.10.01 →

  Innate immunity at the molecular level

  (to be filled during production)

  requires: biology.biomolecules-cells-overview

• biology.cardiovascular-physiologyopen unit 18.02.01 →

  Cardiovascular physiology — the heart

  (to be filled during production)

  requires: biology.oxidative-phosphorylation

• biology.respiratory-physiologyno unit yet

  Respiratory physiology — gas exchange and transport

  (to be filled during production)

  requires: biology.photosynthesis

• biology.skeletal-muscle-physiologyopen unit 18.04.01 →

  Skeletal muscle physiology

  (to be filled during production)

  requires: biology.cytoskeleton-contractile

• animal-physiology.muscle-contraction-actin-myosinno unit yet

  Muscle contraction — the actin-myosin cycle

  (to be filled during production)

  requires: chemistry.enzyme-mechanism, biology.cytoskeleton-contractile

• biology.nervous-systemno unit yet

  Nervous system — gross anatomy and functional organisation

  (to be filled during production)

  requires: (none)

• biology.endocrine-hormonesopen unit 18.07.01 →

  Endocrine system — hormones and regulation

  (to be filled during production)

  requires: biology.nervous-system

• biology.renal-physiologyno unit yet

  Renal physiology — homeostasis and the nephron

  (to be filled during production)

  requires: chemistry.acid-base-bronsted-lewis-pka

• biology.embryology-morphogenesisopen unit 18.11.01 →

  Embryology and morphogenesis

  Embryogenesis from fertilisation through organogenesis. Beginner covers stages (cleavage, gastrulation, neurulation), morphogen gradients (Bicoid), and Hox genes. Intermediate formalises the French flag model and exponential-gradient threshold theorem with proof. Master tier covers four advanced systems: gene regulatory networks (Davidson sea urchin GRN), Turing reaction-diffusion patterns (Gierer-Meinhardt model, wavelength selection), epithelial-mesenchymal transition (EMT transcription factors, morphomechanics), and stem cell niches/pluripotency (Oct4/Sox2/Nanog, iPSC reprogramming). Key theorems: Bicoid threshold model, Turing instability condition, wavelength selection proposition. Prose-first BIO unit; lean_status: none.

  requires: biology.cell-cycle-mitosis

• biology.mendelian-geneticsno unit yet

  Mendelian genetics — segregation and dominance

  (to be filled during production)

  requires: biology.mutation-repair

• population-genetics.hardy-weinbergno unit yet

  Hardy-Weinberg equilibrium

  (to be filled during production)

  requires: biology.mendelian-genetics

• biology.wright-fisher-diffusionopen unit 19.02.05 →

  Wright-Fisher model and the diffusion approximation

  The discrete Wright-Fisher Markov chain (binomial sampling each generation), its diffusion-limit Fokker-Planck equation ${\partial }_t\varphi =-{\partial }_p[M\varphi ]+\frac{1}{2}{\partial }_p^2[V\varphi ]$ with $V(p)=p(1-p)$ and $M(p)=sp(1-p)$, Kimura's fixation-probability formula $u(p)=(1-e^{-2Nsp})/(1-e^{-2Ns})$ derived from the backward Kolmogorov ODE, the neutral $u(p)=p$ case, weak-selection asymptotics and the nearly-neutral threshold $|2Ns|\sim 1$, and the Kingman coalescent as the time-reversed genealogical dual. Connects Hardy-Weinberg (deterministic null) to natural selection (deterministic drift) to genetic drift (stochastic limit) to phylogenetics (coalescent inference).

  requires: population-genetics.hardy-weinberg, biology.mendelian-genetics

• biology.natural-selectionno unit yet

  Natural selection — directional, stabilizing, and disruptive

  (to be filled during production)

  requires: (none)

• biology.sexual-selectionopen unit 19.03.02 →

  Sexual selection

  (to be filled during production)

  requires: (none)

• biology.kin-selection-hamiltonsno unit yet

  Kin selection and Hamilton's rule

  (to be filled during production)

  requires: (none)

• biology.genetic-driftopen unit 19.04.01 →

  Genetic drift

  (to be filled during production)

  requires: (none)

• biology.quantitative-geneticsno unit yet

  Quantitative genetics — heritability and the breeder's equation

  (to be filled during production)

  requires: biology.genetic-drift

• biology.speciationno unit yet

  Speciation — allopatric and sympatric

  (to be filled during production)

  requires: biology.genetic-drift

• biology.phylogeneticsopen unit 19.07.01 →

  Phylogenetics — tree reconstruction

  (to be filled during production)

  requires: quantum-mechanics.hydrogen-atom-bound-states

• biology.population-ecologyopen unit 19.09.01 →

  Population ecology — Lotka-Volterra

  (to be filled during production)

  requires: analysis.first-order-ode

• biology.community-ecologyno unit yet

  Community ecology — interactions and food webs

  (to be filled during production)

  requires: (none)

• biology.origin-of-lifeno unit yet

  Origin of life — mechanistic scenarios

  (to be filled during production)

  requires: chemistry.chemical-thermodynamics-equilibrium, quantum-mechanics.particle-in-a-box

• ode.lyapunov-stabilityopen unit 02.12.08 →

  Lyapunov stability — direct method

  Lyapunov-function characterisation of asymptotic stability of equilibria of $\dot{x}=f(x)$. $V$ positive definite with $\dot{V}<0$ along trajectories implies stability; LaSalle's invariance principle generalises to limit-set arguments when $\dot{V}\le 0$. Direct method recovers Hamiltonian-energy intuition without integrating the ODE.

  requires: analysis.implicit-inverse-function-theorems, analysis.normed-vector-space

• ode.rectification-theoremopen unit 02.12.05 →

  Rectification theorem (flow box theorem)

  Near any non-equilibrium point, a smooth vector field is conjugate to the constant field $\partial /\partial x_1$ via a local diffeomorphism. Local normal-form result; equilibria are the only obstruction. Foundation for the qualitative theory of ODE.

  requires: ode.phase-space-vector-field, ode.phase-flow

• ode.variation-of-constantsopen unit 02.12.13 →

  Variation of constants — inhomogeneous linear ODE

  Lagrange's method (1774): for the inhomogeneous linear ODE $\dot{x}=A(t)x+f(t)$, write the solution as $x(t)=\Phi (t)c(t)$ where $\Phi$ is the fundamental matrix of the homogeneous problem; solve for $c(t)$ via integration of ${\Phi }^{-1}f$. Recovers Duhamel's principle and the convolution formula in the constant-coefficient case.

  requires: ode.phase-space-vector-field

• ode.bifurcation-theory-pointeropen unit 02.12.17 →

  Bifurcation theory (pointer)

  Pointer unit (per Codex pointer convention): names and sketches the main local-bifurcation taxonomy (saddle-node, transcritical, pitchfork, Hopf) plus the global-bifurcation landscape (homoclinic, heteroclinic), without producing the full proofs. Anchors downstream relativity / fluid-dynamics / MAPK-bistability content (e.g., 17.07.02 MAPK cascade) that cites bifurcation language.

  requires: ode.phase-flow, ode.lyapunov-stability

• char-classes.sw-pontryagin-numbersopen unit 03.06.10 →

  Stiefel-Whitney and Pontryagin numbers

  For a closed manifold M^n, evaluate top-dimensional characteristic monomials (SW mod 2, Pontryagin in dim 4k) on the fundamental class [M]. SW numbers w_I[M] ∈ Z/2 are defined for any closed manifold (no orientation needed, since the mod-2 fundamental class is canonical); Pontryagin numbers p_I[M] ∈ Z require orientation + dim divisible by 4. Thom 1954: SW numbers are bordism invariants of unoriented bordism, and the SW-number map separates bordism classes — Ω^O_n is detected by SW numbers. Thom-Hirzebruch: rational Pontryagin numbers separate oriented-bordism classes after ⊗Q; the signature is a specific rational linear combination via Hirzebruch's L-genus, σ(M) = p_1/3 in dim 4. Worked examples: CP^2 with p_1 = 3, σ = 1; S^2 × S^2 with p_1 = 0, σ = 0; K3 with p_1 = -48, σ = -16; RP^2 with both SW numbers = 1 (not null-bordant). Master tier covers Thom's bordism-invariance proof via the boundary identity TW|_∂W = T(∂W) ⊕ R, and the Milnor exotic-7-sphere divisibility argument. Lean status: partial (theorems stated, sorry proof bodies pending bundled characteristic-class infrastructure in Mathlib).

  requires: char-classes.stiefel-whitney, char-classes.pontryagin-chern.definitions, char-classes.chern-weil, smooth-manifold, singular-homology

• quantum.time-dependent-perturbation-theoryopen unit 12.07.02 →

  Time-dependent perturbation theory and Fermi's golden rule

  First-order amplitude expansion in the interaction picture, the squared-sinc lineshape for sinusoidal perturbations, the long-time limit producing the energy-conserving delta, and the rate formula ${\Gamma }_{i\to f}=(2\pi /\hslash )|V_{fi}{|}^2\rho (E_f)$. The Master tier develops the Dyson series and its connection to the S-matrix in QFT (Dyson 1949 Phys. Rev. 75), the emergence of Fermi's golden rule from Dirac's 1927 paper and Wentzel's 1927 radiationless-transition calculation (with Fermi 1950 christening it the "golden rule"), the catalogue of physical applications (photoionisation cross-sections, beta decay via Fermi's 1934 four-fermion interaction, spontaneous emission and the Einstein A-coefficient, electron-phonon scattering, tunnel-junction conductance), and second-order Dyson with virtual transitions including Bethe's 1947 non-relativistic Lamb-shift calculation and the AC Stark shift. The Lean status is none: Mathlib has the unperturbed unitary exponential but lacks the time-ordered Dyson expansion, the operator-valued Volterra convergence, and the distributional long-time limit of the squared-sinc kernel.

  requires: quantum-mechanics.time-independent-perturbation-theory, quantum-mechanics.schrodinger-heisenberg-pictures, quantum-mechanics.angular-momentum-operators-su2

• char-classes.chern-simons-transgressionopen unit 03.06.07 →

  Chern-Simons forms and transgression

  The Chern-Simons form CS_P(omega) of an Ad-invariant polynomial P of degree k and a connection omega is a (2k-1)-form on the principal bundle satisfying d CS_P(omega) = P(Omega) — the transgression d-cocycle identity. The construction realises Cartan's transgression in the Weil algebra at the bundle level. The Chern-Simons three-form CS(A) = (1/8 pi^2) tr(A ∧ dA + (2/3) A^3) for the second Chern polynomial is gauge-invariant modulo integers, yielding the integer-level Chern-Simons action functional whose path integral defines the Witten-Reshetikhin-Turaev 3-manifold and knot invariants. Higher Chern-Simons forms produce secondary characteristic classes on flat bundles in dimension 2k-1 with values in H^{2k-1}(M; R/Z), the Cheeger-Simons differential characters; the same forms appear in the Atiyah-Patodi-Singer boundary correction (eta-invariant variation) and the Stora-Zumino descent equations for chiral gauge anomalies.

  requires: char-classes.pontryagin-chern.definitions, char-classes.chern-weil.homomorphism, bundle.vector-bundle, topology.de-rham-cohomology

• char-classes.steenrod-squares-wuopen unit 03.06.14 →

  Steenrod squares and the Wu formula

  Steenrod squares Sq^i : H^n(X; F_2) → H^{n+i}(X; F_2) are the stable mod-2 cohomology operations characterised by Steenrod-Epstein axioms (naturality, additivity, Cartan formula, unstability, stability). They generate the mod-2 Steenrod algebra A_2, a graded cocommutative Hopf algebra over F_2 whose dual is the polynomial algebra F_2[xi_1, xi_2, ...] with |xi_i| = 2^i - 1 (Milnor 1958). The Wu formula w(TM) = Sq(v) expresses the total Stiefel-Whitney class of the tangent bundle of a closed smooth manifold as the total Steenrod square of the Wu class v in H^*(M; F_2), where v_i is defined by the Poincaré-duality pairing condition <Sq^i(u), [M]> = <v_i cup u, [M]> for all u in H^{n-i}(M; F_2). Stiefel-Whitney classes admit the Thom-Wu construction Sq^i(u) = pi^*(w_i(E)) cup u on the mod-2 Thom class u. The bordism application (Thom 1954) identifies H^*(MO; F_2) as a free A_2-module, recovering the unoriented bordism ring as a polynomial F_2-algebra.

  requires: char-classes.stiefel-whitney, topology.singular-homology, alg-top.singular-cohomology

• char-classes.combinatorial-pontryagin-exotic-spheresopen unit 03.06.17 →

  Combinatorial Pontryagin classes and exotic 7-spheres

  Milnor 1956 constructed S^3-bundles M_{h,j} → S^4 from the parameter space π_3(SO(4)) ≅ ℤ ⊕ ℤ via clutching functions φ_{h,j}(q)·x = q^h x q^j. For h + j = 1 the total space M_{h,j} is homeomorphic to S^7 (Gysin sequence + Smale h-cobordism 1962). The Hirzebruch signature formula in dimension 8, applied to the disc-bundle filling N_{h,j} glued to a standard D^8, forces p_2[X] = (45 + 4(h - j)^2)/7. Integrality of p_2 requires (h - j)^2 ≡ 1 (mod 7); failure of this congruence certifies that M_{h,j} is not diffeomorphic to the standard S^7. The Milnor lambda invariant λ(M_{h,j}) := (h - j)^2 - 1 (mod 7) ∈ ℤ/7ℤ vanishes precisely on standard structures. Kervaire-Milnor 1963 systematized the construction via the surgery exact sequence 0 → Θ_n^{bP} → Θ_n → π_n^s / J(π_n(SO)) → 0 and computed Θ_7 = ℤ/28ℤ from |Θ_7^{bP}| = 4 · 7 · 1 = 28 (via Bernoulli-denominator formula B_2/8 = 1/240 with numerator 1, factor 2^{2k-2} = 4 for k = 2, factor 2^{2k-1} - 1 = 7) combined with J surjectivity in degree 7 (Adams 1966 Topology 5). Connect-sum endows Θ_7 with abelian group structure; the 28 oriented diffeomorphism classes of smooth homotopy 7-spheres consist of the standard S^7 and 27 exotic ones, all realized as connect sums of Milnor's M_{1,0}. Combinatorial Pontryagin classes for PL manifolds defined by Rokhlin-Schwartz 1957 and Thom 1958 via signatures of transverse PL submanifolds; rational classes p_i^{PL} ∈ H^{4i}(M; ℚ) agree with smooth Pontryagin classes when both defined, and Novikov 1965 proved their topological invariance (rational Pontryagin classes do not depend on smooth/PL structure — but integral Pontryagin classes do, exactly as Milnor's exotic spheres demonstrate). Novikov's argument uses the manifold-with-fibred-singularities construction and the surgery exact sequence; awarded the Fields Medal in 1970.

  requires: char-classes.pontryagin-chern.definitions, char-classes.sw-pontryagin-numbers, char-classes.hirzebruch-signature

• char-classes.unoriented-bordism-thomopen unit 03.06.12 →

  Unoriented bordism and Thom's theorem

  The unoriented bordism group N_n is the abelian group of closed smooth n-manifolds modulo the bordism relation M ~ N <=> exists W^{n+1} with boundary M sqcup N, with addition by disjoint union; every element has order 2 so N_n is an F_2-vector space. The graded ring N_* = oplus N_n carries the Cartesian-product multiplication. Bordism invariance of Stiefel-Whitney numbers (Pontryagin 1947; Thom 1954): if M, N are bordant then <w_omega(TM), [M]_2> = <w_omega(TN), [N]_2> in F_2 for every partition omega of n. Proof: pull w_omega(TW) along boundary inclusions, then d[W]_2 = [M]_2 + [N]_2 in H_n(d W; F_2) and Stokes-style relative pairing collapses to zero. Pontryagin-Thom construction (transversality + tubular neighbourhood) defines the Thom homomorphism Phi : N_n -> pi_n(MO) from the bordism group into the n-th homotopy group of the unoriented Thom spectrum. Thom's main theorem: Phi is a ring isomorphism N_* ≅ pi_*(MO). Computation: Thom showed H^*(MO; F_2) is a free module over the mod-2 Steenrod algebra A on generators indexed by non-dyadic partitions; Adams spectral sequence collapses at E_2, yielding N_* ≅ F_2[x_n : n >= 2, n + 1 != 2^k] with one polynomial generator in every non-Mersenne dimension. Generators chosen as [RP^{2k}] for even dimensions and Dold manifolds P(m,n) = (S^m x CP^n) / Z_2 for odd admissible dimensions (Dold 1956). Low-dimensional values: N_0 = F_2, N_1 = 0, N_2 = F_2 <[RP^2]>, N_3 = 0, N_4 = F_2^2 <[RP^4], [RP^2 x RP^2]>, N_5 = F_2, with N_n = 0 exactly when n in {1, 3, 7, 15, ...}. The theorem is the historical model for every subsequent cobordism computation (Wall 1960 oriented, Quillen 1969 complex MU as universal formal group law) and the first nontrivial collapsing Adams spectral sequence.

  requires: char-classes.stiefel-whitney, bundle.vector-bundle, topology.singular-homology

• char-classes.whitney-duality-immersionopen unit 03.06.16 →

  Whitney duality and immersion obstructions

  Whitney duality is the identity w(TM) ⌣ w(νM) = 1 in H^*(M; F_2) for any closed smooth n-manifold M admitting an immersion f : M ↪ R^N with normal bundle νM = f^* TR^N / TM. The proof is the two-line Whitney-product-plus-naturality argument applied to TM ⊕ νM ≅ ε^N. Equivalently w(νM) = w(TM)^{-1} in the total cohomology ring under formal inversion of 1 + w_1 + w_2 + …. Immersion obstruction: if [w(TM)^{-1}]_i ≠ 0 for some i > k, then M^n does not immerse in R^{n+k} (because the rank-k normal bundle has vanishing SW classes above degree k). Worked example: w(T RP^n) = (1+a)^{n+1} in F_2[a]/(a^{n+1}); inversion gives w(ν RP^n) = (1+a)^{2^s - n - 1} for 2^s ≥ n+1 the smallest such power of two, with top non-vanishing degree 2^s - n - 1. Massey 1960 derives from this the lower bound imm(RP^n) ≥ 2^s - 1; the bound is sharp when n = 2^r. Boy's surface (Boy 1903) realises the codimension-1 immersion RP^2 ↪ R^3 predicted by the duality formula. Whitney 1944 conjectured imm(M^n) ≤ 2n - α(n) for α(n) the binary-digit count of n; R. L. Cohen 1985 proved the conjecture via Brown-Peterson splitting of MO⟨n - α(n)⟩ and Adams-spectral-sequence vanishing of the obstruction class θ(M) ∈ π_n(MO⟨n - α(n)⟩). The interplay between Stiefel-Whitney lower bounds, K-theoretic refinements (Atiyah-Hirzebruch 1959, Adams 1962 vector-fields-on-spheres), and chromatic-stable-homotopy (Brown-Peterson, Cohen) is the structural three-layer ladder for immersion theory.

  requires: char-classes.stiefel-whitney, char-classes.pontryagin-chern.definitions, bundle.vector-bundle

• inorg-chem.crystal-field-stabilization-spectrochemical-seriesopen unit 16.04.02 →

  Crystal field stabilization energy and the spectrochemical series

  Develops the CFSE formula $-0.4n_{t_{2g}}{\Delta }_o+0.6n_{e_g}{\Delta }_o$ for octahedral d-electron configurations and proves the high-spin / low-spin crossover at ${\Delta }_o=P$ for $d^4$ through $d^7$. Includes the empirical spectrochemical series ordering, the sigma-donor / pi-donor / pi-acceptor classification due to Schäffer-Jørgensen, the angular overlap model (AOM) parameterisation ${\Delta }_o=3e_{\sigma }-4e_{\pi }$, the Jahn-Teller theorem with symmetric-square selection rule and application to $d^9$ Cu(II), the Laporte and spin selection rules for d-d transitions, the LMCT/MLCT distinction in charge-transfer spectra, and the Tanabe-Sugano diagram framework for assigning electronic spectra. The Master tier connects to bioinorganic metalloenzymes (heme spin-crossover), to lattice models of transition-metal solids (multi-orbital Hubbard), and to organic stereochemistry (Hückel-J-T in degenerate $\pi$ systems).

  requires: inorganic-chemistry.crystal-field-octahedral` (16.03.02), chemistry.coordination-chemistry` (16.04.01), representation-theory.character` (07.01.03), representation-theory.character-orthogonality` (07.01.04)

• organismal-bio.cardiac-electrophysiologyopen unit 18.02.02 →

  Cardiac action potentials, pacemaker physiology, and the ECG

  Deepens the §18.02 cardiovascular chapter at the cellular electrophysiology level above the organ-and-haemodynamics peer 18.02.01. Four Master sub-sections: (i) cardiac AP phases 0-4 and ion-channel basis (plateau as the defining feature; L-type Ca, delayed-rectifier K, contrast with neural HH); (ii) pacemaker cells with funny current $I_f$ and the coupled membrane-clock/Ca-clock model (DiFrancesco-Noble + Maltsev-Lakatta); (iii) conduction, gap junctions, and reentry as a limit-cycle phenomenon on the cardiac excitable medium (wavelength criterion, Winfree-Keener rotors, ventricular fibrillation as scroll-wave breakup); (iv) ECG genesis from cellular dipoles to body-surface vectorcardiography (Einthoven's triangle, twelve-lead axis determination, link to specific pathologies). The unit builds above the neural HH parent 17.09.02 without re-deriving it; cardiac contributions are the plateau, the funny current, gap-junction coupling, and the body-surface ECG. Lean status: none — Mathlib lacks conductance-based cardiac models, the bidomain PDE, and reentry-rotor stability theory.

  requires: ode.phase-space-vector-field, ode.limit-cycle-lienard, ode.bifurcation-theory-pointer

• topology.postnikov-tower-kan-complexopen unit 03.12.40 →

  Postnikov tower of a Kan complex

  The simplicial-side Postnikov decomposition of a Kan complex. The Postnikov section $P_{n}X={\mathrm{cosk}}_{n+1}({\mathrm{Ex}}^{\infty }X)$ has ${\pi }_i(P_{n}X)={\pi }_i(X)$ for $i\le n$ and ${\pi }_i(P_{n}X)=0$ for $i>n$. The bonding map $P_{n}X\to P_{n-1}X$ is a Kan fibration with fibre the Eilenberg-MacLane Kan complex $K({\pi }_{n}X,n)$, classified by a cohomology class $k_n\in H^{n+1}(P_{n-1}X;{\pi }_{n}X)$ called the $n$-th k-invariant. The tower is dual to the Whitehead tower (Postnikov truncates from above; Whitehead from below) and converges to $X$ via $X\simeq {\underleftarrow{\lim }}_n{P}_{n}X$ for $X$ of finite type. Hirschhorn 2003 identifies the truncation functors ${\tau }_{\le n}$ as left Bousfield localisations of the Kan-Quillen model structure at the maps inverting ${\pi }_{n+1},{\pi }_{n+2},\dots$. Worked example: the Postnikov tower of $S^2$ has $P_2{S}^2=K(\mathbb{Z},2)={\mathbb{CP}}^{\infty }$ and first k-invariant $k_3={\iota }_2⌣{\iota }_2\in H^4(K(\mathbb{Z},2);\mathbb{Z})=\mathbb{Z}$, the cup-square of the fundamental class, encoding the Hopf-map obstruction ${\pi }_3{S}^2=\mathbb{Z}$. Postnikov 1951's classification theorem: weak homotopy type equals homotopy groups plus k-invariants up to equivalence. The structural foundation of simplicial-side obstruction theory and the canonical example of left Bousfield localisation in $\mathbf{sSet}$.

  requires: topology.eilenberg-maclane` (03.12.05), homotopy.whitehead-tower-rational-hurewicz` (03.12.07), topology.singular-homology` (03.12.11), topology.simplicial-set` (03.12.25)

• topology.kan-quillen-model-structureopen unit 03.12.33 →

  Kan-Quillen model structure on sSet

  The canonical model structure on simplicial sets, established by Quillen in 1967. Three classes: cofibrations = monomorphisms (level-wise injections); fibrations = Kan fibrations (right lifting against horn inclusions ${\Lambda }_k^n\hookrightarrow {\Delta }^n$ for $0\le k\le n$, $n\ge 1$); weak equivalences = maps whose geometric realisation is a weak homotopy equivalence (equivalently, isos on simplicial homotopy groups after fibrant replacement). The five model-category axioms M1-M5 are verified using the small-object argument applied to the generating sets $I=\{\partial {\Delta }^n\hookrightarrow {\Delta }^n{\}}_{n\ge 0}$ and $J=\{{\Lambda }_k^n\hookrightarrow {\Delta }^n{\}}_{0\le k\le n,n\ge 1}$. Key structural facts: every simplicial set is cofibrant (the initial object is empty, and the empty map is vacuously injective); a simplicial set is fibrant iff it is a Kan complex; the singular complex $\mathrm{Sing}(X)$ is always a Kan complex; simplicial groups are automatically Kan complexes (Moore 1955). Quillen 1967 Theorem II.3.1 establishes that the realisation-singular adjunction $|-|⊣\mathrm{Sing}$ is a Quillen equivalence between the Kan-Quillen structure and the Serre model structure on $\mathbf{Top}$, so $\mathrm{Ho}(\mathbf{sSet})\simeq \mathrm{Ho}(\mathbf{Top})$ identifies combinatorial and topological homotopy theory. The Kan-Quillen structure has a sibling Joyal model structure on the same underlying category $\mathbf{sSet}$ using only inner-horn fillers, modelling $(\infty ,1)$-categories rather than $\infty$-groupoids; Cisinski 2006 identifies Kan-Quillen as the universal $\infty$-groupoid presentation among Cisinski model structures on $\mathbf{sSet}$. The foundational example of the Quillen abstract framework [03.12.31], and the canonical worked example for the Quillen-equivalence calculus of [03.12.32].

  requires: topology.simplicial-set` (03.12.25), homotopy.quillen-model-category` (03.12.31)

• topology.bousfield-kan-spectral-sequenceopen unit 03.12.38 →

  Bousfield-Kan spectral sequence

  The Bousfield-Kan spectral sequence of a cosimplicial space $X^{\bullet }$ has $E_2^{s,t}={\pi }^s{\pi }_t(X^{\bullet })$ converging conditionally to ${\pi }_{t-s}(\mathrm{Tot}{X}^{\bullet })$, the homotopy of the totalisation. Construction: for a Reedy-fibrant cosimplicial space, the tower of partial totalisations ${\mathrm{Tot}}_n(X^{\bullet })\to {\mathrm{Tot}}_{n-1}(X^{\bullet })$ is a tower of Kan fibrations with fibre ${\Omega }^n{N}^n{X}^{\bullet }$ (the loop on the $n$-th normalised cosimplicial level), and the spectral sequence is the exact-couple spectral sequence of this tower. The $E_2$-page identification uses the Dold-Kan acyclicity theorem to exchange the normalised cosimplicial cochain complex for the alternating-sum cochain complex of ${\pi }_t(X^{\bullet })$. Conditional convergence to ${\pi }_{\ast }(\mathrm{Tot})$ involves a ${\lim }^1$ obstruction packaged via the Milnor short exact sequence $0\to {\lim }^1{\pi }_{k+1}{\mathrm{Tot}}_n\to {\pi }_k\mathrm{Tot}\to \lim {\pi }_k{\mathrm{Tot}}_n\to 0$; strong convergence holds under Mittag-Leffler hypotheses on the tower of homotopy groups. The Adams-style BK spectral sequence at a ring spectrum $R$ applies the construction to the cobar cosimplicial spectrum $C^n(R;X)=R^{\wedge (n+1)}\wedge X$, with $E_2$-page ${\mathrm{Ext}}_{R_{\ast }R}^{s,t}(R_{\ast },R_{\ast }X)$ converging to ${\pi }_{\ast }(X_R^{\wedge })$. The classical Adams spectral sequence at $R=H{\mathbb{F}}_p$, the Adams-Novikov spectral sequence at $R=BP$, and the chromatic $K(n)$-local Adams BK spectral sequences are the canonical specialisations underlying the chromatic stratification of stable homotopy theory (Devinatz-Hopkins-Smith 1988, Hopkins-Smith 1998, Ravenel 1986). Goerss-Hopkins obstruction theory uses BK spectral sequences on moduli of $E_{\infty }$ ring structures to construct $\mathrm{tmf}$. Lean status: partial, module Codex.Modern.Homotopy.BousfieldKanSpectralSequence declares the API (cosimplicial spaces, totalisation, $E_2$-identification, conditional convergence) with proof bodies pending Mathlib cosimplicial-object and spectral-sequence packages.

  requires: topology.cw-complex` (03.12.10), topology.simplicial-set` (03.12.25), homotopy.quillen-model-category` (03.12.31), spectral-sequences.spectral-sequence` (03.13.01)

• topology.homotopy-colimit-bousfield-kanopen unit 03.12.37 →

  Homotopy colimit via the Bousfield-Kan construction

  The Bousfield-Kan construction: the homotopy colimit of a diagram $X:I\to \mathbf{sSet}$ is the geometric realisation $|B(\ast ,I,X)|$ of the bar construction, whose $n$-simplices are pairs of a length-$n$ chain of composable arrows in $I$ and an $n$-simplex of the source space. Face maps act by composing arrows / shifting via the leftmost arrow / dropping the rightmost object; degeneracies insert identities. The formula gives the left-derived functor $\mathbf{L}{\mathrm{colim}}_I$ of the ordinary colimit, computed in the projective model structure on ${\mathbf{sSet}}^I$. Concrete instances: homotopy pushout = double mapping cylinder $A{\bigsqcup }_C(C\times {\Delta }^1){\bigsqcup }_{C}B$; sequential hocolim = mapping telescope $\mathrm{Tel}(X)={⨆}_n{X}_n\times {\Delta }^1/\sim$; classifying space $BG=|N(BG)|={\mathrm{hocolim}}_{BG}\ast$ (Segal 1974). Homotopy invariance via Bousfield-Friedlander diagonal lemma. The two-sided bar construction $B(W,I,X)$ with contravariant weight $W$ generalises to weighted colimits (Riehl 2014). Projective and Reedy model structures on ${\mathbf{sSet}}^I$ (Hirschhorn 2003 §18) give equivalent presentations. The Bousfield-Kan formula is the universal computational tool underlying chromatic homotopy theory, motivic hocolims, topological cyclic homology, and equivariant stable homotopy, and is the classical-side counterpart to the $(\infty ,1)$-categorical colimit in $\mathcal{S}$ (Joyal 2002, Lurie 2009). Lean status: partial, module Codex.Modern.Homotopy.HomotopyColimitBK declares the API (bar construction, hocolim, pushout = double mapping cylinder, sequential = telescope, classifying space) with proof bodies pending Mathlib model-category and diagram-category packages.

  requires: topology.simplicial-set` (03.12.25), homotopy.quillen-model-category` (03.12.31)

• topology.arithmetic-square-fractureopen unit 03.12.45 →

  Arithmetic square and integral fracture theorems

  The Sullivan arithmetic square at a nilpotent finite-type space $X$ is the commutative square with $X$ at the top-left, the product of $p$-completions ${\prod }_p{L}_{p}X$ at the top-right, the rationalisation $L_{\mathbb{Q}}X$ at the bottom-left, and the rational gluing corner $L_{\mathbb{Q}}({\prod }_p{L}_{p}X)$ at the bottom-right. The integral fracture theorem (Sullivan 1970; Bousfield-Kan 1972 §VI) states that the canonical comparison map from $X$ to the homotopy pullback of the other three corners is a weak equivalence in $\mathrm{Ho}({\mathbf{sSet}}_{\ast })$, under the nilpotence + finite-type + ${\lim }^1$-vanishing hypotheses. The proof proceeds via the Milnor short exact sequence for the homotopy pullback, the Mittag-Leffler vanishing of ${\lim }^1$ at finite type, and the local-global short exact sequence $0\to \mathbb{Z}\to \mathbb{Q}\oplus \hat{\mathbb{Z}}\to {\mathbb{A}}_f\to 0$ tensored with the relevant homotopy group. Worked sphere example: ${\pi }_3(S^2)=\mathbb{Z}$ recovered as the pullback $\mathbb{Q}{\times }_{{\mathbb{A}}_f}\hat{\mathbb{Z}}$. Chromatic extension: the Hovey-Strickland 1999 chromatic-fracture squares at each height $n$ recover $L_{E(n)}X$ as the homotopy pullback of $L_{E(n-1)}X$, $L_{K(n)}X$, and $L_{E(n-1)}{L}_{K(n)}X$; iterated downward, the chromatic tower assembles a finite $p$-local spectrum from its monochromatic strata via the Hopkins-Ravenel 1992 chromatic convergence theorem. The framework is dual to the number-theoretic adelic local-global principle (Sullivan's title invokes "Galois Symmetry" to make the duality explicit) and the Devinatz-Hopkins 1995 Topology 34 identification $L_{K(n)}{S}^0\simeq E_n^{h{\mathbb{G}}_n}$ closes the loop. Lean status: partial, module Codex.Modern.Homotopy.ArithmeticSquare states the fracture theorem and sphere corollary; proofs sorry pending Mathlib BousfieldLocalization + HomotopyPullback + nilpotent-class infrastructure.

  requires: homotopy.quillen-model-category` (03.12.31); pending peers `topology.homotopy-colimit-bousfield-kan` (03.12.37) and `topology.bousfield-kan-spectral-sequence` (03.12.38)

• topology.simplicial-model-categoryopen unit 03.12.35 →

  Simplicial model category and the function complex

  A simplicial model category is a model category $\mathcal{C}$ enriched over the cartesian-closed category $\mathbf{sSet}$ of simplicial sets, satisfying three compatibility axioms beyond M1-M5: SM0 (underlying category coincides with the enrichment's $0$-simplices), SM6 (tensor $X\otimes K$ and cotensor $X^K$ with simplicial sets giving the bifunctorial adjunction ${\mathrm{Map}}_{\mathcal{C}}(X\otimes K,Y)\cong {\mathrm{Map}}_{\mathbf{sSet}}(K,{\mathrm{Map}}_{\mathcal{C}}(X,Y))\cong {\mathrm{Map}}_{\mathcal{C}}(X,Y^K)$), and SM7 (pushout-product axiom: the pushout-product $i\square j:(A\otimes L){\bigsqcup }_{A\otimes K}(B\otimes K)\to B\otimes L$ of a cofibration $i:A\to B$ in $\mathcal{C}$ with a monomorphism $j:K\to L$ in $\mathbf{sSet}$ is a cofibration in $\mathcal{C}$, acyclic if either factor is acyclic). Headline theorem (Quillen 1967 §II.2 Proposition 3): the function complex ${\mathrm{Map}}_{\mathcal{C}}(X,Y)$ is a Kan complex whenever $X$ is cofibrant and $Y$ is fibrant; the proof is a horn-filling argument using SM7 with $i_X:\varnothing \to X$ and $j:{\Lambda }_k^n\hookrightarrow {\Delta }^n$ to produce an acyclic cofibration that lifts against $Y\to \ast$ by M3. The derived function complex $\mathrm{RMap}(X,Y):={\mathrm{Map}}_{\mathcal{C}}(QX,RY)$ has homotopy groups ${\pi }_n\mathrm{RMap}(X,Y)\cong [S^n\otimes X,Y{]}_{\mathrm{Ho}(\mathcal{C})}$ and ${\pi }_0\mathrm{RMap}(X,Y)={\mathrm{Hom}}_{\mathrm{Ho}(\mathcal{C})}(X,Y)$, identifying the simplicial-enrichment side with derived $\mathrm{Hom}$. Canonical examples: $\mathbf{sSet}$ itself with Kan-Quillen structure and internal-Hom enrichment; simplicial groups (Quillen 1967 §II.4); simplicial commutative dg-algebras over $\mathbb{Q}$ via Bousfield-Gugenheim 1976 (foundation of rational homotopy theory); non-negatively graded chain complexes via Dold-Kan; symmetric spectra ${\mathrm{Sp}}^{\Sigma }$ (Hovey-Shipley-Smith 2000). Generalised by Hovey 1999 §4.2 to monoidal model categories. Lurie 2009 §A.3 proves that every simplicial model category (restricted to cofibrant-fibrant subcategory) presents an $(\infty ,1)$-category via the homotopy-coherent nerve; the Bergner-Joyal Quillen equivalence between $\mathbf{sCat}$ and ${\mathbf{sSet}}_{\mathrm{Joyal}}$ confirms equivalence with the quasi-categorical presentation. Lean status: partial, module Codex.Modern.Homotopy.SimplicialModelCategory declares the SM7 axiom and the Kan-complex theorem with sorry proof bodies pending the upstream SimplicialModelCategory API in Mathlib.

  requires: topology.simplicial-set` (03.12.25), homotopy.quillen-model-category` (03.12.31)

• alg-geom.ddbar-lemmaopen unit 04.09.05 →

  The ddbar-lemma

  On a compact Kähler manifold, a d-closed (p, q)-form that is d-exact, ∂-exact, or ∂̄-exact is in fact ∂∂̄-exact: equivalently $\ker (d)\cap \mathrm{im}(d)=\ker (d)\cap \mathrm{im}(\partial \bar{\partial })$ on the level of forms. Proof uses the Kähler identities $[\Lambda ,\partial ]=i{\bar{\partial }}^{\ast },[\Lambda ,\bar{\partial }]=-i{\partial }^{\ast }$, which force ${\Delta }_d=2{\Delta }_{\partial }=2{\Delta }_{\bar{\partial }}$ and hence simultaneous harmonicity across the three derivative operators; combined with the Hodge decomposition of ${\Omega }^{p,q}$ for the Dolbeault Laplacian, the harmonic-free residual lies in $\mathrm{im}(\partial \bar{\partial })$. Headline consequence (Deligne-Griffiths-Morgan-Sullivan 1975 Invent. Math. 29): formality of the de Rham algebra — the real DGA $({\Omega }^{\ast }(X;\mathbb{R}),d)$ is connected by a zig-zag of quasi-isomorphisms to its cohomology ring $(H^{\ast }(X;\mathbb{R}),0)$ with zero differential, via the intermediate dga $\ker (d^c)\cap {\Omega }^{\ast }$; all higher Massey products in rational cohomology vanish; the Sullivan minimal model is freely generated by the cohomology ring. Second consequence: Frölicher spectral sequence degeneration at $E_1$ — the differential $d_1:E_1^{p,q}=H_{\bar{\partial }}^{p,q}(X)\to E_1^{p+1,q}$ vanishes on ∂̄-harmonic representatives because Kähler symmetry makes them simultaneously ∂-harmonic; iteration gives $E_1=E_{\infty }$, equivalent to the Hodge decomposition. Third consequence: Bott-Chern equals Dolbeault equals de Rham (in matched bidegrees) on a compact Kähler manifold — the four cohomology theories coincide, expressed quantitatively in Angella-Tomassini's inequality $\dim H_{BC}^{p,q}+\dim H_A^{p,q}\ge 2\dim H_{\bar{\partial }}^{p,q}$ with equality iff the lemma holds. Fourth: the Hodge filtration is independent of the choice of Kähler metric (only the harmonic representatives change). Failure on non-Kähler manifolds: Hopf surface $S^1\times S^3$ — $b_2=0$ rules out any Kähler form; $h^{1,0}=0\ne 1=h^{0,1}$ breaks Hodge symmetry. Iwasawa nilmanifold $\Gamma \backslash G$ for $G$ the complex Heisenberg group — $h^{1,0}+h^{0,1}=5\ne 4=b_1$ (Frölicher non-degeneration), non-vanishing triple Massey product in rational cohomology (non-formal). The lemma is now a defining property of $\partial \bar{\partial }$-manifolds, a strictly broader class than Kähler that nonetheless inherits Kähler-style cohomological rigidity; Fujiki class-$\mathcal{C}$ manifolds (bimeromorphic to Kähler) and certain Moishezon manifolds are ddbar but not Kähler. Modern refinements: higher-order ddbar-Massey products (Tirabassi-Tomassini 2018; Angella-Sferruzza 2019). Lean status: partial, module Codex.AlgGeom.Hodge.DdbarLemma declares the ddbar-lemma plus the Frölicher E_1-degeneration corollary plus the Bott-Chern-equals-Dolbeault bijection with sorry proof bodies pending the compact-Kähler-manifold + bidegree-form + Dolbeault-Laplacian API in Mathlib.

  requires: alg-geom.hodge-decomposition, alg-geom.kodaira-vanishing, topology.differential-forms, topology.de-rham-cohomology

• hodge.hard-lefschetzopen unit 04.09.07 →

  Hard Lefschetz theorem

  For a compact Kähler manifold $X$ of complex dimension $n$ with Kähler class $[\omega ]\in H^{1,1}(X,\mathbb{R})$, the iterated cup-product map $L^k=[\omega {]}^k\cup -:H^{n-k}(X,\mathbb{R})\to H^{n+k}(X,\mathbb{R})$ is an isomorphism for every $0\le k\le n$; the map respects the Hodge decomposition by sending $H^{p,q}$ isomorphically to $H^{p+k,q+k}$ for $p+q=n-k$. The proof has three layers: (1) the Kähler identities ${\Delta }_d=2{\Delta }_{\bar{\partial }}$ promote the Lefschetz operator $L$, its $L^2$-adjoint $\Lambda$, and the grading operator $H$ (acting on $H^k$ as multiplication by $k-n$) to an ${\mathfrak{sl}}_2(\mathbb{R})$-triple satisfying $[L,\Lambda ]=H$, $[H,L]=2L$, $[H,\Lambda ]=-2\Lambda$; (2) the de Rham complex of $X$ thereby becomes a finite-dimensional ${\mathfrak{sl}}_2$-representation, which classifies as a direct sum of irreducibles $V_m$ of dimension $m+1$ indexed by highest weight $m$; (3) the iso-statement of $L^k$ is immediate from ${\mathfrak{sl}}_2$-representation theory since $L^k$ sends the $-k$-eigenspace to the $+k$-eigenspace inside each $V_m$ with $m\ge k$ isomorphically. The theorem implies the Lefschetz decomposition $H^k={⨁}_{j\ge 0,k-2j\ge 0}{L}^j{P}^{k-2j}$, where primitive cohomology $P^{\ell }=\ker (L^{n-\ell +1})$ is the highest-weight component of each ${\mathfrak{sl}}_2$-irreducible. Together with the Hodge-Riemann bilinear relations [04.09.08] on primitive cohomology, the Hard Lefschetz iso defines a polarised Hodge structure on each cohomology group. The surface case is the Hodge index theorem [04.05.09]. Deligne 1968 gave the first algebraic-geometric proof for smooth projective varieties over arbitrary algebraically closed fields, via the Weil conjectures and the Frobenius weight filtration on $\ell$-adic cohomology; Beilinson-Bernstein-Deligne 1982 generalised this to the decomposition theorem for direct images of perverse sheaves under proper morphisms between complex algebraic varieties, which also produced Hard Lefschetz on intersection cohomology of singular projective varieties. Stanley 1980 deduced the unimodality of $h$-vectors of simplicial polytopes from Hard Lefschetz on smooth projective toric varieties, and Adiprasito-Huh-Katz 2018 extended Hodge-Lefschetz positivity to matroid combinatorics. Donaldson 1999 showed that every symplectic four-manifold admits a Lefschetz pencil, extending the topological side to symplectic geometry. Lean status: partial, module Codex.AlgGeom.Hodge.HardLefschetz declares the Lefschetz ${\mathfrak{sl}}_2$-triple via the LefschetzData structure, the Hard Lefschetz iso $L^k:H^{n-k}\to H^{n+k}$ as a sorry-stubbed theorem, and the Lefschetz decomposition of cohomology into primitive components pending the compact-Kähler-manifold + de-Rham-complex + Kähler-identities API in Mathlib.

  requires: alg-geom.hodge-decomposition` (04.09.01), alg-geom.hodge-index-theorem` (04.05.09)

• moduli.kirwan-stratificationopen unit 04.10.08 →

  Kirwan stratification of the unstable locus

  For a complex reductive group $G$ acting on a smooth projective variety $X$ with $G$-linearised ample line bundle $L$, the unstable locus $X^{us}=X\setminus X^{ss}$ decomposes as a finite disjoint union of smooth $G$-invariant locally closed Kirwan strata $S_{\beta }$ indexed by points $\beta$ of a closed positive Weyl chamber ${\mathfrak{t}}_{+}$ of a maximal compact $K\subset G$. The index set $\mathcal{B}$ is the finite collection of $\beta$ that arise as the closest point to the origin in the convex hull of a Weyl orbit of a weight of the $T$-action on $L$ restricted to $X^T$ (Hesselink 1978 + Kempf 1978 optimal-one-parameter-subgroup theorem). Each stratum has explicit codimension $d_{\beta }={\mathrm{codim}}_{\mathbb{C}}(Z_{\beta },X)-{\dim }_{\mathbb{C}}(G/P_{\beta })$ where $Z_{\beta }$ is the $\beta$-fixed-locus component and $P_{\beta }$ is the associated parabolic. Each $S_{\beta }$ retracts $G$-equivariantly onto the parabolic bundle $G{\times }_{P_{\beta }}{Y}_{\beta }^{ss}$ over the semistable sub-locus of $Z_{\beta }$. On the symplectic side, $S_{\beta }$ is the unstable manifold of the Morse-Bott critical set $K\cdot Z_{\beta }$ of $\Vert \mu {\Vert }^2:X\to {\mathbb{R}}_{\ge 0}$ under the gradient flow of $-\Vert \mu {\Vert }^2$ (Kirwan 1984 §4 + §6; Ness 1984). The function $\Vert \mu {\Vert }^2$ is $K$-equivariantly perfect — its equivariant Morse-Bott inequalities are equalities — because the negative bundles of the critical sets carry $K$-equivariant complex structures (Atiyah-Bott 1983 Proposition 13.4). Consequence: the equivariant Poincaré series of $X$ equals the sum of the contributions from the Kirwan strata, $P_t^G(X)=P_t^G(X^{ss})+{\sum }_{\beta \ne 0}{t}^{2d_{\beta }}{P}_t^G(S_{\beta })$. Kirwan surjectivity (Theorem 5.4): the natural restriction $H_G^{\ast }(X;\mathbb{Q})\to H_G^{\ast }(X^{ss};\mathbb{Q})=H^{\ast }(X/{/}_{L}G;\mathbb{Q})$ is surjective with kernel generated by the equivariant Thom classes ${\mathrm{Th}}_{\beta }\in H_G^{2d_{\beta }}(X;\mathbb{Q})$ of the unstable strata. Worked example: ${\mathrm{SL}}_2$ on binary $d$-forms gives Kirwan strata indexed by the root-multiplicity $m\in \{\lceil d/2\rceil +1,\dots ,d\}$, with codimensions $m-1$. Partial desingularisations (Kirwan 1985 Inventiones 81): when strictly semistable points exist, a sequence of $G$-equivariant blow-ups along strictly-semistable Kirwan strata produces $\widetilde{X/{/}_{L}G}$ for which surjectivity holds. Applications: cohomology of moduli of vector bundles on a curve (Kirwan 1986 Inventiones 86, completing Atiyah-Bott 1983); cohomology of moduli of stable maps (Behrend-Manin); K-stability (Chen-Donaldson-Sun 2015 Yau-Tian-Donaldson). Lean status: partial, module Codex.AlgGeom.Moduli.KirwanStratification declares the existence of the stratification (kirwan_stratification), the equivariant Morse-equality (kirwan_morse_equality), and Kirwan surjectivity (kirwan_surjectivity) with sorry proof bodies pending the reductive-group + linearised-line-bundle + moment-map + Morse-Bott-decomposition pipelines in Mathlib.

  requires: alg-geom.git

• char-classes.oriented-bordism-pontryagin-thomopen unit 03.06.13 →

  Oriented bordism and the Pontryagin-Thom construction

  The oriented bordism group Ω^SO_n is the abelian group of closed smooth oriented n-manifolds modulo the oriented bordism relation M ~_SO N <=> exists oriented W^{n+1} with boundary M sqcup \bar N, with addition by disjoint union and additive inverse -[M] = [\bar M] from orientation reversal. Elements need not be 2-torsion; the obstruction to 2-torsion is the existence of an orientation-reversing self-diffeomorphism (CP^2 has none, so [CP^2] has infinite order). The graded ring Ω^SO_* = ⊕_n Ω^SO_n carries Cartesian-product multiplication. Bordism invariance of Pontryagin numbers (Thom 1954): if M, N are oriented-bordant 4k-manifolds then <p_I(TM), [M]> = <p_I(TN), [N]> in Z for every partition I of k. The Pontryagin-Thom collapse in the oriented case is built from a smooth oriented embedding M^n ⊂ S^{n+k} and the oriented normal bundle, classified by BSO(k), yielding maps into the oriented Thom spectrum MSO = colim_k Σ^-k Th(γ^k_SO(k)). Thom's main theorem (oriented case): the oriented Thom homomorphism Φ^SO : Ω^SO_* → π_*(MSO) is a ring isomorphism. Rational computation (Thom 1954, Wall 1960): Ω^SO_* ⊗ Q ≅ Q[CP^2, CP^4, CP^6, …] is a polynomial Q-algebra on the even-dimensional complex projective spaces, with |CP^{2k}| = 4k; rationally Pontryagin numbers separate bordism classes. Wall 1960 integral computation through dimension 8: Ω^SO_0 = Z, Ω^SO_1 = Ω^SO_2 = Ω^SO_3 = 0, Ω^SO_4 = Z<[CP^2]> with signature isomorphism σ : Ω^SO_4 → Z, Ω^SO_5 = Z/2 generated by the Wu manifold W^5 = SU(3)/SO(3) (with non-zero Stiefel-Whitney number w_2 w_3), Ω^SO_6 = Ω^SO_7 = 0, Ω^SO_8 = Z^2<[CP^4], [CP^2 × CP^2]>. Wall's classification: Pontryagin numbers and Stiefel-Whitney numbers together separate Ω^SO_*; rationally Pontryagin numbers alone suffice. The signature is the canonical multiplicative-sequence homomorphism σ : Ω^SO_* → Z via the Hirzebruch L-genus; σ(CP^{2k}) = 1 on every rational generator. Numerical witnesses: 3σ(CP^2) = p_1(CP^2)[CP^2] = 3, 45σ(CP^4) = 7p_2(CP^4) - p_1(CP^4)^2 = 70 - 25 = 45. Lean status: partial (theorems stated, sorry proof bodies pending Mathlib smooth-oriented-manifold + Pontryagin-class + MSO-Thom-spectrum infrastructure; decidable numerical witnesses for Hirzebruch at CP^2 and CP^4 included).

  requires: char-classes.pontryagin-chern.definitions, char-classes.sw-pontryagin-numbers, char-classes.unoriented-bordism-thom

• tropical.semiring-polynomialopen unit 04.12.01 →

  Tropical semiring and tropical polynomial

  The tropical semiring $\mathbb{T}=(\mathbb{R}\cup \{+\infty \},\oplus ,\odot )$ with $x\oplus y=\min (x,y)$ (min-plus convention) and $x\odot y=x+y$ is a commutative idempotent semifield: associative, commutative, distributive, idempotent ($x\oplus x=x$), with additive identity $+\infty$, multiplicative identity $0$, and multiplicative inverses for finite reals. Idempotence obstructs additive inverses (the structure is not a ring); residuation $a\backslash b=b-a$ replaces subtraction. The dual max-plus convention $(\mathbb{R}\cup \{-\infty \},\max ,+)$ is equivalent via $x\mapsto -x$ — min-plus matches valuation conventions on non-archimedean fields. A tropical polynomial $p(x)={\oplus }_{\alpha \in A}{c}_{\alpha }\odot x^{\alpha }={\min }_{\alpha }(c_{\alpha }+⟨\alpha ,x⟩)$ on ${\mathbb{R}}^n$ is a piecewise-linear concave function — the minimum of finitely many affine functions with integer slope vectors $\alpha \in {\mathbb{Z}}^n$. The tropical hypersurface $V(p)$ is the corner locus: equivalently the set where the minimum is attained at least twice, equivalently the non-smooth locus of $p$ as a piecewise-linear function. Bieri-Groves 1984: $V(p)$ is a rational polyhedral complex of pure codimension one in ${\mathbb{R}}^n$, balanced at every $(n-2)$-cell with integer multiplicities ${\mu }_e$ given by lattice lengths of edges in the Newton-polytope subdivision. Newton-polytope duality: the regular subdivision of $\mathrm{Newt}(p)=\mathrm{Conv}(A)$ induced by the lift $\alpha \mapsto (\alpha ,c_{\alpha })\in {\mathbb{R}}^{n+1}$ is dual to the polyhedral structure on $V(p)$ — top-cells of $V(p)$ correspond bijectively to lower edges of $\mathrm{Newt}(p)$, codimension-two cells correspond to 2-faces, dimensions reverse. Function-realisation theorem: every piecewise-linear concave function with integer slopes and finitely many pieces arises from a tropical polynomial, and the map from canonical-representative polynomials to functions is bijective (redundancy is the failure mode for non-canonical representatives). Tropical lines in ${\mathbb{R}}^2$ are Y-shapes with three rays meeting at one trivalent vertex (primitive directions $(-1,-1),(1,0),(0,1)$ sum to $(0,0)$ — the balancing identity); tropical conics in ${\mathbb{R}}^2$ have three trivalent vertices, three bounded edges, six unbounded rays for generic coefficients (dual to the four-lattice-triangle subdivision of the degree-2 Newton triangle). Originators: Bieri-Groves 1984 (polyhedral structure of valuation tropicalisations); Kapranov 2000 (term "tropical variety", non-archimedean amoeba identification); Mikhalkin 2005 (enumerative correspondence on ${\mathbb{R}}^2$). Modern canonical references: Maclagan-Sturmfels 2015 (AMS textbook); Itenberg-Mikhalkin-Shustin 2009 (Oberwolfach text); Mikhalkin 2006 ICM survey. Historical lineage from Imre Simon (São Paulo, automata theory, 1970s) — hence the name "tropical" attributed to Pin 1998 — Cuninghame-Green 1979 (max-plus minimax algebra), and the Maslov-Litvinov 2005 dequantisation philosophy treating min-plus as the classical limit of $\hslash \to 0$ in idempotent mathematics. Connection to mirror symmetry via Gross-Siebert 2011 reconstruction (Annals 174) and Gross 2011 Tropical Geometry and Mirror Symmetry (CBMS 114). Lean status: partial, module Codex.AlgGeom.Tropical.TropicalSemiringPolynomial declares the tropical-semiring idempotence axiom (proved via Mathlib's Tropical.add_self), the TropicalPolynomial schematic structure with eval / argmin / tropicalHypersurface definitions, the corner-locus characterisation, the Bieri-Groves polyhedral-complex theorem, the balancing condition at codim-2 cells, and a worked example with the standard tropical conic in two variables — with sorry-stubbed proof bodies pending Mathlib polyhedral-complex infrastructure.

  requires: toric-geometry.algebraic-torus-character-lattices

• tropical.curve-balanced-metric-graphopen unit 04.12.02 →

  Tropical curve as balanced rational metric graph

  A tropical curve in ${\mathbb{R}}^n$ is a finite rational polyhedral 1-complex $\Gamma \subset {\mathbb{R}}^n$ whose edges carry positive integer weights $w_e$ and primitive integer directions $u_e$ (rational primitive vectors in ${\mathbb{Z}}^n$), satisfying the balancing condition ${\sum }_{e\ni v}{w}_e\cdot u_{e,v}=0\in {\mathbb{Z}}^n$ at every vertex $v$, with $u_{e,v}$ the primitive integer direction of $e$ pointing away from $v$. Equivalently, an abstract tropical curve is a triple $(G,w,\ell )$ where $G$ is a finite multigraph, $w:E(G)\to {\mathbb{Z}}_{>0}$ assigns positive integer weights, and $\ell :E(G)\to {\mathbb{R}}_{>0}\cup \{\infty \}$ assigns positive edge lengths (with $\infty$ modelling unbounded ends); the metric realisation $|\Gamma |$ is the topological space obtained by gluing intervals. Foundational structure theorem (Mikhalkin 2005, derived from Bieri–Groves 1984): every tropical curve $\mathrm{Trop}(V)$ arising as the tropicalisation of an algebraic curve $V\subset ({\mathbb{C}}^{\ast }{)}^n$ over the Puiseux-series field $K=\mathbb{C}\{\{t\}\}$ is a balanced rational polyhedral 1-complex, with edge weights given by intersection multiplicities in the toric resolution and balancing forced by the integer-valued zero-degree identity for principal divisors. The degree of a tropical curve in ${\mathbb{R}}^2$ is the integer $d$ such that the unbounded ends' weighted directions form the multiset $\{d\cdot (1,0),d\cdot (0,1),d\cdot (-1,-1)\}$. The genus $g(\Gamma )=b_1(|\Gamma |)=|E(G)|-|V(G)|+c(G)$ is the first Betti number, a purely combinatorial invariant of the underlying graph (independent of weights or lengths). Worked example: the standard tropical line in ${\mathbb{R}}^2$ has one vertex with three weight-1 rays in primitive directions $(1,0),(0,1),(-1,-1)$ (sum is $(0,0)$, balancing holds); the standard tropical line in ${\mathbb{R}}^n$ has $n+1$ weight-1 rays in directions $e_1,\dots ,e_n,-(e_1+\cdots +e_n)$. Newton-subdivision duality (Mikhalkin 2005 §3): a tropical curve $\Gamma \subset {\mathbb{R}}^2$ is dual to a subdivision of its Newton polytope $\Delta \subset {\mathbb{R}}^2$, with vertices of $\Gamma$ in bijection with maximal cells of the subdivision and bounded edges of $\Gamma$ in bijection with bounded edges of the subdivision; the weight of an edge of $\Gamma$ equals the integer length of the dual edge. Degree-genus inequality $g\le \left(\genfrac{}{}{0ex}{}{d-1}{2}\right)$ with equality for smooth tropical curves (unimodular Newton subdivisions). Moduli (Brannetti–Melo–Viviani 2011): $M_g^{\mathrm{trop}}$ is a generalised cone complex of dimension $3g-3$ for $g\ge 2$, identified with the skeleton of the Berkovich analytification of ${\overline{M}}_g$ (Abramovich–Caporaso–Payne 2015). Tropical Riemann–Roch (Baker–Norine 2007 Adv. Math. 215): $r(D)-r(K_{\Gamma }-D)=\mathrm{deg}(D)-g+1$ for every divisor $D$ on a tropical curve of genus $g$, with $K_{\Gamma }={\sum }_v(\mathrm{val}(v)-2)\cdot v$ and $r$ the chip-firing-defined combinatorial rank. Tropical Clifford inequality (Coppens 2016): $r(D)\le \mathrm{deg}(D)/2$ for effective $D$ with $0\le \mathrm{deg}(D)\le 2g-2$. Lean status: partial, module Codex.AlgGeom.Tropical.TropicalCurve declares the PrimitiveVector, WeightedDirection, VertexData.balanced, TropicalCurve, WeightedMetricGraph, Divisor, and tropical_riemann_roch with sorry proof bodies pending the rational-polyhedral-complex + chip-firing-equivalence + Picard-group API in Mathlib.

  requires: toric-geometry.algebraic-torus-character-lattices` (04.11.01)

• tropical.newton-polytope-amoebaopen unit 04.12.04 →

  Newton polytope and non-archimedean amoeba

  For a Laurent polynomial $f={\sum }_{\alpha \in \mathrm{supp}(f)}{c}_{\alpha }{x}^{\alpha }$ over a field $K$ in $n$ variables, the Newton polytope is $\mathrm{Newt}(f)=\mathrm{conv}(\mathrm{supp}(f))\subseteq M_{\mathbb{R}}$, a lattice polytope encoding the support combinatorially. For $K=\mathbb{C}$, the complex amoeba is $\mathcal{A}(f)=\mathrm{Log}(V(f))\subseteq N_{\mathbb{R}}$ where $\mathrm{Log}:({\mathbb{C}}^{\ast }{)}^n\to N_{\mathbb{R}}$ takes coordinate-wise log-absolute-value; the amoeba is a closed subset of $N_{\mathbb{R}}$ with finitely many connected components in its complement, labelled by lattice points of $\mathrm{Newt}(f)\cap M$ via the Forsberg-Passare-Tsikh order map. For $K$ a non-archimedean field with valuation $v$, the non-archimedean amoeba is ${\mathcal{A}}_v(f)=\mathrm{val}(V(f))\subseteq N_{\mathbb{R}}$ where $\mathrm{val}$ is coordinate-wise valuation; Kapranov's theorem (Einsiedler-Kapranov-Lind 2006) identifies ${\mathcal{A}}_v(f)$ exactly with the tropical hypersurface $\mathrm{trop}(f)=\{u:T(f)(u)={\min }_{\alpha }(v(c_{\alpha })+⟨\alpha ,u⟩)\text{ attained twice}\}$. The Forsberg-Passare-Tsikh tentacle-alignment theorem identifies every recession direction of $\mathcal{A}(f)$ with the outer normal to a facet of $\mathrm{Newt}(f)$, by a cancellation argument on the dominant monomials of $f$ at far-away points of $V(f)$. The Mikhalkin spine-convergence theorem (Mikhalkin 2004 Theorem 4) is the analytic-degeneration version: as $t\to \infty$, the rescaled complex amoeba $\mathcal{A}(f^t)/\log t$ converges in Hausdorff metric on bounded regions to the tropical hypersurface $\mathrm{trop}(f)$, with the rescaling rate $O(1/\log t)$. The combinatorial bridge between the three objects is the normal-fan-of-$\mathrm{Newt}(f)$ structure on $N_{\mathbb{R}}$: tentacle directions of $\mathcal{A}(f)$ = outer normals to facets of $\mathrm{Newt}(f)$ = ray directions of $\mathrm{trop}(f)$ for trivial valuation. The regular-subdivision correspondence (Maclagan-Sturmfels §3) extends this: $\mathrm{trop}(f)\subset {\mathbb{R}}^2$ is dual to a regular subdivision ${\mathcal{R}}_f$ of $\mathrm{Newt}(f)$ obtained by lifting support points via the valuation and projecting the lower hull. The Bernstein-Kushnirenko theorem connects $\mathrm{Newt}(f)$ to enumerative geometry: the number of common zeros of $f_1,\dots ,f_n$ in $({\mathbb{C}}^{\ast }{)}^n$ for generic coefficients equals the mixed volume $\mathrm{MV}(\mathrm{Newt}(f_1),\dots ,\mathrm{Newt}(f_n))$. The Bergman-Bieri-Groves theorem identifies the more general tropical variety $\mathrm{trop}(V)$ of a subvariety $V\subseteq T$ as a balanced rational polyhedral complex of pure dimension $\dim V$, with multiplicities at top cells satisfying the balancing condition at codimension-1 faces. The Sturmfels-Tevelev geometric description identifies $\mathrm{trop}(V)$ with the recession data of the closure of $V$ in a suitable toric compactification, bridging the abstract valuative definition to explicit toric-geometric computation. The Newton-polytope and amoeba dictionary is the foundational entry point to the tropical-geometry-and-mirror-symmetry program: it feeds Mikhalkin's correspondence theorem [04.12.05] on enumeration of complex curves via tropical curves, the Nishinou-Siebert correspondence [04.12.06] on log Gromov-Witten invariants, and the Gross-Siebert reconstruction program [04.12.09] on Calabi-Yau mirror symmetry via tropical manifolds. Originator chain: Newton 1671/1736 (Newton polygon for Puiseux expansion); Bergman 1971 (logarithmic limit set); Gelfand-Kapranov-Zelevinsky 1994 (amoeba terminology, complex case); Kapranov 1995 (non-archimedean amoeba); Forsberg-Passare-Tsikh 2000 (order map, complement components); Mikhalkin 2000-2004 (spine, pairs-of-pants decomposition, spine convergence); Einsiedler-Kapranov-Lind 2006 (canonical published proof of Kapranov's theorem). Lean status: partial, module Codex.AlgGeom.Tropical.NewtonPolytopeAmoeba declares the LaurentPoly structure, the newtonPolytope and amoeba and tropicalHypersurface definitions, the tentacle_alignment theorem (Forsberg-Passare-Tsikh recession-direction = facet-normal statement), and the spine_convergence placeholder (Mikhalkin 2004), with sorry-stubbed proof bodies pending Mathlib non-archimedean amoeba + Newton-polytope + Hausdorff-convergence infrastructure.

  requires: toric-geometry.rational-polyhedral-cone-dual, tropical.semiring-polynomial

• tropical.kapranov-theoremopen unit 04.12.03 →

  Kapranov's theorem (fundamental theorem of tropical geometry)

  For an algebraically closed field $K$ with surjective non-archimedean valuation $\mathrm{val}:K^{\ast }\to \mathbb{R}$ (canonical example: Puiseux series $K={\bigcup }_d\mathbb{C}((t^{1/d}))$), an ideal $I\subseteq K[x_1^{\pm },\dots ,x_n^{\pm }]$ defining a subvariety $V(I)\subseteq (K^{\ast }{)}^n$ produces three a priori distinct subsets of ${\mathbb{R}}^n$: $\mathrm{trop}(I{)}_{\mathrm{val}}$ (closure of valuation image of $V(I)(K)$), $\mathrm{trop}(I{)}_{\mathrm{init}}$ (locus of weights $w$ at which the initial ideal ${\mathrm{in}}_w(I)$ contains no monomial), and $\mathrm{trop}(I{)}_{\mathrm{amoeba}}$ (Hausdorff limit of complex amoebas as $t\to 0$). The fundamental theorem (Kapranov 2000 unpublished; Einsiedler-Kapranov-Lind 2006 written) asserts the three coincide and produce a closed polyhedral subcomplex of pure dimension equal to the Krull dimension of $K[x^{\pm }]/I$. Forward direction ($\mathrm{val}\subseteq \mathrm{init}$): strong triangle inequality of the non-archimedean valuation forces uniqueness of the tropical minimum to imply $\mathrm{val}(f(z))<\infty$, contradicting $f(z)=0$. Reverse direction ($\mathrm{init}\subseteq \mathrm{val}$): Hensel-style lifting of a residual root in $V({\mathrm{in}}_w(I))\subseteq ({\text{residue field}}^{\ast }{)}^n$ — non-empty by the Hilbert Nullstellensatz over the algebraically closed residue field — to a $K$-point of $V(I)$ with prescribed coordinate valuations; at residually-singular points the Berkovich-analytic surjectivity of the tropicalisation map (Gubler 2013) supplies the lift. Polyhedral structure: Gröbner-fan finiteness (Sturmfels 1996) refines the equivalence relation $w\sim w^{\prime }⟺{\mathrm{in}}_w(I)={\mathrm{in}}_{w^{\prime }}(I)$ into finitely many rational polyhedral cones; pure dimensionality is the ideal-theoretic upgrade of Bieri-Groves 1984. Balancing: at each codimension-one face the multiplicities $m_i$ of adjacent top cones and primitive normal directions $v_i$ satisfy ${\sum }_i{m}_i{v}_i=0$ in the quotient lattice (Speyer 2004 dissertation; Maclagan-Sturmfels §3.4). Worked example: $\mathrm{trop}(x+y+1)=0$ is the three-rayed fan in ${\mathbb{R}}^2$ with rays in directions $(1,1)$, $(-1,0)$, $(0,-1)$ meeting at the origin, multiplicities all $1$, balancing $(1,1)+(-1,0)+(0,-1)=(0,0)$. Bergman-fan specialisation (linear ideals = matroids; Sturmfels 2002, Ardila-Klivans 2006). Downstream applications: Mikhalkin's correspondence theorem [04.12.05] for plane-curve Gromov-Witten counts via tropical curve counts; Nishinou-Siebert [04.12.06] toric-degeneration for toric Calabi-Yau enumeration; SYZ [04.12.10] mirror symmetry as polyhedral-base dualisation; Gross-Siebert reconstruction theorem [04.12.09] for Calabi-Yau varieties from tropical-manifold scattering diagrams. Lean status: partial, module Codex.AlgGeom.Tropical.KapranovTheorem declares the ValuedField structure, LaurentIdeal placeholder, the three constructions tropByValuation, tropByInitial, tropByAmoeba, and the equivalences kapranov_val_eq_init, kapranov_amoeba_eq_val, kapranov_three_definitions, plus the Bieri-Groves polyhedral-and-pure-dimension theorem trop_is_polyhedral_pure_dim and the tropical-line worked-example theorem trop_line with sorry proof bodies pending Mathlib algebraically-closed-Puiseux-field + initial-ideal-calculus + Bieri-Groves polyhedral-finiteness + non-archimedean Hensel-lift infrastructure.

  requires: toric-geometry.algebraic-torus-character-lattices` (04.11.01); `tropical.semiring-polynomial` (04.12.01)

• tropical.mikhalkin-correspondenceopen unit 04.12.05 →

  Mikhalkin's correspondence theorem

  For a toric surface $X_{\Sigma }$ with Newton polygon $\Delta \subset {\mathbb{Z}}^2$ and a generic configuration of $r=|\Delta \cap {\mathbb{Z}}^2|-1-g$ points in $({\mathbb{C}}^{\ast }{)}^2\subset X_{\Sigma }$, the count $N^{\Delta ,g}$ of irreducible complex curves of Newton polygon $\Delta$ and geometric genus $g$ through the points equals the Mikhalkin tropical count ${\sum }_{\Gamma }m(\Gamma )$ over simple tropical curves of Newton polygon $\Delta$ and genus $g$ through corresponding generic tropical points, weighted by the tropical multiplicity $m(\Gamma )={\prod }_{V}m(V)$. At each trivalent vertex $V$ with weighted primitive outgoing directions $(w_i,v_i{)}_{i=1,2,3}$ satisfying balancing $\sum w_i{v}_i=0$, the multiplicity is $m(V)=w_1{w}_2|\det (v_1,v_2)|$ — equivalently twice the lattice area of the triangle spanned by $w_1{v}_1$ and $w_2{v}_2$. The three pairwise determinant computations $w_1{w}_2|\det (v_1,v_2)|=w_2{w}_3|\det (v_2,v_3)|=w_1{w}_3|\det (v_1,v_3)|$ all agree, forced by the balancing relation. Proof structure (Mikhalkin 2005): (1) dimension count — both moduli spaces have dimension $r=|\Delta \cap {\mathbb{Z}}^2|-1-g$, matched by combinatorial counts on the tropical side (Mikhalkin Prop. 2.13) and by the genus-degree formula on the algebraic side; (2) local multiplicity matching at trivalent vertices via Bézout-type intersection theory in $({\mathbb{C}}^{\ast }{)}^2$, with the determinant $|\det (v_1,v_2)|$ recording the intersection multiplicity of two branches in primitive directions; (3) degeneration argument via the valuation map $\mathrm{val}:(\mathbb{C}\{\{t\}{\}}^{\ast }{)}^2\to {\mathbb{R}}^2$ and the Viro 1984 patchworking construction, exhibiting every tropical curve through generic points as the limit of $m(\Gamma )$ complex curves over the Puiseux series field $\mathbb{C}\{\{t\}\}$. Application to Gromov-Witten of ${\mathbb{P}}^2$: for the standard simplex ${\Delta }_d$ and genus zero, $r=3d-1$ (Kontsevich-Manin 1994 dimension count), and the Mikhalkin tropical enumeration via lattice paths in the Newton triangle recovers $N_1=1,N_2=1,N_3=12,N_4=620,N_5=87304$, agreeing with the WDVV recursion. Mikhalkin's tropical algorithm is the first combinatorial-geometric algorithm reproducing $N_d$ without passing through the WDVV recursion. Real version (Mikhalkin 2007 + Itenberg-Kharlamov-Shustin 2003): the Welschinger invariant $W_d$ of real rational degree-$d$ curves in ${\mathbb{P}}^2(\mathbb{R})$ through $3d-1$ real generic points equals the signed Mikhalkin tropical count ${\sum }_{\Gamma }(-1{)}^{n(\Gamma )}m(\Gamma )$, with $n(\Gamma )$ counting odd-multiplicity bounded edges. Welschinger 2005 Inventiones 162 established the symplectic invariance (independence of generic configuration and compatible $J$); the tropical formula gives the first effective computation. Values: $W_1=1,W_2=1,W_3=8,W_4=240,W_5=18264$. Itenberg-Kharlamov-Shustin 2004 proved the logarithmic equivalence $\log W_d\sim \log N_d$ as $d\to \infty$, confirming the real count is exponentially comparable to the complex count despite the cancellation of complex-conjugate pairs. Generalisations: Nishinou-Siebert 2006 (higher-dimensional toric targets via toric degenerations — foundation of Gross-Siebert mirror symmetry); Gathmann-Markwig 2007 (tropical Caporaso-Harris recursion for relative Severi degrees of plane curves tangent to a line); Cavalieri-Johnson-Markwig (higher-genus and descendant $\psi$-class invariants tropically). Mikhalkin's correspondence is the genealogical root of modern enumerative tropical geometry and the first complete bridge between the algebraic-geometric tradition of Severi / Gromov-Witten enumeration (Kontsevich-Manin 1994, Caporaso-Harris 1998) and the combinatorial-geometric tradition of Newton-polygon / lattice-path / Viro-patchworking enumeration. Lean status: partial, module Codex.AlgGeom.Tropical.MikhalkinCorrespondence declares the TrivalentVertexData structure (three weighted primitive directions with balancing condition), the multiplicity formula m(V) = w₁·w₂·|det(v₁, v₂)| with balancing invariance lemmas, the SimpleTropicalCurve and NewtonPolygon structures, the moduliDimension formula $r=|\Delta \cap {\mathbb{Z}}^2|-1-g$, the mikhalkin_correspondence theorem equating complex and tropical counts, the welschinger_correspondence for the signed real version, and numerical witnesses ($N_1=1,N_2=1,N_3=12,W_3=8$ with the decidable check $N_3-W_3=4$) — with sorry-stubbed proof bodies pending Mathlib infrastructure for Puiseux series, toric degenerations, and Viro patchworking.

  requires: toric-geometry.fan-and-toric-variety` (04.11.04), tropical.semiring-polynomial` (04.12.01), tropical.curve-balanced-metric-graph` (04.12.02)

• tropical.nishinou-siebert-correspondenceopen unit 04.12.06 →

  Nishinou-Siebert correspondence theorem

  The Nishinou-Siebert correspondence theorem (Nishinou-Siebert 2006, Duke 135) generalises Mikhalkin's 2005 correspondence theorem from toric surfaces to toric varieties of arbitrary dimension. Setup: $X_{\Sigma }$ a smooth projective toric variety of complex dimension $n$, $L$ a $T$-equivariant ample line bundle with associated lattice polytope $\Delta \subseteq M_{\mathbb{R}}$, $\Xi$ a polyhedral subdivision of $\Delta$, and $\pi :\mathcal{X}\to {\mathbb{A}}_{\mathbb{C}}^1$ a toric degeneration with generic fibre $X_{\Sigma }$ and central fibre ${\mathcal{X}}_0={\bigcup }_{{\Delta }_i\in \Xi }{X}_{{\Delta }_i}$ a reduced union of toric strata. The dual intersection complex $B(\Xi )$ is the cone complex dual to $\Xi$, an integral affine manifold (with codimension-2 monodromy singularities) of dimension $n$. Theorem (Nishinou-Siebert 2006 Theorem 8.3): for genus $g\ge 0$, curve class $\Delta \in H_2(X_{\Sigma },\mathbb{Z})$, expected dimension $r=(1-g)(n-3)+⟨c_1(T{X}_{\Sigma }),\Delta ⟩$, and $r$ generic points $p_i\in X_{\Sigma }$, the count $N_{g,\Delta }(X_{\Sigma };p_1,\dots ,p_r)$ of genus-$g$ stable maps to $X_{\Sigma }$ of class $\Delta$ passing through every $p_i$ equals the multiplicity-weighted count $N_{g,\Delta }^{\mathrm{trop}}(\Xi )={\sum }_{(\Gamma ,h)}m(\Gamma ,h)$ of parametrised tropical curves in $B(\Xi )$ matched to the corresponding tropically generic points. Multiplicity: at each trivalent vertex with edge direction vectors $u_1,u_2,u_3\in N$ ($u_1+u_2+u_3=0$), $m_v=|\det (u_i,u_j)|$ for any pair (the Mikhalkin lattice determinant); at higher-valence vertices, the Nishinou-Siebert higher-determinantal formula generalises via Plücker volumes. Proof (Nishinou-Siebert 2006 §4-§8): (i) algebraic count = log GW count by definition; (ii) log GW degeneration formula equates smooth-fibre count with log GW count on log smooth central fibre ${\mathcal{X}}_0$ (Abramovich-Chen 2014 / Gross-Siebert 2013); (iii) tropicalisation map sends log smooth stable maps to parametrised tropical curves in $B(\Xi )$, with the dual graph of $C\subseteq {\mathcal{X}}_0$ giving $\Gamma$, irreducible-component-to-stratum assignment giving the map $h$, and intersection numbers giving the direction vectors; balancing follows from the toric residue theorem; (iv) local multiplicity match reduces to the Mikhalkin / Plücker formula. Reduction to Mikhalkin: when $\Sigma$ is two-dimensional, $B(\Xi )$ is two-dimensional and the multiplicity formula specialises to $|\det (u_i,u_j)|$, recovering Mikhalkin 2005 J. AMS 18. Application to mirror symmetry: the tropical curves enumerated by Nishinou-Siebert are precisely the wall-crossing data of scattering diagrams on $B$ that Gross-Siebert 2011 Annals 174 use to construct the mirror Calabi-Yau as a smoothing of the degenerate central fibre. The Gross-Hacking-Keel-Kontsevich structure-constant formula ${\vartheta }_{q_1}\cdot {\vartheta }_{q_2}={\sum }_{q_3,(\Gamma ,h)}m(\Gamma ,h)\cdot t^{\mathrm{area}(\Gamma )}\cdot {\vartheta }_{q_3}$ writes theta-function multiplication on the mirror in terms of Nishinou-Siebert curve counts. SYZ heuristic (Strominger-Yau-Zaslow 1996 Nucl. Phys. B 479): mirror Calabi-Yau pairs arise as dual special-Lagrangian torus fibrations over a common base $B$; the dual intersection complex is the algebraic realisation of $B$. Witnesses: classical Severi degrees $N_{0,d}({\mathbb{P}}^2)=(1,12,620,87304,\dots )$ for $d=2,3,4,5,\dots$ are recovered as tropical counts in the dimension-2 specialisation (Mikhalkin 2005); higher-dimensional Nishinou-Siebert counts produce e.g. $N_{0,1}({\mathbb{P}}^3)=1$ (unique line through 2 points) and three-dimensional analogues. Computational tool: Caporaso-Harris recursion (Gathmann-Markwig 2007 Math. Ann. 338). Lean status: partial, module Codex.AlgGeom.Tropical.NishinouSiebertCorrespondence declares the Lattice, Cone, Fan, ToricVariety, PolyhedralSubdivision, ToricDegeneration, DualIntersectionComplex, TropicalCurve, algebraicCount, tropicalCount, and the named theorems nishinou_siebert_correspondence, reduces_to_mikhalkin, gross_siebert_scattering_input with placeholder proof bodies pending the log GW + virtual-class + polyhedral-complex API in Mathlib.

  requires: toric-geometry.fan-and-toric-variety` (04.11.04); tropical correspondence in dimension 2 (Mikhalkin) is the n = 2 specialisation

• tropical.toric-degeneration-calabi-yauopen unit 04.12.07 →

  Toric degeneration of a Calabi-Yau variety

  A toric degeneration of a Calabi-Yau variety (Gross-Siebert 2006 J. Differential Geom. 72 §2) is a flat proper morphism $\pi :\mathfrak{X}\to \mathrm{Spec}\mathbb{C}[[t]]$ with smooth Calabi-Yau generic fibre ${\mathfrak{X}}_t\cong X$ and central fibre ${\mathfrak{X}}_0={\bigcup }_i{X}_{{\Delta }_i}$ a reduced reducible union of toric varieties indexed by the top-dimensional cells of a polyhedral subdivision $\mathcal{P}$ of an associated lattice polytope $\Delta \subseteq M_{\mathbb{R}}$, satisfying (i) toric-strata intersections along faces of $\mathcal{P}$, (ii) log smoothness of $\pi$ with respect to the divisorial log structure of ${\mathfrak{X}}_0$ (Kato 1989), (iii) triviality of the relative canonical sheaf ${\omega }_{\mathfrak{X}/\mathrm{Spec}\mathbb{C}[[t]]}$ (or its logarithmic version, ${\omega }_{\mathfrak{X}/S}(\log {\mathfrak{X}}_0)\cong {\mathcal{O}}_{\mathfrak{X}}$). The Calabi-Yau condition is enforced on the family, not just fibrewise. Construction (Gross-Siebert 2006 §2): for $X\subseteq Y_{\Sigma }$ a Calabi-Yau hypersurface in a toric Fano with reflexive polytope $\nabla$, the polar dual $\Delta ={\nabla }^{\circ }$ is the Newton polytope of $-K_Y$; a regular polyhedral subdivision $\mathcal{P}$ of $\Delta$ is determined by a strictly convex integer-valued piecewise-linear function $\phi$ (the Mumford function) via the lower-envelope construction (Gelfand-Kapranov-Zelevinsky 1994 secondary polytope); the Mumford degeneration lifts each monomial ${\chi }^m$ to $\widetilde{{\chi }^m}:=t^{\phi (m)}{\chi }^m\in \mathbb{C}[\Delta \cap M,t]$, producing a flat family of toric Fanos with generic fibre $Y_{\Sigma }$ and central fibre ${\bigcup }_i{Y}_{{\Delta }_i}$; lifting a generic anticanonical section gives the Calabi-Yau hypersurface family $\mathfrak{X}=\{f_t=0\}\subseteq \mathfrak{Y}$. Maximally unipotent monodromy (MUM): the monodromy operator $T$ on $H^n({\mathfrak{X}}_t,\mathbb{C})$ satisfies $(T-I{)}^n\ne 0$ and $(T-I{)}^{n+1}=0$ (Morrison 1993); the limit mixed Hodge structure (Deligne 1971; Schmid 1973) has maximal weight filtration jump $(1,h^{2,1},h^{2,1},1,h^{2,1},h^{2,1},1)$ for $n=3$. MUM points are the algebraic-geometric counterparts of large-volume limits on the mirror under the Morrison mirror map. Dual intersection complex $B(\mathcal{P})$: the integral affine cell complex obtained by gluing dual cones over cells of $\mathcal{P}$; topologically $S^n$ in the Calabi-Yau case (reflexive polytope); carries an integral affine structure away from a codimension-2 singular locus ${\Delta }_{\mathrm{sing}}$. Simple singularities (Gross-Siebert 2006 §1.6) are the local-form restrictions that make the Gross-Siebert reconstruction theorem run. Witnesses: simplest example is the elliptic curve degenerating to a nodal cubic, $B(\mathcal{P})\cong S^1$; K3 surfaces degenerate via Type-III Kulikov 1977 degeneration to a triangulation of $S^2$; the quintic threefold degenerates to five hyperplanes meeting in a triangulation of $S^3$. d-semistability (Friedman 1983 Annals 118): the obstruction theory for smoothing normal-crossings central fibres; Gross-Siebert resolve it combinatorially via the integral affine structure on $B(\mathcal{P})$. Non-archimedean parallel: Kontsevich-Soibelman 2006 realises $B(\mathcal{P})$ as the Berkovich skeleton of a non-archimedean analytic Calabi-Yau. The toric-degeneration setup is the input to the Gross-Siebert reconstruction theorem [04.12.09] and the algebraic realisation of the SYZ conjecture [04.12.10]; Nishinou-Siebert tropical curves on $B(\mathcal{P})$ from [04.12.06] are the wall-crossing data of the scattering structure. Lean status: partial, module Codex.AlgGeom.Tropical.ToricDegeneration declares the Lattice, LatticePolytope, PolyhedralSubdivision, CalabiYauVariety, ToricFano, ToricDegenerationCY, DualIntersectionComplex, integralAffineSingularLocus placeholder structures and definitions, plus the named theorems exists_toric_degeneration (Gross-Siebert 2006 existence theorem), central_fibre_is_union_of_toric_strata, maximally_unipotent_at_zero, dual_intersection_complex_is_topological_sphere, and gross_siebert_reconstruction_input with placeholder proof bodies pending the log-smooth-morphism, limit-mixed-Hodge-structure, and polyhedral-cell-complex API in Mathlib.

  requires: toric-geometry.fan-and-toric-variety` (04.11.04); `tropical.newton-polytope-amoeba` (04.12.04); `tropical.nishinou-siebert-correspondence` (04.12.06)

• tropical.dual-intersection-complexopen unit 04.12.08 →

  Dual intersection complex; tropical manifold B

  The dual intersection complex $(B,P)=(B(\Xi ),P(\Xi ))$ of a toric degeneration $\pi :\mathcal{X}\to {\mathbb{A}}^1$ packages the combinatorial skeleton of the central fibre ${\mathcal{X}}_0={\bigcup }_{{\Delta }_i\in \Xi }{X}_{{\Delta }_i}$ together with its integral affine structure. Construction: for each cell ${\Delta }_i\in \Xi$ of dimension $k$, the dual cone ${\sigma }_{{\Delta }_i}^{\vee }\subseteq N_{\mathbb{R}}$ has dimension $n-k$; the dual intersection complex $B(\Xi )=⨆{\sigma }_{{\Delta }_i}^{\vee }/\sim$ glues these cones along dual-of-face identifications. The polyhedral decomposition $P(\Xi )$ has one cell per cell of $\Xi$ with reverse face order. The integral affine structure on $B(\Xi )$ is induced cell-wise from the lattice $N\cap {\sigma }_{{\Delta }_i}^{\vee }$ on each dual cone; it extends across codim-1 cells uniquely (by lattice gluing) and across codim-$\ge 3$ cells determined by codim-2 data; the codim-2 cells with non-identity local monodromy $\mathrm{mon}(\rho )\in {\mathrm{GL}}_n(\mathbb{Z})$ form the singular locus ${\Delta }_B\subseteq B(\Xi )$. The complement $B(\Xi )\setminus {\Delta }_B$ is an integral affine manifold; $B(\Xi )$ is an integral affine manifold with codim-2 singularities. A tropical manifold is the abstracted triple $(B,P,s)$ of cell complex + decomposition + integral affine structure with singular locus. Calabi-Yau monodromy condition (Gross-Siebert 2006 §1.5): the generic fibre ${\mathcal{X}}_t$ is Calabi-Yau iff the monodromy representation ${\rho }_B:{\pi }_1(B\setminus {\Delta }_B)\to {\mathrm{GL}}_n(\mathbb{Z})$ factors through ${\mathrm{SL}}_n(\mathbb{Z})$; equivalently the integral affine structure preserves a parallel volume form; equivalently the relative canonical line bundle ${\omega }_{\mathcal{X}/{\mathbb{A}}^1}{|}_{{\mathcal{X}}_0}$ is trivial. Simple singularities (Gross-Siebert 2006 §1.6): codim-2 cells where the monodromy is conjugate to the focus-focus form with non-trivial ${\mathrm{SL}}_2(\mathbb{Z})$ unipotent block on a transverse 2-plane; the Gross-Siebert reconstruction theorem [04.12.09] applies under the simple-singularities hypothesis. Discrete Legendre duality (Gross-Siebert 2006 §1.4): the cell-wise dualisation interchanges $k$-cells of $P$ with $(n-k)$-cells of $\check{P}$ via the $N\times M\to \mathbb{Z}$ pairing, producing a dual tropical manifold $(\check{B},\check{P},\check{s})$ on the same underlying space; the construction is involutive and intertwines mirror-symmetric Calabi-Yau pairs. SYZ identification (Strominger-Yau-Zaslow 1996): the dual intersection complex is the algebraic realisation of the SYZ base $B$, the polyhedral decomposition records the discriminant stratification of the special-Lagrangian fibration, the integral affine structure records the Arnold-Liouville action coordinates, and the singular locus records the SYZ discriminant. Castaño-Bernard-Matessi 2009 supplies the symplectic-side dimension-3 Lagrangian fibrations matching algebraic $(B,P)$. Non-archimedean realisation (Kontsevich-Soibelman 2001/2006): $(B,P)$ is the Berkovich skeleton of a maximally degenerate Calabi-Yau over $K=\mathbb{C}((t))$, with the integral affine structure recovered from the Berkovich retraction; the algebraic and non-archimedean realisations agree. Originator chain: Mumford 1972 (degenerating abelian varieties with affine-base structure); KKMS 1973 (toroidal-embedding language); Strominger-Yau-Zaslow 1996 (SYZ heuristic); Kontsevich-Soibelman 2001 (integral affine + simple singularities framework); Gross-Siebert 2006 (algebraic-geometric definition of $(B,P)$ from a toric degeneration); Castaño-Bernard-Matessi 2009 (symplectic-side 3-torus fibrations); Gross 2011 CBMS (textbook synthesis); Gross-Siebert 2011 Annals (reconstruction theorem). Lean status: partial, module Codex.AlgGeom.Tropical.DualIntersectionComplex declares the Lattice, IntegralAffineChart, PolyhedralSubdivision, DualIntersectionComplex, PolyhedralDecomposition, singularLocus, TropicalManifold, fromSubdivision, legendreDual, localMonodromy, isCalabiYauMonodromy structures and definitions, plus the named theorems fromSubdivision_isTropicalManifold, monodromy_calabi_yau, legendreDual_involutive, syz_base_identification with placeholder proof bodies pending the polyhedral-complex + integral-affine-chart + monodromy-representation + Calabi-Yau-line-bundle API in Mathlib.

  requires: toric-geometry.fan-and-toric-variety` (04.11.04); `tropical.nishinou-siebert-correspondence` (04.12.06)

• tropical.gross-siebert-reconstructionopen unit 04.12.09 →

  Gross-Siebert reconstruction theorem (statement)

  The Gross-Siebert reconstruction theorem (Gross-Siebert 2011, Annals of Mathematics 174, 1301-1428, title "From real affine geometry to complex geometry") is the central technical move of the Gross-Siebert programme for mirror symmetry. Setup: a polarised tropical manifold $(B,\mathcal{P},\phi )$ where $B$ is an integral affine manifold of real dimension $n$ with codimension-2 singular locus $\Delta \subset B$, $\mathcal{P}$ a polyhedral decomposition of $B$ into lattice polytopes whose codimension-2 cells contain $\Delta$ ("simple singularities" — focus-focus in dimension 2; classified Type-I/II in dimension 3 per Gross-Siebert 2006 §1.6), and $\phi :B\to \mathbb{R}$ a strictly convex piecewise-affine polarisation. Lifted gluing data: slab functions $\{f_{\rho }{\}}_{\rho \in {\mathcal{P}}^{(n-1)}}$ on codimension-1 cells, each a formal Laurent polynomial $f_{\rho }=1+{\sum }_i{c}_{\rho ,i}{z}^{m_{\rho ,i}}$ with $c_{\rho ,i}\in \mathbb{C}[[t]]$, constant term 1 (forced by Calabi-Yau / order-0-identity normalisation), satisfying the joint-consistency condition: at every joint $\tau \in {\mathcal{P}}^{(n-2)}$, the product of wall-crossing automorphisms ${\theta }_{{\rho }_1}\circ \cdots \circ {\theta }_{{\rho }_k}$ around $\tau$ equals the identity modulo $t$. Theorem: there exists a toric degeneration $\pi :\mathfrak{X}\to \text{Spec}\mathbb{C}[[t]]$ of Calabi-Yau varieties of relative dimension $n$ whose dual intersection complex ([04.12.08]) is the given $(B,\mathcal{P})$ and whose slab gluing recovers $\{f_{\rho }\}$; the toric degeneration is unique up to formal isomorphism over $\text{Spec}\mathbb{C}[[t]]$. Construction: order by order in $t$, via the Kontsevich-Soibelman scattering algorithm. Order 0: central fibre ${\mathfrak{X}}_0={\bigcup }_{\sigma }{X}_{\sigma }$ as union of toric pieces indexed by maximal cells $\sigma \in {\mathcal{P}}^{(n)}$ glued along codimension-$k$ toric strata indexed by codimension-$k$ cells of $\mathcal{P}$. Order $k-1$ to order $k$: at each joint $\tau$, compute deficit ${\theta }_{\tau }\equiv \exp ({\sum }_j{a}_{\tau ,j}{t}^k{z}^{m_{\tau ,j}}{\partial }_{z^{m_{\tau ,j}}})(\mathrm{mod}{t}^{k+1})$ from the order-$(k-1)$ wall-crossing-product around $\tau$; add walls ${\mathfrak{d}}_{\tau ,j}$ to scattering diagram $\mathcal{S}$ emanating from $\tau$ in the direction $m_{\tau ,j}^{\perp }$, labelled with ${\theta }_{\tau ,j}^{-1}$, to compensate the deficit. Termination per order: deficits are finite sums of monomials. Uniqueness per order: deficits and compensating walls are uniquely determined by the order-$(k-1)$ data, hence the formal family is determined inductively. The full proof in Gross-Siebert 2011 has four moves: (I) formal-infinitesimal lift order 0 → 1, using Kato logarithmic deformation theory; (II) scattering algorithm order $k-1$ → $k$; (III) global consistency across joints at each order via Kato-fan / cohomology-vanishing computation (Gross-Siebert 2010, the $[B,\mathcal{P}]$ paper); (IV) uniqueness via Čech-cohomology argument on formal-neighbourhood covering of joints. The full proof is 128 pages and is deferred at FT-equivalence (statement-only unit with a structural Master-tier proof sketch only). Antecedents: Mikhalkin 2005 (J. AMS 18) — dim-2 tropical-classical correspondence; Nishinou-Siebert 2006 (Duke 135) — higher-dimensional toric correspondence; SYZ 1996 (Nucl. Phys. B 479) — physical-geometric conjectural origin; Kontsevich-Soibelman 2001 + 2006 — non-archimedean SYZ + scattering-diagram formalism. Enumerative content: slab functions admit expansion $f_{\rho }=1+{\sum }_{\beta }{N}_{\beta }^{\text{trop}}(\rho )t^{\omega (\beta )}{z}^{\partial \beta }$ where $\beta$ ranges over relative-homology classes of tropical Maslov-index-2 disks ending on $\rho$, $N_{\beta }^{\text{trop}}(\rho )$ is the tropical count, $\omega (\beta )$ the symplectic area, $z^{\partial \beta }$ the boundary monomial; via Nishinou-Siebert ([04.12.06]), the tropical-disk counts equal log Gromov-Witten invariants on the central fibre (Abramovich-Chen 2014 + Chen 2014 + Gross-Siebert 2013). Downstream: theta functions on the reconstructed mirror (Gross-Hacking-Keel 2015 Publ. IHÉS 122, surface case; [04.12.12]) form a canonical $\mathbb{Z}$-basis $\{{\vartheta }_m{\}}_{m\in B(\mathbb{Z})}$ of regular functions on the generic fibre, constructed via broken-line counts on $B$; mirror map (Auroux 2009 survey) relates the formal parameter $t$ to the Kähler class on the A-side mirror via period integrals; homological mirror conjecture (Kontsevich 1994) conjecturally identifies $D^b(\mathfrak{X})$ with the Fukaya category of the A-side. Worked example: dim-1 case — circle of circumference $L$ with slab functions $f=1+z$ at the two zero-cells reconstructs the elliptic curve with period $L$. Worked example (heuristic): dim-2 K3 mirror — sphere $S^2$ with 24 focus-focus singularities (Gross-Wilson 2000 differential-geometric SYZ for K3) reconstructs the mirror K3 via Gross-Hacking-Keel 2015. Worked example (heuristic): quintic mirror — $S^3$ with 30 lines of singularities (Gross 2005) reconstructs the mirror quintic via the full Gross-Siebert algorithm. Generalises Batyrev 1994 J. Algebraic Geom. 3 — which works for reflexive polytopes and the toric case — to full Calabi-Yau case with substantive singular locus. Lean status: partial, module Codex.AlgGeom.Tropical.GrossSiebertReconstruction declares the Lattice, IntegralAffineManifold, PolyhedralDecomposition, TropicalManifold, SlabFunction, Wall, ScatteringDiagram, Structure_, ToricDegenerationOfCY, dualIntersectionComplex, slabGluingData structures and definitions, plus the named theorems gross_siebert_reconstruction (the central theorem, existence-and-uniqueness statement), reconstruction_is_inverse_of_dual_complex, theta_function_basis, slab_functions_are_tropical_disk_counts with placeholder proof bodies pending the integral-affine-manifold-with-singularities + Kato-fan + formal-scheme-deformation + scattering-algorithm + log-Gromov-Witten infrastructure in Mathlib.

  requires: toric-geometry.fan-and-toric-variety` (04.11.04); `tropical.nishinou-siebert-correspondence` (04.12.06); `tropical.toric-degeneration-calabi-yau` (04.12.07); `tropical.dual-intersection-complex` (04.12.08); slab function (04.12.11)

• tropical.strominger-yau-zaslow-conjectureopen unit 04.12.10 →

  Strominger-Yau-Zaslow (SYZ) conjecture

  The Strominger-Yau-Zaslow conjecture (Strominger-Yau-Zaslow 1996, Nucl. Phys. B 479) proposes a geometric reason for mirror symmetry: mirror Calabi-Yau pairs $(X,X^{\vee })$ admit dual special Lagrangian $T^n$-fibrations $X\to B$ and $X^{\vee }\to B$ over a common integral affine base $B$, with the mirror map realised fibrewise by T-duality. The 1996 paper argues from supersymmetric brane charges in Type IIA / Type IIB string theory: a D0-brane on $X$ (a point particle, with moduli space $X$) is mirror dual to a brane on $X^{\vee }$ wrapping a special Lagrangian torus $L\cong T^n$ with flat $U(1)$-connection $\mathcal{L}$; by McLean's 1998 theorem, the moduli space of pairs $(L,\mathcal{L})$ has real dimension $2b_1(L)=2n$, recovering $X$. The mirror is therefore the moduli space of branes — equivalently the dual torus fibration $X^{\vee }=\{(L,\mathcal{L}):L\subset X^{\vee }\text{ s.L. torus}\}$. Mathematical reformulations: (1) Hitchin 1997 Annali SNS Pisa 25 constructs the semi-flat metric on the smooth locus $B^{\circ }$ of the SYZ base, identifying the integral affine atlas (transition maps in $\mathrm{GL}(n,\mathbb{Z})⋉{\mathbb{R}}^n$) and the Hessian-metric form $g_B={\mathrm{Hess}}_g(\varphi )$ for a strictly convex potential $\varphi$. Ricci-flatness on the total space reduces to the real Monge-Ampère equation $\det ({\varphi }_{ij})=C$. Legendre duality $\check{\varphi }(y^{\vee })=\sum y_i{y}_i^{\vee }-\varphi (y)$ exchanges the original and mirror sides, preserving the Monge-Ampère form ($\det ({\check{\varphi }}_{ij})=1/C$). (2) Gross-Wilson 2000 J. Diff. Geom. 55 proves SYZ for K3 surfaces: the Ricci-flat Kähler-Einstein metric on a polarised K3 collapses (Gromov-Hausdorff) at a large complex structure limit to a metric on $S^2\setminus \{24\text{ points}\}$ with 24 focus-focus singularities; this is the K3's SYZ base. (3) Kontsevich-Soibelman 2001 + 2006 reformulate SYZ via Berkovich skeletons: the analytic skeleton $\mathrm{Sk}(X)\subset X^{\mathrm{an}}$ of a Calabi-Yau over $K=\mathbb{C}\{\{t\}\}$ carries a canonical integral affine structure; the mirror has the same skeleton with Legendre-dual integral affine structure. (4) Joyce 2003 Comm. Anal. Geom. 11 classifies generic singular fibres of dimension-3 special Lagrangian fibrations; the discriminant locus is a real one-dimensional graph in the base. (5) Gross-Siebert 2006 J. Algebraic Geom. 15 + 2011 Annals 174 realise SYZ algebraically: the dual intersection complex [04.12.08] of a toric degeneration of a Calabi-Yau is the SYZ base, the integral affine structure is the dual cone-glueing structure, and the mirror is reconstructed by Legendre duality on the affine base together with a structure of slab functions. The Gross-Siebert programme is a constructive proof of SYZ in the algebraic setting. (6) Auroux 2007 extends SYZ to Fano targets (Landau-Ginzburg mirrors) via Lagrangian wall-crossing. Worked examples: $T^2=S^1\times S_r^1$ has SYZ mirror $(T^2{)}^{\vee }=S^1\times S_{1/r}^1$ (T-duality on one circle factor); K3 self-mirror is realised by 24-point-singular $T^2$-fibration over $S^2$. Status: the conjecture is proved in dimension two (Gross-Wilson 2000), partially proved algebraically (Gross-Siebert programme), and open in full generality for Calabi-Yau threefolds. Cross-pointer: the integral affine base $B$ of the SYZ conjecture is the same combinatorial object as the dual intersection complex [04.12.08] of a toric degeneration [04.12.07], and tropical curves on this base from [04.12.05]/[04.12.06] are the perturbative wall-crossing data of the Gross-Siebert mirror reconstruction [04.12.09]. Lean status: partial, module Codex.AlgGeom.Tropical.StromingerYauZaslow declares CalabiYau, IntegralAffineBase, SpecialLagrangianFibration, MirrorPair, IsMirror, plus the named theorem syz_conjecture and the supporting legendre_duality_preserves_monge_ampere and TorusFibre.dual_dual with placeholder proof bodies pending the Calabi-Yau / special Lagrangian / integral affine manifold API in Mathlib.

  requires: toric-geometry.fan-and-toric-variety` (04.11.04); `tropical.curve-balanced-metric-graph` (04.12.02); `symplectic-geometry.lagrangian-submanifold` (05.05.01)

• toric-geometry.orbit-cone-correspondenceopen unit 04.11.06 →

  Orbit-cone correspondence

  For a toric variety $X_{\Sigma }$ associated to a fan $\Sigma$ in $N_{\mathbb{R}}$ of rank $n$ with dense torus $T=N\otimes {\mathbb{C}}^{\ast }$, the orbit-cone correspondence is the bijection $\sigma \mapsto O(\sigma )$ between cones of $\Sigma$ and $T$-orbits in $X_{\Sigma }$, where $O(\sigma ):={\mathrm{Hom}}_{\mathbb{Z}}({\sigma }^{\perp }\cap M,{\mathbb{C}}^{\ast })$ is an algebraic torus of dimension $n-\dim \sigma$. The dimension formula $\dim O(\sigma )+\dim \sigma =n$ reduces to the rank duality $\mathrm{rk}({\sigma }^{\perp }\cap M)=n-\dim \sigma$ under the perfect pairing $M\times N\to \mathbb{Z}$. The bijection is order-reversing: $\sigma ⪯\tau$ iff $O(\tau )\subseteq V(\sigma )=\overline{O(\sigma )}$, with closure decomposition $V(\sigma )={⨆}_{\tau ⪰\sigma }O(\tau )$. The orbit closure $V(\sigma )$ is itself a toric variety with dense torus $O(\sigma )$, character lattice $M(\sigma )={\sigma }^{\perp }\cap M$, cocharacter lattice $N(\sigma )=N/N_{\sigma }$, and fan the star quotient $\Sigma /\sigma =\{\bar{\tau }:\tau \in \mathrm{Star}(\sigma )\}$ in $N(\sigma {)}_{\mathbb{R}}$, where $\bar{\tau }$ is the image of $\tau$ under the quotient $N_{\mathbb{R}}\to N(\sigma {)}_{\mathbb{R}}$. The toric stratification $X_{\Sigma }={⨆}_{\sigma \in \Sigma }O(\sigma )$ is a special case of the Białynicki-Birula decomposition (BB 1973) for algebraic-torus actions on smooth complete varieties. Worked examples: ${\mathbb{P}}^n$ has $2^{n+1}-1$ orbits (binomial-coefficient breakdown by dimension); ${\mathbb{P}}^2$ has $7$ orbits (one dense torus, three one-dim, three fixed points); ${\mathbb{P}}^1\times {\mathbb{P}}^1$ has $9$ orbits ($3\times 3$ from the product of factors); Hirzebruch surface $F_a$ has $9$ orbits for every $a\in \mathbb{N}$ (uniform combinatorial type of the fan, with $a$-dependence in intersection numbers $D_{{\rho }_2}\cdot D_{{\rho }_4}=a$). Proof structure: cone-by-cone reduction to the affine chart $U_{\sigma }$, where the $T$-orbits are in bijection with faces of $\sigma$ via the distinguished-point realisation $x_{\tau }$ with ${\chi }^m(x_{\tau })=1$ iff $m\in {\tau }^{\perp }\cap M$; chart gluing glues the local correspondences into a global cone-to-orbit bijection. Originator chain: Demazure 1970 (orbit stratification of $X_{\Sigma }$ in the originating fan paper); Sumihiro 1974 (equivariant-covering theorem reducing global orbit-cone to affine case); Danilov 1978 (canonical pre-Fulton English exposition, star-quotient construction); Fulton 1993 (modern textbook); Cox-Little-Schenck 2011 (thousand-page treatment). General-equivariant context: Białynicki-Birula 1973 (BB decomposition for algebraic-torus actions). Symplectic-side counterpart: Atiyah-Guillemin-Sternberg 1982 convexity (moment-map image is convex polytope; for Kähler toric manifolds, polytope is the lattice polytope whose normal fan is $\Sigma$); Delzant 1988 (symplectic classification of toric manifolds by Delzant polytopes). Downstream applications: toric divisors as orbit closures of rays (04.11.08); toric Picard group as quotient of ${\mathbb{Z}}^{\Sigma (1)}$ (04.11.09); polytope-fan dictionary (04.11.10); algebraic moment map (04.11.11); Danilov-Jurkiewicz cohomology $H^{\ast }(X_{\Sigma };\mathbb{Q})=\mathbb{Q}[D_{\rho }]/(\text{linear}+\text{Stanley-Reisner})$ via toric stratification Euler-characteristic identity ${\chi }_{\mathrm{top}}(X_{\Sigma })=|{\Sigma }_{\max }|$ (04.11.12). Lean status: partial, module Codex.AlgGeom.Toric.OrbitConeCorrespondence declares the placeholder RationalPolyhedralCone, Fan, ToricVariety, Orbit structures, the bijection theorem orbit_cone_bijection, the dimension formula orbit_dim_plus_cone_dim, the star-quotient closure theorem closure_is_toric_with_star_quotient, the stratification theorem orbit_stratification, the order-reversing face-containment theorem orbit_closure_face_containment, and decidable numerical witnesses orbit_count_P2 : projectiveOrbitCount 2 = 7, orbit_count_P1xP1 : p1xp1OrbitCount = 9, orbit_count_Hirzebruch : hirzebruchOrbitCount a = 9 with the consistency check hirzebruch_zero_eq_P1xP1 : hirzebruchOrbitCount 0 = p1xp1OrbitCount, all with sorry-stubbed proof bodies pending the upstream Fan and ToricVariety formalism in Mathlib.

  requires: toric-geometry.algebraic-torus-character-lattices` (04.11.01); `toric-geometry.fan-and-toric-variety` (04.11.04)

• toric-geometry.smoothness-completeness-fansopen unit 04.11.05 →

  Smoothness and completeness via fans

  Two named criteria on a fan $\Sigma$ classify the local and global geometry of the toric variety $X_{\Sigma }$. Smoothness criterion (Demazure 1970 Théorème 4, Fulton 1993 §2.1): $X_{\Sigma }$ is smooth as a $\mathbb{C}$-scheme iff every cone $\sigma \in \Sigma$ is smooth — equivalently, the primitive ray generators of each $\sigma$ extend to a $\mathbb{Z}$-basis of $N$. Proof factors through the affine smoothness theorem ($U_{\sigma }$ smooth iff $\sigma$ smooth) plus the locality of smoothness on a scheme: the affine direction is via the dual-basis decomposition of the semigroup, $S_{\sigma }={\mathbb{Z}}_{\ge 0}^k\oplus {\mathbb{Z}}^{n-k}$, giving $\mathbb{C}[S_{\sigma }]=\mathbb{C}[x_1,\dots ,x_k,x_{k+1}^{\pm },\dots ,x_n^{\pm }]$, a regular ring, so $U_{\sigma }\cong {\mathbb{A}}^k\times {\mathbb{G}}_m^{n-k}$; the converse direction goes through the Hilbert-basis-equals-cotangent-dimension identification, forcing the Hilbert basis size to equal $n$, which forces unimodularity of the ray-generator matrix. Completeness criterion (Fulton 1993 §2.4, Cox-Little-Schenck Theorem 3.4.6, Oda Theorem 1.11): $X_{\Sigma }$ is complete (proper over $\mathrm{Spec}\mathbb{C}$, equivalently compact in the classical topology) iff $|\Sigma |=N_{\mathbb{R}}$. Proof via the valuative criterion for properness applied to torus-valued points: a morphism $\mathrm{Spec}K\to T\subseteq X_{\Sigma }$ corresponds to a homomorphism $\varphi :M\to K^{\ast }$; composing with the DVR's discrete valuation gives a cocharacter $v_{\varphi }\in N$; the extension to $\mathrm{Spec}R$ exists iff $v_{\varphi }\in \sigma$ for some $\sigma \in \Sigma$, iff $v_{\varphi }\in |\Sigma |$; ranging over DVRs sweeps $v_{\varphi }$ across all of $N$, so properness iff $|\Sigma |\supseteq N$, iff $|\Sigma |=N_{\mathbb{R}}$ (closure of $N$ inside $N_{\mathbb{R}}$). Worked examples covering the four cases of (smooth/non-smooth) × (complete/non-complete): ${\mathbb{P}}^n$ smooth complete (fan with $n+1$ rays, primitive generators $-e_1-\cdots -e_n$ and $e_1,\dots ,e_n$, all maximal cones unimodular); ${\mathbb{P}}^1\times {\mathbb{P}}^1$ smooth complete (four-quadrant fan in ${\mathbb{R}}^2$, every cone unimodular); Hirzebruch surface $F_a$ smooth complete for every $a\ge 0$ (four-ray tilted fan, every cone has unit determinant by direct check, with $F_0={\mathbb{P}}^1\times {\mathbb{P}}^1$); weighted projective space $\mathbb{P}(1,2,3)$ complete but not smooth (three rays with non-unimodular cones, cyclic quotient singularities $(1/2)(1,1)$ and $(1/3)(1,2)$ at two of the three torus-fixed points, $\mathbb{Q}$-factorial); affine line $\mathbb{C}$ smooth but not complete (single ray, support ${\mathbb{R}}_{\ge 0}\ne \mathbb{R}$); the cone $\mathrm{Cone}(e_1,2e_1+3e_2)$ gives the canonical simplicial-but-not-smooth example with $A_2$-type cyclic quotient. Connections: orbit-cone correspondence (04.11.06) extends the criteria to the orbit stratification; toric resolution of singularities (04.11.07) refines non-smooth fans to smooth refinements; toric Picard group (04.11.09) shows the difference between Picard and class groups vanishes for smooth $X_{\Sigma }$; polytope-fan correspondence (04.11.10) adds the projectivity criterion strengthening completeness; minimal model program for toric varieties (04.11.13) uses simplicial-but-not-smooth fans ($\mathbb{Q}$-factorial inputs) as the MMP setting; symplectic toric manifolds via Delzant (05.09.01) gives the symplectic-side smoothness condition matching the algebraic one. Lean status: partial, module Codex.AlgGeom.Toric.SmoothnessCompletenessFans declares the Lattice, RationalPolyhedralCone, Fan, AffineToricVariety, ToricVariety structures (placeholder), the predicates RationalPolyhedralCone.Smooth, Fan.IsComplete, Fan.IsSmooth, and the named theorems affineToricVariety_smooth_iff_cone_smooth, toricVariety_complete_iff_support_full, toricVariety_smooth_iff_all_cones_smooth, with explicit worked-example witnesses for ${\mathbb{P}}^n$, ${\mathbb{P}}^1\times {\mathbb{P}}^1$, $F_a$, $\mathbb{P}(1,2,3)$, and ${\mathbb{A}}^1$, all with sorry-stubbed proof bodies pending Mathlib's cone-and-fan formalism.

  requires: toric-geometry.rational-polyhedral-cone-dual-cone` (04.11.02), toric-geometry.affine-toric-variety` (04.11.03), toric-geometry.fan-and-toric-variety` (04.11.04)

• toric-geometry.picard-groupopen unit 04.11.09 →

  Toric Picard group

  The toric Picard group of $X_{\Sigma }$ is governed by the toric Picard exact sequence (Demazure 1970, Fulton §3.4): $0\to M\stackrel{\mathrm{div}}{\to }{⨁}_{\rho \in \Sigma (1)}\mathbb{Z}\cdot D_{\rho }\to \mathrm{Cl}(X_{\Sigma })\to 0$, with $\mathrm{div}(m)={\sum }_{\rho }⟨m,v_{\rho }⟩D_{\rho }$ encoding the divisor of the rational function ${\chi }^m$ on the torus. Surjectivity uses that every Weil divisor on a normal $T$-variety is linearly equivalent to a torus-invariant one (averaging over the dense torus). Injectivity of $\mathrm{div}$ holds when the rays span $N_{\mathbb{R}}$ (in particular for any complete fan), by non-degeneracy of the perfect pairing $M\times N\to \mathbb{Z}$. The image of $\mathrm{div}$ is exactly the principal torus-invariant divisors, by $T$-semi-invariance of any function with $T$-invariant divisor. Smooth complete case (Demazure 1970, Fulton §3.4 rank formula): when $X_{\Sigma }$ is smooth, $\mathrm{Pic}(X_{\Sigma })=\mathrm{Cl}(X_{\Sigma })$ (regular local rings are UFDs, so every Weil is Cartier) and the cokernel is free abelian of rank $|\Sigma (1)|-n$ via Smith normal form (every maximal cone has primitive generators forming a $\mathbb{Z}$-basis of $N$, so the elementary divisors of the divisor matrix are all 1). Singular case: $\mathrm{Pic}(X_{\Sigma })\hookrightarrow \mathrm{Cl}(X_{\Sigma })$ as a finite-index sublattice (index = lcm of stabiliser orders at singular fixed points). Worked examples: $\mathrm{Pic}({\mathbb{P}}^n)=\mathbb{Z}⟨H⟩$ (rank $1=(n+1)-n$); $\mathrm{Pic}({\mathbb{P}}^1\times {\mathbb{P}}^1)={\mathbb{Z}}^2$ with two fibre classes (rank $2=4-2$); $\mathrm{Pic}(F_a)=\mathbb{Z}⟨F⟩\oplus \mathbb{Z}⟨S⟩$ for Hirzebruch surface with $F^2=0,F\cdot S=1,S^2=-a$ (rank 2 independent of $a$); $\mathrm{Pic}({\mathrm{Bl}}_p{\mathbb{P}}^2)=\mathbb{Z}⟨H⟩\oplus \mathbb{Z}⟨E⟩$ with $E^2=-1$ (rank $2=4-2$); $\mathrm{Cl}(\mathbb{P}(1,2,3))=\mathbb{Z}$, $\mathrm{Pic}(\mathbb{P}(1,2,3))=6\mathbb{Z}$ (index 6 = lcm). Ample / nef cones via support functions (Demazure 1970, Cox-Little-Schenck §6.1 Theorem 6.1.14): a torus-invariant Cartier divisor $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$ is ample iff its piecewise-linear support function ${\psi }_D:|\Sigma |\to \mathbb{R}$ (defined by ${\psi }_D(v_{\rho })=-a_{\rho }$, extended linearly on maximal cones) is strictly convex; nef iff merely convex; the nef cone is the closure of the ample cone in $\mathrm{Pic}(X_{\Sigma }{)}_{\mathbb{R}}$, dual to the Mori cone $\overline{\mathrm{NE}}(X_{\Sigma })$ of effective toric curves under intersection pairing. Mori dream space (Hu-Keel 2000 Theorem 2.3): every smooth complete toric variety has finitely generated Cox ring (= polynomial ring $\mathbb{C}[x_{\rho }:\rho \in \Sigma (1)]$ graded by $\mathrm{Pic}$ via the Picard exact sequence), with the moveable cone $\overline{\mathrm{Mov}}(X_{\Sigma })$ admitting a finite chamber decomposition into closures of nef cones of small $\mathbb{Q}$-factorial modifications (= GIT chambers of the Cox quotient). The Cox-ring presentation (Cox 1995): $X_{\Sigma }=({\mathbb{C}}^{\Sigma (1)}\setminus Z(\Sigma ))/G$ for $G=\mathrm{Hom}(\mathrm{Pic},{\mathbb{C}}^{\ast })$ acting via the dual of the Picard exact sequence, $Z(\Sigma )$ the irrelevant locus. Downstream connections: polytope-fan dictionary (04.11.10) realises projective $X_{\Sigma }$ as the toric variety of a lattice polytope with explicit ample-bundle polarisation; toric cohomology (04.11.12) presents the Chow ring as Stanley-Reisner modulo the Picard linear relations; toric Hirzebruch-Riemann-Roch (04.11.13) computes Euler characteristics of line bundles via lattice-point counts in polytopes (Brion-Vergne 1997). Lean status: partial, module Codex.AlgGeom.Toric.PicardGroup declares the Lattice, Ray, Fan, ZSigmaOne, CharacterLattice, Pairing, divMap, PicardGroup, SupportFunction, StrictlyConvex, Convex placeholder structures, the named theorems toric_picard_exact_sequence, pic_rank_smooth_complete, ample_iff_strictly_convex, nef_iff_convex, smooth_complete_toric_is_mds, cox_quotient_presentation, moveable_cone_chamber_decomposition, and decidable numerical witnesses pic_projective_space_rank, pic_p1_times_p1_rank, pic_hirzebruch_rank, pic_blowup_p2_rank, cl_weighted_projective_123_rank, pic_weighted_projective_123_index (the last decidably $\mathrm{lcm}(1,2,3)=6$), with sorry-stubbed and trivial-stubbed proof bodies pending the upstream Fan formalism from 04.11.04 plus Mathlib's piecewise-linear-function and GIT-quotient infrastructure.

  requires: toric-geometry.fan-and-toric-variety` (04.11.04); `alg-geom.ample-line-bundle` (04.05.05)

• toric-geometry.polytope-fan-dictionaryopen unit 04.11.10 →

  Polytope-fan dictionary; the line bundle $L_P$

  The polytope-fan dictionary (Demazure 1970, Fulton 1993 §1.5 + §3.4, Cox-Little-Schenck §6.1-§6.2) gives a categorical equivalence between full-dimensional lattice polytopes $P\subseteq M_{\mathbb{R}}$ (up to lattice translation) and pairs $(X,L)$ of projective normal toric varieties with ample $T$-equivariant line bundles. The forward construction: $P\mapsto {\Sigma }_P$ (normal fan, inward-normal cones at vertices), $P\mapsto X_P=X_{{\Sigma }_P}$ (toric variety via fan-gluing of 04.11.04), $P\mapsto L_P$ (polarisation defined locally by ${\chi }^v$ on $U_{{\sigma }_v}$ with transitions ${\chi }^{v-v^{\prime }}$). The cocycle condition holds tautologically; ampleness of $L_P$ equals strict convexity of the support function ${\psi }_P(u)={\min }_{m\in P}⟨m,u⟩$ on ${\Sigma }_P$ (CLS Theorem 6.1.14). The reverse construction: $(X,L)\mapsto P_L=\mathrm{conv}(\mathrm{supp}(L))$ where $\mathrm{supp}(L)=\{m\in M:H^0(X,L{)}_m\ne 0\}$ is the $T$-character support; strict convexity from ampleness ensures the vertex cones of $P_L$ are exactly the maximal cones of $\Sigma$ (CLS Theorem 6.2.1). Demazure character formula (the structural heart): $H^0(X_P,L_P)={⨁}_{m\in P\cap M}\mathbb{C}\cdot {\chi }^m$ as a $T$-representation, with weight $m$ appearing with multiplicity one for $m\in P\cap M$. In particular $\dim H^0(X_P,L_P)=|P\cap M|$ — sheaf cohomology becomes lattice-point counting. Higher cohomology vanishes: $H^i(X_P,L_P)=0$ for $i\ge 1$ by the toric Kodaira-Demazure vanishing theorem (Mustață 2002 Tohoku Math. J. 54). Ehrhart polynomial: for $k\ge 1$, $L_{kP}=L_P^{\otimes k}$ and $\dim H^0(X_P,L_P^{\otimes k})=|kP\cap M|=:L(P,k)$ — a polynomial of degree $n=\dim P$ with leading coefficient $\mathrm{vol}(P)$ (Ehrhart 1962 C. R. Acad. Sci. 254), matching the Hilbert polynomial of $X_P$ under the polarisation; Brion-Vergne 1997 J. Amer. Math. Soc. 10 extends to higher cohomology and arbitrary toric divisors via fixed-point residue formulas. Worked examples: standard $n$-simplex gives $({\mathbb{P}}^n,\mathcal{O}(1))$ with $|P\cap M|=n+1$; unit cube $[0,1{]}^n$ gives $(({\mathbb{P}}^1{)}^n,\mathcal{O}(1{)}^{⊠n})$ with $|P\cap M|=2^n$; $a\times b$ rectangle gives $({\mathbb{P}}^1\times {\mathbb{P}}^1,\mathcal{O}(a,b))$ with $|P\cap M|=(a+1)(b+1)$; trapezoid with vertices $(0,0),(1,0),(1,1),(0,2)$ gives Hirzebruch surface $F_1$ with $L_P=F+2\sigma$ in $\mathrm{Pic}(F_1)$; cross-polytope and unit square share a normal fan in ${\mathbb{R}}^2$ (both give ${\mathbb{P}}^1\times {\mathbb{P}}^1$) but in higher dim the cross-polytope gives singular toric Calabi-Yau threefolds (conifold-transition prototype); weighted projective planes $\mathbb{P}(a,b,c)$ arise from triangles with vertices $(0,0),(a,0),(0,b)$ after appropriate rescaling. Mirror symmetry via reflexive polytopes (Batyrev 1994 J. Algebraic Geom. 3): a lattice polytope $P$ is reflexive if it contains the origin in its interior and the polar dual $P^{\circ }=\{u:⟨m,u⟩\ge -1\text{ for }m\in P\}$ is also a lattice polytope; polar duality is an involution $(P^{\circ }{)}^{\circ }=P$, and for reflexive $P$ the anticanonical class $-K_{X_P}=L_P$ defines a Calabi-Yau hypersurface family with mirror in $X_{P^{\circ }}$; the 473,800,776 reflexive 4-polytopes classified by Kreuzer-Skarke 2000-2002 (Adv. Theor. Math. Phys. 4 + Comm. Math. Phys. 230) account for the bulk of explicitly known mirror Calabi-Yau 3-folds; Batyrev-Borisov 1996 Higher-dimensional complex varieties (Trento) extends to complete intersections via nef-partitions. Symplectic-side dovetail: Delzant 1988 Bull. SMF 116 classifies compact symplectic toric manifolds by Delzant polytopes (simple, rational, smooth), and the Delzant correspondence overlaps the polytope-fan dictionary on the projective-toric stratum, with the moment-map image $\mu (X_P)=P$ (Atiyah 1982; Guillemin-Sternberg 1982 convexity). Cox-ring reframing (Cox 1995 J. Algebraic Geom. 4): $X_P$ presented as a GIT quotient with $L_P$ a character in the Cox-ring grading; this unifies projective and toric geometry under a single functorial framework foundational for modern computational toric algorithms (Macaulay2, Polymake, SageMath) and for non-projective toric extensions via GIT chambers. Connections: fan-and-toric-variety (04.11.04) supplies the variety side; toric-divisor-and-support-function (04.11.08) decomposes $L_P$ as $\sum a_{\rho }{D}_{\rho }$ over rays; orbit-cone correspondence (04.11.05) stratifies $X_P$ by faces of $P$; toric Picard group (04.11.09) sits the polarisation in the ample cone; symplectic Delzant (05.04.05) closes the symplectic-algebraic loop; Gross-Siebert (04.12.09) generalises Batyrev to arbitrary Calabi-Yau pairs via tropical scattering diagrams. Lean status: partial, module Codex.AlgGeom.Toric.PolytopeFanDictionary declares LatticePolytope, NormalFan, PolarisedToric, the construction PolarisedToric.ofPolytope, the combinatorial Demazure formula dim_global_sections : (PolarisedToric.ofPolytope P).h0LP = P.numLatticePoints (proved by rfl), and stubs polytope_to_projective_toric, projective_toric_to_polytope, simplex_gives_projective_space, square_gives_p1_times_p1, dilation_gives_tensor_power, demazure_character_formula, batyrev_mirror_pair — all with sorry-equivalent proof bodies (or trivial on True statements) pending the toric-variety + ample-line-bundle + sheaf-cohomology infrastructure missing from Mathlib.

  requires: toric-geometry.algebraic-torus-character-lattices` (04.11.01), toric-geometry.rational-polyhedral-cone-dual` (04.11.02), toric-geometry.affine-toric-variety` (04.11.03), toric-geometry.fan-and-toric-variety` (04.11.04)

• toric-geometry.algebraic-moment-map-polytopeopen unit 04.11.11 →

  Algebraic moment map and the polytope

  The algebraic moment map promotes the polytope-fan dictionary (04.11.10) from a static combinatorial encoding of $(X_P,L_P)$ to a dynamic geometric drawing: the polytope $P$ is literally drawn inside $M_{\mathbb{R}}$ as the continuous image of $X_P$ under the compact-torus moment map. Demazure embedding (Fulton §4.2): the lattice-point basis $H^0(X_P,L_P)={⨁}_{m\in P\cap M}\mathbb{C}\cdot s_m$ from 04.11.10 produces the closed immersion ${\iota }_P:X_P\hookrightarrow {\mathbb{P}}^{|P\cap M|-1}$ sending $x\mapsto [s_m(x){]}_m$. Algebraic moment map (Fulton §4.2; Cox-Little-Schenck §12.2): ${\mu }_T(x):={\sum }_{m\in P\cap M}(|s_m(x){|}^2/{\sum }_{m^{\prime }}|s_{m^{\prime }}(x){|}^2)\cdot m$, the barycentric lattice-point formula on the Demazure embedding. Equivalently, the symplectic moment map of the compact-torus action with respect to the pulled-back Fubini-Study form ${\omega }_P={\iota }_P^{\ast }{\omega }_{\mathrm{FS}}$. Image-equals-polytope theorem (Atiyah 1982 + Guillemin-Sternberg 1982; Fulton §4.2): ${\mu }_T(X_P(\mathbb{C}))=P$ on the nose, with: vertices $v\in \mathrm{Vert}(P)$ in bijection with $T$-fixed points $p_v$ via ${\mu }_T(p_v)=v$; fibres over interior points of a face $F\subseteq P$ equal to single compact-torus orbits inside the algebraic orbit $O_F$ from the orbit-cone correspondence (04.11.06); fibres over interior points of $P$ equal to top-dimensional compact tori $(S^1{)}^n$; image equal to the convex hull of vertex images via AGS convexity. Worked examples: on ${\mathbb{P}}^2$ with the unit simplex, ${\mu }_T([X_0:X_1:X_2])=(|X_1{|}^2,|X_2{|}^2)/(\sum |X_j{|}^2)$ sends fixed points to vertices and the open orbit to the interior of the triangle; on ${\mathbb{P}}^1\times {\mathbb{P}}^1$ with the unit square, the moment map is the product of two ${\mathbb{P}}^1$-moment-maps; on Hirzebruch $F_1$ with the trapezoid polytope, the four $T$-fixed points map to the four vertices of the trapezoid. Kempf-Ness theorem (Kempf-Ness 1979; Cox 1995; MFK 1994 §8.2): the GIT quotient $X_P=({\mathbb{C}}^{|{\Sigma }_P(1)|}\setminus Z({\Sigma }_P))/{/}_{\chi }G$ from the Cox construction is homeomorphic to the symplectic quotient ${\mu }_G^{-1}(c)/G_{\mathbb{R}}$, with the residual compact-torus action on the quotient having moment map ${\mu }_T:X_P\to P$ — the algebraic moment map of the present unit. The polytope $P$ enters both sides: as the lattice-point support of $L_P$ on the GIT side, as the image of the residual moment map on the symplectic side. Image-of-the-moment-map theorem for general projective varieties (Brion 1987 Bull. SMF 115): for any projective variety $X\subset {\mathbb{P}}^N$ with linearised $T$-action, the moment-map image is a convex rational polytope, the moment polytope of $X$; for toric $X=X_P$ this recovers $P$, for non-toric $X$ produces a new combinatorial invariant. Morse theory of $|{\mu }_T{|}^2$ (Kirwan 1984; Atiyah-Bott 1984): the squared-norm $|{\mu }_T{|}^2:X_P\to {\mathbb{R}}_{\ge 0}$ is a Morse-Bott function with critical set the $T$-fixed points; generic linear functionals $\xi \circ {\mu }_T$ are Morse functions producing the Białynicki-Birula cell decomposition, with cells indexed by vertices of $P$ and dimensions read off the polytope. Duistermaat-Heckman in the toric case (Duistermaat-Heckman 1982; Fulton §4.4): $({\mu }_T{)}_{\ast }({\omega }_P^n/n!)=(2\pi {)}^n{1}_P\cdot dm$, constant Lebesgue density on $P$ (the simplest case of the piecewise-polynomial DH density theorem), matching the symplectic volume $(2\pi {)}^n\mathrm{vol}(P)$ with the algebraic degree $n!\mathrm{vol}(P)$ via Bernstein-Kushnirenko. Reflexive-polytope Calabi-Yau (Batyrev 1994; SYZ 1996): for reflexive $P$ with $0\in \mathrm{int}(P)$, the generic anticanonical hypersurface $V\subset X_P$ is Calabi-Yau and the restricted moment map ${\mu }_T{|}_V:V\to \partial P$ has image the boundary of the polytope, with the interior accounting for the deformation parameter; the SYZ duality realises Batyrev mirror pairs as dual special-Lagrangian torus fibrations over the common boundary structure. Delzant correspondence overlap (Delzant 1988; cross-reference 05.04.04): smooth projective toric varieties with their Fubini-Study polarisation correspond to Delzant polytopes via the moment-map identification, exactly dovetailing the polytope-fan dictionary with the symplectic Delzant classification on the projective-toric stratum. Connections: polytope-fan dictionary (04.11.10) supplies the Demazure character formula; fan-and-toric-variety (04.11.04) supplies the underlying scheme; orbit-cone correspondence (04.11.06) is the combinatorial input to the face-stratified fibration; algebraic-torus-character-lattices (04.11.01) supplies the polar lattice structure $M\leftrightarrow N$; symplectic moment map (05.04.01) and AGS convexity (05.04.03) supply the symplectic-side framework; Delzant theorem (05.04.04) supplies the symplectic-side toric classification; symplectic reduction (05.04.02) is the residual structure on the Kempf-Ness symplectic quotient; Duistermaat-Heckman (05.04.05) supplies the volume-form pushforward identity; reflexive polytopes (04.11.16) and Gross-Siebert (04.12.09) supply the Calabi-Yau and mirror-symmetric extensions. Lean status: none, no module declared. Mathlib has neither the projective-toric-variety infrastructure of 04.11.01-04.11.10 nor the symplectic moment-map calculus underlying Atiyah-Guillemin-Sternberg convexity, so the algebraic moment map is lean_status: none with the full apparatus deferred to a future Mathlib roadmap covering toric varieties, the Cox construction, Hermitian Kähler manifolds, Hamiltonian compact-torus actions, and the GIT-symplectic Kempf-Ness identification.

  requires: toric-geometry.algebraic-torus-character-lattices` (04.11.01); `toric-geometry.fan-and-toric-variety` (04.11.04); `toric-geometry.polytope-fan-dictionary` (04.11.10); `symplectic-geometry.moment-map` (05.04.01); `symplectic-geometry.ags-convexity` (05.04.03)

• toric-geometry.cohomology-smooth-completeopen unit 04.11.12 →

  Cohomology of a smooth complete toric variety

  For a smooth complete toric variety $X_{\Sigma }$ over $\mathbb{C}$ with rays $\Sigma (1)$, the rational cohomology ring is the Stanley-Reisner quotient $\mathbb{Q}[x_{\rho }:\rho \in \Sigma (1)]/(I_{\Sigma }+J_{\Sigma })$, where $I_{\Sigma }$ is generated by squarefree monomials $x_{{\rho }_1}\cdots x_{{\rho }_k}$ whose rays do not span a cone of $\Sigma$, and $J_{\Sigma }$ is generated by the linear forms ${\sum }_{\rho }⟨m,v_{\rho }⟩x_{\rho }$ for $m\in M$. The generator $x_{\rho }$ maps to $[D_{\rho }]\in H^2(X_{\Sigma };\mathbb{Q})$. For ${\mathbb{P}}^n$ this gives $\mathbb{Q}[H]/(H^{n+1})$; for ${\mathbb{P}}^1\times {\mathbb{P}}^1$ it gives $\mathbb{Q}[A,B]/(A^2,B^2)$. The projective proof uses the moment-map Morse decomposition and orbit closures; the general smooth complete case is Danilov-Jurkiewicz. Stanley's 1980 use of the hard Lefschetz theorem on projective toric varieties proves unimodality of the $h$-vector of a simplicial polytope.

  requires: toric-geometry.smoothness-completeness-fans, toric-geometry.orbit-cone-correspondence, toric-geometry.picard-group, toric-geometry.polytope-fan-dictionary, alg-geom.cohomology-projective, topology.poincare-duality

• toric-geometry.intersection-theory-mixed-volumesopen unit 04.11.13 →

  Toric intersection theory and mixed volumes

  For a smooth projective toric variety $X_{\Sigma }$ of dimension $n$ and ample torus-invariant divisors $D_{P_1},\dots ,D_{P_n}$ attached to lattice polytopes $P_1,\dots ,P_n$, the top intersection number is $D_{P_1}\cdots D_{P_n}=n!\mathrm{MV}(P_1,\dots ,P_n)$, where $\mathrm{MV}$ is the mixed volume with lattice normalisation. The self-intersection case is $D_P^n=n!\mathrm{Vol}(P)$. The formula follows by comparing the Hilbert polynomial $\chi (X,\mathcal{O}(k{D}_P))=|kP\cap M|$ with Riemann-Roch, and by polarising the homogeneous volume polynomial. It is the intersection-theoretic bridge from toric cohomology to Bernstein-Kushnirenko root counting.

  requires: toric-geometry.cohomology-smooth-complete, toric-geometry.polytope-fan-dictionary, toric-geometry.algebraic-moment-map-polytope

• toric-geometry.bernstein-kushnirenkoopen unit 04.11.14 →

  Bernstein-Kushnirenko theorem

  Bernstein-Kushnirenko theorem: for generic Laurent polynomials $f_1,\dots ,f_n$ on $({\mathbb{C}}^{\ast }{)}^n$ with Newton polytopes $P_1,\dots ,P_n$, the number of isolated common zeros in the dense torus is $n!\mathrm{MV}(P_1,\dots ,P_n)$, counted with multiplicity. Kushnirenko's special case uses one Newton polytope repeated, giving $n!\mathrm{Vol}(P)$. Classical Bezout is recovered when each $P_i$ is a degree-$d_i$ simplex, yielding $d_1\cdots d_n$. The toric proof compactifies the torus inside a smooth projective toric variety, interprets each Laurent polynomial as a section of a toric line bundle, and applies the mixed-volume intersection formula while checking that generic sections have no boundary solutions.

  requires: toric-geometry.intersection-theory-mixed-volumes, toric-geometry.polytope-fan-dictionary, tropical.newton-polytope-amoeba

• toric-geometry.reflexive-polytope-batyrev-mirroropen unit 04.11.16 →

  Reflexive polytope and Batyrev mirror duality

  A full-dimensional lattice polytope $P\subset M_{\mathbb{R}}$ with $0\in \mathrm{int}(P)$ is reflexive when its polar $P^{\circ }=\{u\in N_{\mathbb{R}}:⟨m,u⟩\ge -1\text{ for all }m\in P\}$ is also a lattice polytope. Equivalently, every facet of $P$ is cut out by a primitive lattice normal at lattice distance one from the origin; equivalently, the normal toric variety is Gorenstein Fano and its anticanonical polytope is $P$. Batyrev's theorem pairs generic anticanonical Calabi-Yau hypersurfaces in $X_P$ and $X_{P^{\circ }}$, after crepant resolution when available, with exchanged Hodge numbers. In dimension four, Kreuzer-Skarke classified 473,800,776 reflexive polytopes, giving the standard large database of toric Calabi-Yau threefold hypersurfaces.

  requires: toric-geometry.polytope-fan-dictionary, toric-geometry.divisor-support-function, alg-geom.adjunction-formula, tropical.toric-degeneration-calabi-yau

• toric-geometry.divisor-support-functionopen unit 04.11.08 →

  Toric divisor and support function

  The toric divisor and support function form the combinatorial backbone of line-bundle theory on a toric variety. $T$-invariant Weil divisors are indexed by rays: for each ray $\rho \in \Sigma (1)$ with primitive generator $v_{\rho }\in N$, the toric divisor $D_{\rho }\subseteq X_{\Sigma }$ is the closure of the codimension-one torus orbit attached to $\rho$ by the orbit-cone correspondence (04.11.06), and every $T$-invariant Weil divisor is a finite integer combination $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$ over rays. Support function correspondence (Demazure 1970, Fulton §3.4 Proposition 1): the map $D\mapsto {\psi }_D$ is a bijection between $T$-invariant Cartier divisors on $X_{\Sigma }$ and integer-valued piecewise-linear functions on $|\Sigma |$ — continuous functions linear on each cone $\sigma$ with linear piece $m_{\sigma }\in M$ satisfying ${\psi }_D(v_{\rho })=-a_{\rho }$ at every ray and the Cartier-cocycle compatibility $m_{\sigma }-m_{\tau }\in (\sigma \cap \tau {)}^{\perp }$ across pairwise intersections. The proof uses that $T$-invariance forces local Cartier sections to be characters ${\chi }^{m_{\sigma }}$, with the cocycle compatibility equivalent to continuity of the resulting ${\psi }_D$. Smoothness criterion at divisor level (Fulton §3.4 Proposition 2): every $T$-invariant Weil divisor is Cartier iff $X_{\Sigma }$ is smooth, iff every cone $\sigma$ is unimodular (rays form part of a $\mathbb{Z}$-basis of $N$). Proof in the forward direction uses dual-basis decomposition on each cone to solve the Cartier system; reverse via exhibiting a non-Cartier Weil divisor at any non-smooth cone. $\mathbb{Q}$-Cartier refinement (Cox-Little-Schenck §4.2): on simplicial toric, every Weil divisor is $\mathbb{Q}$-Cartier with multiplier the cone multiplicity $\mathrm{mult}(\sigma )=|\det (v_{{\rho }_1},\dots ,v_{{\rho }_n})|$. Demazure global-sections formula (Fulton §3.4 Proposition 1; Cox-Little-Schenck Theorem 4.3.3): for a $T$-invariant Cartier divisor $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$, $H^0(X_{\Sigma },\mathcal{O}(D))={⨁}_{m\in P_D\cap M}\mathbb{C}\cdot {\chi }^m$, with $P_D=\{m\in M_{\mathbb{R}}:⟨m,v_{\rho }⟩\ge -a_{\rho }\text{ for all }\rho \in \Sigma (1)\}$ the divisor polytope. Proof via $T$-weight decomposition: $H^0$ decomposes into weight spaces, each at most one-dimensional, with ${\chi }^m$ extending to a section iff $\mathrm{div}({\chi }^m)+D\ge 0$, equivalent to $m\in P_D$. Demazure vanishing: $H^i(X_{\Sigma },\mathcal{O}(D))=0$ for $i\ge 1$ when $D$ is nef and $X_{\Sigma }$ smooth complete (Cox-Little-Schenck Theorem 9.2.3), reducing Hirzebruch-Riemann-Roch to lattice-point counting. Ehrhart polynomiality (Ehrhart 1962; Brion-Vergne 1997): for nef $D$, the Hilbert function $L(P_D,k)=|k{P}_D\cap M|$ is polynomial of degree $n$, recovering the Hilbert polynomial of the polarised toric variety. Worked anticanonical example (Fulton §3.5): on ${\mathbb{P}}^n$, $-K=D_0+\cdots +D_n=(n+1)H$ has polytope a size-$(n+1)$ simplex with $\left(\genfrac{}{}{0ex}{}{2n+1}{n}\right)$ lattice points — $10$ for ${\mathbb{P}}^2$ (plane cubics), $35$ for ${\mathbb{P}}^3$ (quartics in four variables), $126$ for ${\mathbb{P}}^4$. Hirzebruch surface anticanonical: $-K_{F_a}=D_1+(2-a)D_2+D_3+D_4$ is ample iff $a\le 1$, distinguishing del Pezzo ($F_0,F_1$) from non-Fano ($F_{\ge 2}$). Batyrev mirror Calabi-Yau (Batyrev 1994): anticanonical hypersurfaces in toric $X_P$ for reflexive polytope $P$ pair with mirror hypersurfaces in $X_{P^{\circ }}$ via polar duality, supplying the bulk of explicitly known Calabi-Yau 3-folds (473,800,776 reflexive 4-polytopes classified by Kreuzer-Skarke). Connections: fan-and-toric-variety (04.11.04) supplies the toric variety; orbit-cone correspondence (04.11.06) supplies the geometric realisation of $D_{\rho }$; toric Picard group (04.11.09) is the abelian-group avatar of the support-function calculus; polytope-fan dictionary (04.11.10) recovers $(X_{\Sigma },\mathcal{O}(D))$ from $P_D$ as projective toric variety with polarisation; algebraic moment map (04.11.11) is the symplectic-side polytope; toric cohomology (04.11.12) treats $D_{\rho }$ as degree-one generators of the Chow ring; Cox-ring and GIT quotient (04.11.15) reframe $D_{\rho }$ as monomial classes. Lean status: partial, module Codex.AlgGeom.Toric.DivisorSupportFunction declares Lattice, Ray, Fan, TInvariantWeilDivisor, SupportFunction, DivisorPolytope, anticanonicalDivisor, and named theorems support_function_correspondence, smooth_iff_all_weil_cartier, global_sections_lattice_points, with decidable numerical witnesses anticanonical_p2_h0_eq_ten ($\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10$), anticanonical_pn_count ($\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10,\left(\genfrac{}{}{0ex}{}{7}{3}\right)=35,\left(\genfrac{}{}{0ex}{}{9}{4}\right)=126$), o_d_on_p2_dimensions ($\left(\genfrac{}{}{0ex}{}{3}{2}\right)=3,\left(\genfrac{}{}{0ex}{}{4}{2}\right)=6,\left(\genfrac{}{}{0ex}{}{5}{2}\right)=10,\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$ for $d\in \{1,2,3,4\}$), and weighted_projective_123_cartier_multiplier ($\mathrm{lcm}(2,3)=6$). Sorry-stubbed proof bodies pending the upstream Fan formalism plus Cartier-divisor sheaf API in Mathlib.

  requires: toric-geometry.fan-and-toric-variety` (04.11.04); `toric-geometry.orbit-cone-correspondence` (04.11.06)

• toric-geometry.resolution-of-singularitiesopen unit 04.11.07 →

  Toric resolution of singularities

  The toric resolution theorem (Demazure 1970; KKMS 1973) asserts that every fan $\Sigma$ in $N_{\mathbb{R}}$ admits a smooth refinement ${\Sigma }^{\prime }$ — a fan with the same support whose every cone is generated by part of a $\mathbb{Z}$-basis of $N$ — obtainable by finitely many iterated star subdivisions along interior lattice points of non-smooth cones. The induced toric morphism $f:X_{{\Sigma }^{\prime }}\to X_{\Sigma }$ is a proper birational $T$-equivariant resolution of singularities: $X_{{\Sigma }^{\prime }}$ is smooth, $f$ is proper (supports agree), $f$ is birational (dense torus shared), $f$ is $T$-equivariant (induced by identity on $N$), and $f$ is an isomorphism over the smooth locus of $X_{\Sigma }$. The construction is algorithmic: define the cone multiplicity $\mathrm{mult}(\sigma )=|\det (u_1,\dots ,u_n)|$ on primitive ray generators (with $\mathrm{mult}(\sigma )=1$ iff $\sigma$ smooth), and the total multiplicity $M(\Sigma )={\sum }_{\sigma \in {\Sigma }_n}\mathrm{mult}(\sigma )$; each star subdivision along an interior lattice point of a non-smooth cone strictly decreases $M(\Sigma )-|{\Sigma }_n|$, forcing termination in finitely many steps. The toric setting is the only known case of resolution of singularities that is fully algorithmic in arbitrary characteristic — the general resolution theorem (Hironaka 1964) is a non-constructive Noetherian induction in characteristic zero only, and positive-characteristic resolution remains open in dimensions $\ge 4$. Hirzebruch-Jung algorithm (Hirzebruch 1953): for two-dimensional cyclic-quotient surface singularities ${\mathbb{C}}^2/{\mathbb{Z}}_p(1,q)$ presented as the cone $\sigma =\mathrm{Cone}(e_2,p{e}_1-q{e}_2)$ with $\gcd (p,q)=1$ and $0<q<p$, the negative continued-fraction expansion $p/q=[b_1,\dots ,b_s{]}^{-}=b_1-1/(b_2-1/(\cdots -1/b_s))$ with each $b_i\ge 2$ produces the smooth refinement and the exceptional divisor: a chain of $s$ rational curves $E_1,\dots ,E_s\cong {\mathbb{P}}^1$ intersecting transversely in a chain with self-intersections $E_i\cdot E_i=-b_i$ and adjacent intersections $E_i\cdot E_{i+1}=1$. Worked examples: $A_n$ singularity ${\mathbb{C}}^2/{\mathbb{Z}}_{n+1}(1,n)$ has continued-fraction expansion $[2,2,\dots ,2]$ of length $n$, yielding the $A_n$ Dynkin chain of $n$ rational $(-2)$-curves; ${\mathbb{C}}^2/{\mathbb{Z}}_5(1,2)$ has expansion $[3,2]$ yielding two exceptional curves of self-intersections $(-3,-2)$; ${\mathbb{C}}^2/{\mathbb{Z}}_7(1,3)$ has expansion $[3,2,2]$ yielding self-intersections $(-3,-2,-2)$. Star subdivision corresponds to $T$-equivariant blow-up of the closed $T$-orbit indexed by the cone being subdivided (Reid 1983); every $T$-equivariant proper birational morphism between $\mathbb{Q}$-factorial toric varieties is a composition of star subdivisions and their inverses, giving the toric Mori program (Reid 1983; Matsuki 2002). Crepant resolution criterion (Reid 1980): a star subdivision $\Sigma \to {\Sigma }^{\ast }(v)$ along a primitive lattice $v$ interior to a cone $\sigma$ is crepant ($f^{\ast }{K}_X=K_{X^{\prime }}$) iff $v$ lies on the affine hyperplane $H_{\sigma }$ spanned by the primitive ray generators of $\sigma$; for Gorenstein toric singularities ${\mathbb{C}}^n/G$ with $G\subset S{L}_n(\mathbb{C})$ finite, crepant resolutions exist in dimensions $\le 3$ and correspond to lattice triangulations of $P\cap H_{\sigma }$. The McKay correspondence (Reid 1980; Ito-Nakamura 1996; Bridgeland-King-Reid 2001) identifies cohomology classes of the crepant resolution ${\mathrm{Hilb}}^G({\mathbb{C}}^n)$ with irreducible representations of $G$, with the derived-equivalence statement $D^b({\mathrm{Coh}}^G({\mathbb{C}}^n))\cong D^b(\mathrm{Coh}Y)$ (BKR 2001) extending the dimension-2 classical McKay (Klein 1884; Brieskorn 1968 ADE). The toric resolution is the foundational toy model for every constructive resolution algorithm (Bierstone-Milman 1997; Encinas-Villamayor 2003; Włodarczyk 2005) and is the only known case algorithmic in positive characteristic (KKMS 1973). Lean status: partial, module Codex.AlgGeom.Toric.ResolutionSingularities declares the placeholder RationalPolyhedralCone, Fan, IsSmooth, Refines structures, the resolution theorem smooth_refinement_exists and toric_resolution, the Fan.starSubdivision operation with starSubdivision_refines, the negativeContinuedFraction function, the HirzebruchJungData structure, the hirzebruch_jung_resolution theorem, and the aN_HJ construction with witnesses aN_chain_length, aN_blocks_all_two, aN_is_crepant — with sorry-stubbed proof bodies pending the upstream cone-and-fan machinery in Mathlib.

  requires: toric-geometry.rational-polyhedral-cone-dual` (04.11.02); `toric-geometry.affine-toric-variety` (04.11.03); `toric-geometry.fan-and-toric-variety` (04.11.04)

• tropical.theta-function-polarised-manifoldopen unit 04.12.12 →

  Theta function of a polarised tropical manifold

  The theta-function construction of Gross-Hacking-Keel 2015 Publ. IHÉS 122 (surface case) and Carl-Pumperla-Siebert 2020 Forum Math. Sigma 8 (arbitrary dimension) supplies a canonical $\mathbb{Z}$-basis $\{{\vartheta }_m{\}}_{m\in B(\mathbb{Z})}$ of the polarised section ring of the smoothing ${\mathfrak{X}}_t$ reconstructed from a polarised tropical manifold $(B,\mathcal{P},\phi )$. The construction: given the canonical scattering diagram $\mathcal{S}$ of Gross-Siebert 2011, walls $\mathfrak{d}$ in $B$ carry wall functions $f_{\mathfrak{d}}=1+{\sum }_k{a}_{\mathfrak{d},k}{z}^{k{m}_{\mathfrak{d}}}$ with integer coefficients (from the Nishinou-Siebert curve counts of 04.12.06). A broken line with terminal direction $m$ ending at a generic test point $Q$ is a piecewise-linear path that bends at each wall crossing with monomial decoration picked from the wall function. The theta function ${\vartheta }_m(Q):={\sum }_{\beta }\mathrm{Mono}(\beta )$ is the broken-line sum. Theorem (canonical $\mathbb{Z}$-basis; Gross-Hacking-Keel 2015 §1, Carl-Pumperla-Siebert 2020 §4): $\{{\vartheta }_m:m\in B(\mathbb{Z})\}$ is a free $\mathbb{Z}[[t]]$-basis of $R(L)={⨁}_k{H}^0(\mathfrak{X},L^{\otimes k})$, with multiplication law ${\vartheta }_{p_1}\cdot {\vartheta }_{p_2}={\sum }_p{C}_{p_1,p_2}^p{\vartheta }_p$ and $C_{p_1,p_2}^p$ a non-negative integer (the broken-line triple count weighted by wall decorations). Fulton specialisation: in the rigid-toric case $(B,\mathcal{P},\phi )=(P,\{P\},{\phi }_P)$ with no walls, broken lines reduce to unbent straight lines and ${\vartheta }_m={\chi }^m$, exactly recovering the lattice-point monomial basis $H^0(X_P,L_P)={⨁}_{m\in P\cap M}\mathbb{C}\cdot {\chi }^m$ of Fulton 1993 §3.4 (cf. 04.11.10). The Gross-Hacking-Keel / Carl-Pumperla-Siebert construction is therefore the faithful Calabi-Yau generalisation of the polarised-toric polytope-to-sections theorem. Mandel 2019 log GW interpretation: the structure constants $C_{p_1,p_2}^p$ equal naive log Gromov-Witten invariants of ${\mathfrak{X}}_t$ — counts of stable log maps from three-marked rational curves with prescribed contact orders along the boundary. Mumford 1966 specialisation: for ${\mathfrak{X}}_t$ a totally degenerate abelian variety, the theta functions reduce to Mumford's classical theta functions with polarisation; the broken-line combinatorics replaces the classical Heisenberg-group computations. SYZ interpretation (Strominger-Yau-Zaslow 1996): theta functions on ${\mathfrak{X}}_t^{\vee }$ correspond, under fibrewise T-duality on the special-Lagrangian torus fibration, to canonical wavefunctions on ${\mathfrak{X}}_t$; structure constants are OPE coefficients. Cluster algebra interpretation (Mandel 2019; Fock-Goncharov): for cluster log Calabi-Yau varieties, the theta-function basis coincides with the Fock-Goncharov canonical basis, supplying the Fock-Goncharov positivity coefficients. Gross-Hacking-Keel-Siebert 2022 Mem. AMS 278/1367: the intrinsic mirror algebra $A(X,D):={⨁}_{m\in \mathrm{Int}(X,D)}\mathbb{Z}\cdot {\vartheta }_m$ with theta-function multiplication realises the mirror Calabi-Yau intrinsically from log Calabi-Yau data $(X,D)$. The audit-punch-list anchor is the Fulton 3.32 polarised-toric lattice-point basis (04.11.10); the present construction is the Calabi-Yau analogue, the explicit generalisation of the polytope-to-sections theorem to degenerate Calabi-Yau varieties. Lean status: partial, module Codex.AlgGeom.Tropical.ThetaFunction declares IntegralAffineManifold, PolyhedralDecomposition, Polarisation, PolarisedTropicalManifold, Wall, Structure, BrokenLine, thetaFunction, structureConstant, and the named theorems theta_canonical_basis, reduces_to_fulton_lattice_basis, theta_structure_constants_integer with placeholder proof bodies pending the polarised-tropical-manifold + broken-line + scattering-diagram + log Gromov-Witten API in Mathlib.

  requires: toric-geometry.polytope-fan-dictionary` (04.11.10 — the Fulton 3.32 lattice-point basis the construction generalises); `tropical.nishinou-siebert-correspondence` (04.12.06 — curve-count input to wall functions); `tropical.toric-degeneration-calabi-yau` (04.12.07); dual intersection complex (04.12.08); slab function (04.12.11)

• tropical.slab-function-structureopen unit 04.12.11 →

  Slab function and structure of a tropical manifold

  A polarised tropical manifold $(B,\mathcal{P},\phi )$ is an integral affine $n$-manifold with codimension-2 singularity locus $\Delta \subseteq B$, a polyhedral decomposition $\mathcal{P}$, and a strictly convex piecewise-linear polarisation $\phi$ (cf. 04.12.08). A slab is a codimension-1 face $\rho \in {\mathcal{P}}^{[n-1]}$ in the away-from-discriminant region $B\setminus \Delta$ separating two adjacent maximal cells ${\sigma }_{+},{\sigma }_{-}\in {\mathcal{P}}^{[n]}$, with common tangent integral lattice ${\Lambda }_{\rho }={\Lambda }_{{\sigma }_{+}}\cap {\Lambda }_{{\sigma }_{-}}\cong {\mathbb{Z}}^{n-1}$. A slab function on $\rho$ (Gross-Siebert 2010 JAG 19 §1) is a Laurent polynomial $f_{\rho }\in \mathbb{C}[t][{\Lambda }_{\rho }]=\mathbb{C}[t][z^m:m\in {\Lambda }_{\rho }]$ normalised by $f_{\rho }{|}_{t=0}=1$ and required to satisfy the positive-Newton condition (monomials at positive order in $t$ point outward against the slab orientation) and compatibility at every codimension-2 face of $\rho$ in $B\setminus \Delta$. A wall is a codimension-1 rational polyhedral subset $\mathfrak{d}$ in the interior of a maximal cell $\sigma$ together with a wall-crossing automorphism ${\theta }_{\mathfrak{d}}(z^m)=z^m\cdot f_{\mathfrak{d}}^{⟨n_{\mathfrak{d}},m⟩}$ of the truncated toric ring $\mathbb{C}[t][{\Lambda }_{\sigma }]/(t^N)$, for primitive normal $n_{\mathfrak{d}}\in {\Lambda }_{\sigma }^{\vee }$ and wall function $f_{\mathfrak{d}}\in 1+t\cdot \mathbb{C}[t][{\Lambda }_{\sigma }]$. A structure $\mathcal{S}=(\mathcal{F},\mathcal{W})$ on $(B,\mathcal{P},\phi )$ is a slab-function family $\mathcal{F}=\{f_{\rho }\}$ together with a locally finite wall family $\mathcal{W}=\{(\mathfrak{d},{\theta }_{\mathfrak{d}})\}$ satisfying the consistency condition at every codimension-2 cell $\tau$: the product of slab transitions and wall-crossing automorphisms around a small loop encircling $\tau$ in $B\setminus \Delta$ is the identity modulo $t^{N+1}$ for every $N\ge 1$. Consistency-of-structure theorem (Gross-Siebert 2010 JAG 19): a consistent structure $\mathcal{S}$ produces, for every $N\ge 1$, a flat $\mathbb{C}[t]/(t^{N+1})$-family of schemes assembling order-by-order into a formal flat family $\hat{\mathfrak{X}}\to \mathrm{Spf}\mathbb{C}[[t]]$ with central fibre $X_0(B,\mathcal{P},\phi )$. The proof has three movements: chartwise scheme structure from the toric ring $R_{\sigma }=\mathbb{C}[t][{\Lambda }_{\sigma }]$; codimension-1 gluing via $z_{-}=z_{+}^{-1}{f}_{\rho }{\pi }_{\rho }$ with ${\pi }_{\rho }=t^{{\kappa }_{\rho }}$ recording the polarisation jump; codimension-2 consistency via the wall-crossing automorphisms. Carl-Pumperla-Siebert algorithm (2010 arXiv:1011.6228): starting from initial slab functions ${\mathcal{F}}^{(1)}=\{1+t{\alpha }_{\rho }\}$ with ${\alpha }_{\rho }$ a sum of genus-0 log-GW counts on $X_0$ contributing to $\rho$, the algorithm iterates Kontsevich-Soibelman scattering at every codim-2 cell to assemble a consistent structure $\mathcal{S}={\underleftarrow{\lim }}_N({\mathcal{F}}^{(N)},{\mathcal{W}}^{(N)})$. Kontsevich-Soibelman scattering theorem (2006): two walls with non-proportional primitive normals admit a unique locally finite completion to a consistent system. Worked examples: at the simplest substantive level, the one-dimensional interval split into two segments has one slab with slab function $f_{\rho }=1+tz$ producing a one-parameter family; the focus-focus singularity of an integral affine surface has two slabs ${\rho }_1,{\rho }_2$ with order-1 functions $f_{{\rho }_i}=1+t{z}^{m_i}$ and an order-1 wall $f_{\mathfrak{d}}=1+t{z}^{m_1+m_2}$ correcting the cocycle. Toric specialisation: when $(B,\mathcal{P},\phi )$ is the polarised tropical manifold of a lattice polytope $P$, all slab functions reduce to $f_{\rho }=1$, the wall set is empty, and the reconstruction theorem applied to this identity structure recovers $(X_P,L_P)$ from 04.11.10. Connection: the apparatus is the codimension-1 enhancement of the dual intersection complex (04.12.08); it is the input to the Gross-Siebert reconstruction theorem (04.12.09); it determines the broken-line theta functions (04.12.12) on a polarised mirror Calabi-Yau; and it consumes the log-Gromov-Witten correspondence (04.12.06) at order 1 and order $N\ge 2$ via Carl-Pumperla-Siebert. Originator chain: Mumford 1972 Compositio Math. 24 (Tate-curve prototype); Kempf-Knudsen-Mumford-Saint-Donat 1973 Toroidal Embeddings I (toric-pasting precedent); Strominger-Yau-Zaslow 1996 (SYZ heuristic); Kontsevich-Soibelman 2006 (non-archimedean SYZ, scattering); Gross-Siebert 2006 JDG 72 (log Calabi-Yau central fibre, $(B,\mathcal{P})$); Gross-Siebert 2010 JAG 19 (slab functions, structure $\mathcal{S}$, consistency-of-structure theorem); Gross-Siebert 2011 Annals 174 (algebraisation, reconstruction theorem); Carl-Pumperla-Siebert 2010 (scattering algorithm from log-GW counts); Gross-Hacking-Keel 2015 Publ. IHÉS 122 (two-dimensional log-Calabi-Yau realisation). Lean status: partial, module Codex.AlgGeom.Tropical.SlabFunction declares Lattice, PolarisedTropicalManifold, Slab, Wall, SlabFunction, Structure, toricCentralFibre, orderwiseFamily, polarisationHeight, and the named theorems structure_glues_orderwise (Gross-Siebert 2010 consistency-of-structure), polarisation_compatibility (compatibility of ${\pi }_{\rho }=t^{{\kappa }_{\rho }}$ with the polarisation $\phi$), and reconstruction_input (the formal-family output is the input to the Gross-Siebert reconstruction theorem of 04.12.09) — with sorry-equivalent (rfl-on-placeholder) proof bodies pending the polyhedral + integral-affine + Laurent + wall-crossing infrastructure in Mathlib.

  requires: toric-geometry.fan-and-toric-variety` (04.11.04); `toric-geometry.polytope-fan-dictionary` (04.11.10); `tropical.nishinou-siebert-correspondence` (04.12.06); toric degeneration (04.12.07); dual intersection complex (04.12.08)

• tropical.period-integral-mirror-mapopen unit 04.12.13 →

  Period integral and the mirror map (pointer)

  The period integral and mirror-map story for Calabi-Yau mirror symmetry, presented as a Master-tier pointer unit bridging the Gross-Siebert tropical-mirror-symmetry programme (04.12.07/09/12) to the Hodge-theoretic foundations of chapter 04.09. For the Gross-Siebert reconstructed toric degeneration $\pi :\mathfrak{X}\to \mathrm{Spec}\mathbb{C}[[t]]$ of Calabi-Yau varieties from a polarised tropical manifold $(B,\mathcal{P},\phi )$, the relative holomorphic volume form ${\Omega }_t$ produces period integrals ${\Pi }_i(t)={\int }_{{\gamma }_i}{\Omega }_t$ along a basis of cycles ${\gamma }_i\in H_n({\mathfrak{X}}_{\eta },\mathbb{Z})$. The periods satisfy a Picard-Fuchs equation $L\cdot \Pi (t)=0$ where $L={\theta }^r+a_{r-1}(t){\theta }^{r-1}+\cdots +a_0(t)$ is the linear ODE in the logarithmic derivative $\theta =td/dt$ derived from the Gauss-Manin connection on the relative de Rham cohomology $R^n{\pi }_{\ast }{\Omega }_{\mathfrak{X}/B}^{\bullet }$. Large-complex-structure limit (LCSL) at $t=0$: the monodromy $T$ is maximally unipotent ($N=\log T$ nilpotent of order $n+1$), the weight filtration $W_{\bullet }$ has maximum length $2n+1$, and the limiting mixed Hodge structure of Schmid 1973 / Deligne 1997 gives a canonical pair of periods ${\Pi }_0(t)$ (regular, ${\Pi }_0(0)\ne 0$) and ${\Pi }_1(t)={\Pi }_0(t)\log (t)+R(t)$ (logarithmic) constructed from generators of ${\mathrm{Gr}}_0^W$ and ${\mathrm{Gr}}_2^W$. Mirror map. The canonical coordinate $q(t):=\exp ({\Pi }_1(t)/{\Pi }_0(t))$ is the mirror map: it satisfies $q(t)=t\cdot (1+r_{1}t+r_2{t}^2+\cdots )$ near the LCSL, and identifies the algebraic-geometric parameter $t$ on the B-side with the symplectic Kähler parameter $q$ on the A-side mirror via the Strominger-Yau-Zaslow correspondence (04.12.10). Tropical-disk-count interpretation (Gross 2011 Lecture 7). The higher-order coefficients $r_k$ of the mirror map are computed by tropical-disk count generating series on the polarised tropical manifold $(B,\mathcal{P},\phi )$ via the Nishinou-Siebert correspondence (04.12.06): each $r_k$ is a count of Maslov-index-2 tropical disks with prescribed combinatorial type, identified with log Gromov-Witten invariants on the central fibre via Nishinou-Siebert and with Gromov-Witten invariants on the generic fibre via the deformation invariance of log GW theory. Quintic example (Candelas-de la Ossa-Green-Parkes 1991). For the mirror quintic of the quintic threefold $X\subset {\mathbb{P}}^4$, the Picard-Fuchs operator is of order $4$, the LCSL coincides with a specific point in the one-parameter mirror moduli, and the leading coefficient $r_1$ of the mirror map equals (a rational multiple of) the degree-1 Gromov-Witten count $n_1(X)=2875$ (the number of lines on the quintic, classically known). The systematic identification of all $r_k$ with Gromov-Witten invariants is the BCOV recursion (Bershadsky-Cecotti-Ooguri-Vafa 1994) and was rigorously confirmed by Givental 1996 and Lian-Liu-Yau 1997. GKZ side. For Calabi-Yau hypersurfaces in toric varieties, the Picard-Fuchs operator is a reduction of the Gelfand-Kapranov-Zelevinsky hypergeometric system ${\mathcal{H}}_A(\beta )$ attached to the reflexive polytope $A\subset {\mathbb{Z}}^n$ with Calabi-Yau parameter $\beta =(-1,0,\dots ,0)$; the reduction via the principal $A$-determinant ideal is the content of GKZ 1994 Ch. 10-11, Hosono-Klemm-Theisen-Yau 1995, Hosono-Lian-Yau 1996, and the algorithmic completion of Doran-Morrison 2010. The bridge from the algebraic-geometric GKZ side to the tropical Gross-Siebert side is provided by the Nishinou-Siebert correspondence and is the foundational content of the Gross 2011 Lecture 7 anchor. Three structural bridges of the pointer unit. Bridge one: from Gross-Siebert reconstruction (04.12.09) to the period integral via the relative holomorphic volume form. Bridge two: from the period integral to the Picard-Fuchs equation via the Gauss-Manin connection and the Frobenius method at the LCSL. Bridge three: from the mirror-map coefficients back to the tropical side via the Nishinou-Siebert correspondence applied to the slab functions. The unit is a Master-tier pointer ($\ge 6000$ words per CYCLE_4_STYLE_PARITY_PLAN §5.1; Beginner $\le 600$w; Intermediate $\le 2000$w by pointer-unit interpretation) deferring the full Hodge-theoretic content to chapter 04.09 and to the monograph literature. Originator chain: Schmid 1973 Invent. Math. 22 (variation of Hodge structure, nilpotent-orbit theorem); Griffiths 1968 (period mappings); Candelas-de la Ossa-Green-Parkes 1991 (mirror-symmetry prediction); Morrison 1993 (Mathematicians' guide; LCSL); GKZ 1994 (GKZ hypergeometric system); Hosono-Klemm-Theisen-Yau 1995, Hosono-Lian-Yau 1996 (GKZ-Picard-Fuchs systematic reduction); Strominger-Yau-Zaslow 1996 (SYZ geometric origin); Givental 1996, Lian-Liu-Yau 1997 (rigorous Gromov-Witten matching); Deligne 1997 (limiting mixed Hodge structure); Cox-Katz 1999, Voisin 1996 (canonical monographs); Gross-Siebert 2011 (reconstruction theorem); Gross 2011 CBMS Lecture 7 (tropical-disk-count interpretation); Doran-Morrison 2010 (algorithmic completion). Lean status: partial, module Codex.AlgGeom.Tropical.PeriodIntegral declares PeriodVector, picardFuchsOperator, the named theorems mirror_map_asymptotic and period_picard_fuchs_annihilation with sorry-equivalent placeholder bodies pending Mathlib's holomorphic-volume-form + Gauss-Manin-connection + limiting-mixed-Hodge-structure + GKZ-hypergeometric-system + tropical-disk-count infrastructure.

  requires: algebraic-geometry.kodaira-embedding` (04.09.11 — polarisation + ample line bundles on the Calabi-Yau family); `tropical.toric-degeneration-calabi-yau` (04.12.07); `tropical.gross-siebert-reconstruction` (04.12.09 — the family on which periods are computed); `tropical.theta-function-polarised-manifold` (04.12.12 — broken-line parallel to the period reconstruction)

• chemistry.molecular-spectroscopy-fundamentalsopen unit 14.12.01 →

  UV-Vis, IR, and NMR — fundamentals of molecular spectroscopy

  Foundations of molecular spectroscopy unifying UV-Vis, IR/Raman, and NMR techniques around a single quantum-mechanical principle: a photon is absorbed when its energy matches an allowed gap between two stationary states of the molecule, with rate governed by the transition dipole moment ${\mathbf{\mu }}_{fi}=⟨f|\hat{\mathbf{\mu }}|i⟩$ via Fermi's golden rule (see [12.07.02]). UV-Vis spectroscopy probes electronic transitions (2-6 eV photons); the Beer-Lambert law $A=\epsilon c\ell$ — derivable from a first-order linear ODE on intensity in the linear-response regime — relates absorbance to concentration with the molar absorptivity $\epsilon$ identified with the transition rate. Vibronic-band shape is governed by the Franck-Condon principle, with intensity proportional to the squared vibrational-overlap integral $|⟨{\chi }_f|{\chi }_i⟩{|}^2$; for harmonic potentials with displacement $\Delta x$ the intensity distribution follows a Poisson law $P(v_f)=e^{-S}{S}^{v_f}/v_f!$ in the Huang-Rhys factor $S=(\Delta x/x_0{)}^2/2$. Chromophore concept transferable across molecules; HOMO-LUMO gaps scale as $1/L^2$ for conjugated polyenes (free-electron-in-a-box approximation); transition-metal d-d transitions in the visible due to crystal-field splitting ${\Delta }_o$ (see [16.04.02]); charge-transfer bands much more intense than d-d because not parity-forbidden. Vibrational spectroscopy probes normal modes of the mass-weighted Hessian; a molecule with $N$ atoms has $3N-6$ (non-linear) or $3N-5$ (linear) modes. Harmonic-oscillator energies $E_n=\hslash \omega (n+1/2)$ with selection rule $\Delta n=\pm 1$; anharmonic Morse-potential corrections $E_n=\hslash {\omega }_e(n+1/2)-\hslash {\omega }_e{x}_e(n+1/2{)}^2$ make overtones weakly allowed. IR-active modes require $\partial \mathbf{\mu }/\partial Q\ne 0$; Raman-active modes require $\partial \mathbf{\alpha }/\partial Q\ne 0$. Mutual-exclusion principle: in centrosymmetric molecules, no mode is both IR- and Raman-active because dipole is $u$-parity and polarisability is $g$-parity. Group-theoretic selection rule via direct-product decomposition ${\Gamma }_f\otimes {\Gamma }_{\mu }\otimes {\Gamma }_i$ contains the trivial representation ${\Gamma }_1$; for $C_{2v}$ formaldehyde, $\pi \to {\pi }^{\ast }$ ($B_1\otimes B_1=A_1$, $z$-polarised, allowed); $n\to {\pi }^{\ast }$ ($B_2\otimes B_1=A_2$, none of ${\mu }_x,{\mu }_y,{\mu }_z$, forbidden, weakly observed via vibronic coupling). Group-frequency analysis: C=O at 1715 cm${}^{-1}$ in ketones, 1735 in esters, 1690 in conjugated, 1640 in amides — chemical environment modulates bond strength and frequency. NMR spectroscopy probes nuclear-spin states in external field $B_0$ via Larmor frequency ${\omega }_0=\gamma B_0$. Two key chemical observables: chemical shift $\delta$ in ppm relative to TMS reflecting local electronic shielding $\sigma$ via $B_{\text{eff}}=B_0(1-\sigma )$; J-coupling Hamiltonian ${\hat{H}}_J=2\pi J{\hat{\mathbf{I}}}_1\cdot {\hat{\mathbf{I}}}_2$ mediated by Fermi-contact interaction through bonding electrons, with Karplus relation ${}^{3}J(\varphi )=A{\cos }^2\varphi +B\cos \varphi +C$ linking vicinal couplings to dihedral angles. Bloch equations $d\mathbf{M}/dt=\gamma \mathbf{M}\times \mathbf{B}-(M_x\hat{x}+M_y\hat{y})/T_2-(M_z-M_0)\hat{z}/T_1$ govern macroscopic-magnetisation dynamics: Larmor precession plus phenomenological relaxation, yielding Lorentzian lineshape of half-width $1/T_2$. FT-NMR revolution (Ernst 1966) increased sensitivity by $\sqrt{N}$ Felgett advantage, enabling routine ${}^{13}$C NMR despite 1.1% natural abundance and reduced gyromagnetic ratio. 2D pulse sequences (COSY, HSQC, HMBC, NOESY, TOCSY) generate molecular-connectivity maps via Fourier transform of two evolution times; Wuthrich 2002 extended to biological-macromolecule structure determination. Modern frontiers include hyperpolarisation (DNP, parahydrogen-induced polarisation, optical pumping) amplifying NMR sensitivity by $10^2$-$10^4$; single-molecule fluorescence and super-resolution microscopy (Moerner-Hell-Betzig 2014 Nobel); ultrafast pump-probe spectroscopy reaching femtosecond timescales (Zewail 1999 Nobel) and attosecond pulses (Krausz-Agostini-L'Huillier 2023 Nobel); 2D-IR (Hochstrasser, Tokmakoff) and 2D electronic spectroscopy applying the 2D-NMR concept to optical regime. Connections: [12.07.02] time-dependent perturbation theory + Fermi golden rule (transition-rate basis); [16.04.02] CFSE + spectrochemical series (UV-Vis of transition-metal complexes via d-d transitions); [15.11.01] NMR organic chemistry (structure-elucidation companion). Lean status: none, with substantive mathlib_gap describing three independent formalisation targets — transition dipole moments as matrix elements of position operator in bra-ket inner-product API; Beer-Lambert law in measure-theoretic form derivable from Bochner integral over photon-absorption probabilities; Bloch equations as first-order linear ODE on magnetisation vector $\mathbf{M}(t)\in {\mathbb{R}}^3$ with existence-uniqueness from Picard-Lindelöf already in Mathlib. The (3) ODE-foundation route is the most accessible formalisation target; (1) and (2) need new analytic foundations.

  requires: chemistry.hydrogen-atom-quantum-chemistry` (14.04.01); `chemistry.mo-theory-homonuclear-diatomics` (14.05.02); `chemistry.hybridization-valence-bond` (14.02.02)

• number-theory.riemann-zetaopen unit 21.03.01 →

  Riemann zeta function $\zeta (s)$

  The Riemann zeta function is the foundational analytic object of analytic number theory. Definition by Dirichlet series $\zeta (s)={\sum }_{n=1}^{\infty }1/n^s$ on the half-plane $\mathrm{Re}(s)>1$; equivalent Euler product $\zeta (s)={\prod }_p(1-p^{-s}{)}^{-1}$ over primes (Euler 1737, encoding unique factorisation in $\mathbb{Z}$); analytic continuation to a meromorphic function on $\mathbb{C}$ with a single simple pole at $s=1$ of residue $1$. The signature theorem is the functional equation in symmetric form $\xi (s)=\xi (1-s)$ where the completed function $\xi (s)=\frac{1}{2}s(s-1){\pi }^{-s/2}\Gamma (s/2)\zeta (s)$ is entire of order $1$; equivalently in asymmetric form $\zeta (s)=2^s{\pi }^{s-1}\sin (\pi s/2)\Gamma (1-s)\zeta (1-s)$. Two proofs (both Riemann 1859): (i) contour deformation via the Hankel integral $\zeta (s)=-\Gamma (1-s)(2\pi i{)}^{-1}{\int }_C(-z{)}^{s-1}/(e^z-1)dz$, providing analytic continuation in one step; (ii) theta-function symmetry via the Mellin transform ${\pi }^{-s/2}\Gamma (s/2)\zeta (s)={\int }_0^{\infty }\psi (x)x^{s/2-1}dx$ where $\psi (x)={\sum }_{n=1}^{\infty }{e}^{-\pi n^{2}x}$ and the Jacobi transformation $\theta (1/x)=x^{1/2}\theta (x)$ (a Poisson summation identity on the Gaussian) produces $s\leftrightarrow 1-s$ symmetry of the integrand. Trivial zeros $\zeta (-2k)=0$ for $k\ge 1$, forced by the entirety of $\xi$ cancelling the simple poles of $\Gamma (s/2)$ at negative even integers. Special values $\zeta (2k)=(-1{)}^{k+1}(2\pi {)}^{2k}{B}_{2k}/(2(2k)!)$ for positive integer $k$, including $\zeta (2)={\pi }^2/6$ (Euler 1735 Basel problem) and $\zeta (4)={\pi }^4/90$; $\zeta (0)=-1/2$, $\zeta (-1)=-1/12$, $\zeta (1-2n)=-B_{2n}/(2n)$. Hadamard product $\xi (s)=e^{a+bs}{\prod }_{\rho }(1-s/\rho )e^{s/\rho }$ over non-trivial zeros (Riemann 1859 stated, Hadamard 1893 proved). Explicit formula (Riemann-von Mangoldt) $\psi (x)=x-{\sum }_{\rho }{x}^{\rho }/\rho -\log (2\pi )-\frac{1}{2}\log (1-x^{-2})$ connecting Chebyshev's $\psi$ to non-trivial zeros. Prime number theorem (Hadamard 1896 + de la Vallée Poussin 1896 independently), equivalent to $\zeta (s)\ne 0$ on $\mathrm{Re}(s)=1$. Riemann hypothesis (Riemann 1859, statement only): all non-trivial zeros lie on the critical line $\mathrm{Re}(s)=1/2$. Open since 1859; Hilbert problem 8; Clay Millennium Prize (Bombieri 2000). Partial results: Hardy 1914 (infinitely many zeros on the line), Selberg 1942 (positive proportion), Levinson 1974 ($\ge 1/3$), Conrey 1989 ($\ge 2/5$). Random-matrix heuristic (Montgomery 1973, Odlyzko 1987-): local zero statistics match GUE eigenvalue statistics. Function-field analogue proved by Weil 1948. Chapter-opening unit of Section 21.03 L-functions; forward-connects to Dirichlet $L$-functions (21.03.02), Dedekind/Hecke/Artin $L$-functions (21.03.03), modular forms (21.04.01), Hecke operators (21.04.02), and the Langlands programme (21.10.01). Originator chain: Euler 1735 Basel ($\zeta (2)={\pi }^2/6$); Euler 1737 Var. obs. (Euler product); Riemann 1859 Monatsber. (complex-variable treatment, analytic continuation, functional equation, Hadamard product, explicit formula, RH); Hadamard 1893 (factorisation theorem); Mangoldt 1895 (explicit-formula rigour); Hadamard 1896 + de la Vallée Poussin 1896 (PNT); Hardy 1914 (infinitely many critical-line zeros); Selberg 1942 (positive proportion); Weil 1948 (function-field analogue); Montgomery 1973 + Odlyzko 1987 (random-matrix heuristic). Lean status: partial, module Codex.NumberTheory.LFunctions.RiemannZeta declares riemannZeta : ℂ → ℂ, riemannZeta_eq_dirichlet_series, riemannZeta_residue_at_one, riemannZeta_meromorphic, zeta_euler_product, zeta_nonzero_re_gt_one, completedZeta, completedZeta_functional_equation, zeta_functional_equation, zeta_trivial_zeros, criticalStrip, criticalLine, RiemannHypothesis — all sorry-stubbed pending Mathlib infrastructure on Dirichlet-series meromorphic continuation, Poisson summation on $\mathbb{Z}\subset \mathbb{R}$, and Mellin-transform $\Gamma$-twist pairing.

  requires: complex-analysis.analytic-continuation` (06.01.04); `complex-analysis.gamma-function` (06.01.15); `analysis.infinite-series-convergence` (02.03.03)

• number-theory.modular-forms-sl2-zopen unit 21.04.01 →

  Modular Forms on ${\mathrm{SL}}_2(\mathbb{Z})$

  Foundational definitional unit for Section 21.04 modular forms. A modular form of weight $k$ for ${\mathrm{SL}}_2(\mathbb{Z})$ is a holomorphic function $f:\mathbb{H}\to \mathbb{C}$ on the upper half-plane satisfying the transformation law $f((a\tau +b)/(c\tau +d))=(c\tau +d{)}^{k}f(\tau )$ for every $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{SL}}_2(\mathbb{Z})$, together with holomorphy at the cusp $\infty$ (equivalently, the Fourier expansion $f(\tau )={\sum }_{n\ge 0}{a}_n{q}^n$ in $q=e^{2\pi i\tau }$ has no negative-index terms). A cusp form has additional vanishing $a_0=0$. The space $M_k$ of weight-$k$ modular forms is finite-dimensional, with dimension formula $\dim M_k=\lfloor k/12\rfloor +1$ for $k$ even and $k\not\equiv 2(\mathrm{mod}12)$, $\lfloor k/12\rfloor$ for $k\equiv 2(\mathrm{mod}12)$, and $0$ for $k$ odd. Foundational examples: Eisenstein series $E_k=1-(2k/B_k){\sum }_{n\ge 1}{\sigma }_{k-1}(n)q^n$ for even $k\ge 4$ (lattice sum $G_k(\tau )={\sum }_{(m,n)\ne (0,0)}(m\tau +n{)}^{-k}$ normalised by $2\zeta (k)$); modular discriminant $\Delta =(E_4^3-E_6^2)/1728=q{\prod }_{n\ge 1}(1-q^n{)}^{24}$ of weight 12, the smallest non-zero cusp form, with Fourier coefficients $\tau (n)$ the Ramanujan tau function; Klein $j$-invariant $j=E_4^3/\Delta =1/q+744+196884q+\cdots$, the generator of the field of meromorphic modular functions on $X(1)$. Signature theorem is the structure theorem $M_{\ast }({\mathrm{SL}}_2(\mathbb{Z}))={⨁}_{k\ge 0}{M}_k\cong \mathbb{C}[E_4,E_6]$ as graded $\mathbb{C}$-algebras with $\mathrm{deg}{E}_4=4,\mathrm{deg}{E}_6=6$ (Hecke 1937; Serre VII §3 Theorem 4). The proof uses the valence formula ${\nu }_{\infty }(f)+\frac{1}{2}{\nu }_i(f)+\frac{1}{3}{\nu }_{\rho }(f)+{\sum }_P{\nu }_P(f)=k/12$ derived from the residue theorem on the standard fundamental domain $F=\{|\tau |\ge 1,|\mathrm{Re}\tau |\le 1/2\}$. The Petersson inner product $⟨f,g⟩={\int }_{F}f\bar{g}{y}^k{y}^{-2}dxdy$ makes $S_k$ into a finite-dimensional Hilbert space on which the Hecke algebra acts self-adjointly; simultaneous Hecke eigenforms give an orthogonal basis with multiplicative Fourier coefficients and attached $L$-function $L(f,s)=\sum a_n/n^s$ with Euler product. The Ramanujan-Petersson conjecture $|a_p|\le 2p^{(k-1)/2}$ (Petersson 1939) is proved by Deligne 1974 Publ. Math. IHES 43 via the Weil conjectures for Kuga-Sato varieties. Forward-connects to Hecke operators and Hecke algebra (21.04.02), Eichler-Shimura correspondence (21.04.03), $\ell$-adic Galois representations (21.05.01), modularity theorem and BSD (21.06.01), Langlands programme (21.10.01); backward-connects to Möbius transformations (06.01.08), holomorphic function (06.01.01), Riemann surfaces (06.03.01), Riemann zeta function (21.03.01). Lean status: partial, module Codex.NumberTheory.ModularForms.SL2Z declares UpperHalfPlane, ModularGroup, mobius, automorphyFactor, ModularForm k, CuspForm k, eisensteinSeries k, modularDiscriminant, gradedRingOfModularForms, plus theorems modular_ring_polynomial_in_E4_E6, discriminant_via_E4_E6, dim_formula_even, dim_formula_odd, eisenstein_constant_term_one — all sorry-stubbed pending Mathlib infrastructure on lattice-sum convergence, holomorphy of Eisenstein series, the valence formula via residue calculus on the fundamental domain, and the Hecke-Serre polynomial-ring identification.

  requires: linalg.linear-transformation-rank-nullity; complex-analysis.holomorphic-function; complex-analysis.mobius-transformations

• number-theory.dirichlet-l-functionsopen unit 21.03.02 →

  Dirichlet $L$-functions $L(s,\chi )$

  The Dirichlet $L$-function $L(s,\chi )={\sum }_{n\ge 1}\chi (n)/n^s$ on the half-plane $\mathrm{Re}(s)>1$ is the originator and prototype of analytic number theory. Dirichlet 1837 Abh. Königl. Preuss. Akad. introduces the Dirichlet character $\chi :(\mathbb{Z}/m{)}^{\times }\to {\mathbb{C}}^{\times }$ as a homomorphism of the finite multiplicative group of units, extended to $\mathbb{Z}$ by zero on integers sharing a factor with $m$, completely multiplicative and $m$-periodic. The Euler product $L(s,\chi )={\prod }_p(1-\chi (p)p^{-s}{)}^{-1}$ for $\mathrm{Re}(s)>1$ encodes complete multiplicativity. Principal character ${\chi }_0$: $L(s,{\chi }_0)=\zeta (s){\prod }_{p\mid m}(1-p^{-s})$, inheriting the simple pole at $s=1$ from $\zeta$ with residue $\phi (m)/m$. Non-principal characters: $L(s,\chi )$ is entire on $\mathbb{C}$ (Hecke 1918). Signature theorem (non-vanishing at $s=1$, Dirichlet 1837): $L(1,\chi )\ne 0$ for every non-principal $\chi$; proof splits into the complex-character case (Landau's theorem on Dirichlet series with non-negative coefficients applied to ${\prod }_{\chi }L(s,\chi )$) and the real-character case (class-number formula $L(1,\chi )=2\pi h(D)/(w_D\sqrt{|D|})$ for $D<0$ or $L(1,\chi )=2\pi h(D)\log \epsilon /\sqrt{|D|}$ for $D>0$, both strictly positive). Dirichlet's theorem on primes in arithmetic progressions (Dirichlet 1837): for every modulus $m$ and residue $a$ coprime to $m$, the arithmetic progression $a,a+m,a+2m,\dots$ contains infinitely many primes; quantitative PNT (de la Vallée Poussin 1896) gives $\pi (x;m,a)\sim x/(\phi (m)\log x)$. The proof reduces via orthogonality of characters and the logarithmic Euler product to the divergence $\log L(s,{\chi }_0)\to +\infty$ at $s=1^{+}$ — the non-vanishing $L(1,\chi )\ne 0$ being the only analytic input. Functional equation (Hecke 1918 Math. Z. 1): for primitive $\chi$ of conductor $m$ with parity $\mathfrak{a}=(1-\chi (-1))/2$, the completed $L$-function $\Lambda (s,\chi )=(m/\pi {)}^{(s+\mathfrak{a})/2}\Gamma ((s+\mathfrak{a})/2)L(s,\chi )$ satisfies $\Lambda (s,\chi )=W(\chi )\Lambda (1-s,\overline{\chi })$ with root number $W(\chi )=\tau (\chi )/(i^{\mathfrak{a}}\sqrt{m})$ and Gauss sum $\tau (\chi )={\sum }_a\chi (a)e^{2\pi ia/m}$, $|\tau (\chi )|=\sqrt{m}$. Proof via Mellin transform of the Dirichlet theta function and Poisson summation. Exceptional Siegel zero (Page 1935, Siegel 1935): real $\chi$ has at most one real zero $\beta <1$ within $1-O(1/\log m)$ of $1$; ineffective Siegel bound $1-\beta {\gg }_{\epsilon }{m}^{-\epsilon }$. Linnik's theorem (Linnik 1944): least prime $p\equiv a\mathrm{mod}m$ is $\ll m^L$, current $L\le 5$ (Xylouris 2011), conjecturally $L=1+\epsilon$ under GRH. Modern reformulations: Hecke 1918 generalises to characters of the idele class group of number fields, giving Hecke $L$-functions; Artin 1923/1930 generalises further to $L$-functions of Galois representations, with one-dimensional Artin = Dirichlet by class field theory and Artin reciprocity (Artin 1927); Tate 1950 PhD thesis recasts everything as global zeta integrals over ${\mathbb{A}}_{\mathbb{Q}}^{\times }$ against Schwartz-Bruhat functions — the ${\mathrm{GL}}_1/\mathbb{Q}$ prototype of the Langlands programme (Bump 1997). The Dirichlet $L$-function is the foundational unit of Section 21.03; sibling $\zeta$ (21.03.01), forward Dedekind/Hecke/Artin $L$ (21.03.03), forward modular forms (21.04.01) and Hecke operators (21.04.02), and lateral character of a finite group (07.01.03). Originator chain: Euler 1737 ($\zeta$ Euler product); Dirichlet 1837 (characters + $L$-series + non-vanishing + arithmetic progression theorem); Dirichlet 1839/40 (class-number formula); Riemann 1859 (functional equation for $\zeta$, analytic-continuation method generalised by Hecke); Hadamard-de la Vallée Poussin 1896 (PNT and zero-free region); Hecke 1918, 1920 (Hecke $L$ + functional equation); Artin 1923, 1930 (Artin $L$); Page 1935, Siegel 1935 (Siegel zero); Linnik 1944 (least prime); Selberg 1949 (elementary proof of PNT in arithmetic progressions); Tate 1950 (adelic Tate thesis). Lean status: partial, module Codex.NumberTheory.LFunctions.DirichletL declares structure DirichletCharacter, def DirichletL, theorem L_nonvanish_at_one, theorem dirichlet_progression, plus orthogonality lemmas and the witness L_one_chi_one_mod_four = π/4 (Leibniz series), all sorry-stubbed pending Mathlib infrastructure on Dirichlet-series meromorphic continuation, Landau theorem on positive-coefficient series, and the class-number formula for real characters.

  requires: algebra.group` (01.02.01 — the multiplicative group $(\mathbb{Z}/m)^\times$); `algebra.subgroup-coset-quotient` (01.02.02 — quotient and finite-abelian-group machinery underlying $(\mathbb{Z}/m)^\times$); `complex-analysis.complex-numbers-euler` (02.09.01 — codomain $\mathbb{C}^\times$ and Mellin / contour-integral background); sibling `number-theory.riemann-zeta` (21.03.01 — principal-character $L$-function inherits the analytic continuation of $\zeta$)

• tropical.log-gromov-witten-invariantsopen unit 04.12.15 →

  Log Gromov-Witten Invariants (pointer)

  Pointer unit recording the foundational log Gromov-Witten package consumed by [04.12.06] Nishinou-Siebert correspondence and [04.12.09] Gross-Siebert reconstruction. A log Gromov-Witten invariant is an integer count of stable log maps from a log smooth curve to a log smooth target, weighted by the virtual fundamental class on the moduli stack with perfect obstruction theory. Key data: a fine saturated (fs) log scheme $(X,M_X)$ in the sense of Kato 1989; a log smooth morphism $f:(C,M_C)\to (X,M_X)$ from a log smooth pointed curve; contact-order data $\mathbf{c}=(c_1,\dots ,c_m)$ at the marked points recording tangency to the boundary divisor; the moduli stack ${\overline{\mathcal{M}}}_{g,m}^{\log }(X/S,\beta ,\mathbf{c})$ of stable log maps of fixed genus, marking, class, and contact data, with its perfect obstruction theory and virtual fundamental class $[{\overline{\mathcal{M}}}^{\log }{]}^{\mathrm{vir}}$ of expected dimension $(1-g)(n-3)+⟨c_1(T{X}^{\log }),\beta ⟩+m$. The log GW invariant is the integral $⟨{\tau }_{a_1}({\gamma }_1)\cdots {\tau }_{a_m}({\gamma }_m){⟩}_{g,\beta ,\mathbf{c}}^{\log }={\int }_{[{\overline{\mathcal{M}}}^{\log }{]}^{\mathrm{vir}}}\prod {\psi }_i^{a_i}{\mathrm{ev}}_i^{\ast }({\gamma }_i)$. Foundational theorems: (1) the Abramovich-Chen vs Gross-Siebert equivalence (Abramovich-Chen-Marcus-Wise 2017), proving that the two parallel constructions of the moduli stack agree as Deligne-Mumford stacks with matching obstruction theories, so the resulting log GW invariants are well-defined; (2) the degeneration formula (Gross-Siebert 2013, Theorem 0.3): for a flat family $\pi :\mathcal{X}\to {\mathbb{A}}^1$ with smooth generic fibre and log smooth central fibre, $N_{g,\beta }({\mathcal{X}}_t)={\sum }_{\tau }{N}_{g,\beta }^{\log ,\tau }({\mathcal{X}}_0)$, the sum over decoration types $\tau$ of dual graphs with contact-order labels; (3) the Nishinou-Siebert application (Nishinou-Siebert 2006, [04.12.06]): for a toric degeneration of a smooth toric variety, the central-fibre log GW counts equal a multiplicity-weighted tropical-curve count on the dual intersection complex $B(\Xi )$. Master-tier sections: §1 foundational log-scheme setup after Kato 1989 (fs log structures, log smooth morphisms, log differentials, chart theory, basic monoid construction); §2 the two parallel constructions of Abramovich-Chen vs Gross-Siebert with the 2017 equivalence theorem; §3 the Nishinou-Siebert correspondence as a tropical computation via the degeneration formula. Forward-connects to [04.12.06] Nishinou-Siebert, [04.12.09] Gross-Siebert reconstruction, [04.12.10] SYZ, [04.12.13] period integral, plus a future Gromov-Witten foundations unit (not yet shipped; the Behrend-Fantechi virtual-class machinery is sketched here in lieu). Lean status partial with module Codex.AlgGeom.Tropical.LogGromovWitten: declares FsLogStructure, LogScheme, LogSmoothMap, LogSmoothCurve, StableLogMap, ModuliStableLogMaps, virtualFundamentalClass, logGWInvariant, plus theorems logGW_well_defined, degeneration_formula, nishinou_siebert_application — all sorry-stubbed pending Mathlib infrastructure on log geometry, moduli of stable log maps, perfect obstruction theory in the log category, and the virtual fundamental class.

  requires: tropical.nishinou-siebert-correspondence` (04.12.06 — uses log GW theory in its proof); `tropical.toric-degeneration-calabi-yau` (04.12.07); `tropical.gross-siebert-reconstruction` (04.12.09 — consumes log GW as scattering input)

• number-theory.hecke-operatorsopen unit 21.04.02 →

  Hecke Operators and Hecke Algebra

  The Hecke operator $T_p$ at a prime $p$ acting on $M_k({\mathrm{SL}}_2(\mathbb{Z}))$ is the sub-lattice average $(T_{p}f)(z)=p^{k-1}f(pz)+p^{-1}{\sum }_{b=0}^{p-1}f((z+b)/p)$, equivalently the sum over coset representatives $f{|}_k\delta$ for $\delta$ ranging over $\Gamma \backslash M_2(\mathbb{Z}{)}_{\det =p}$. The operators commute pairwise, satisfy the multiplicativity $T_{mn}=T_m{T}_n$ for $\gcd (m,n)=1$ and the prime-power recursion $T_p{T}_{p^n}=T_{p^{n+1}}+p^{k-1}{T}_{p^{n-1}}$, and generate the commutative Hecke algebra $\mathbb{T}=\mathbb{C}[T_n:n\ge 1]\subseteq {\mathrm{End}}_{\mathbb{C}}(M_k(\Gamma ))$. Signature theorem (Hecke 1936): for a normalised Hecke eigenform $f$ with $a_1(f)=1$, the eigenvalue ${\lambda }_n$ of $T_n$ equals the $n$-th Fourier coefficient $a_n(f)$; consequently the Dirichlet series $L(f,s)=\sum a_n{n}^{-s}$ acquires the Euler product ${\prod }_p(1-a_p{p}^{-s}+p^{k-1-2s}{)}^{-1}$, with quadratic local factors whose coefficients are the eigenvalue $a_p$ and the weight-dependent constant $p^{k-1}$. Petersson self-adjointness (1939): the Hecke operators are self-adjoint with respect to the Petersson inner product $⟨f,g{⟩}_{\mathrm{Pet}}={\int }_{\Gamma \backslash \mathbb{H}}f\bar{g}{y}^{k-2}dxdy$, yielding an orthonormal basis of simultaneous Hecke eigenforms on $S_k(\Gamma )$. Atkin-Lehner 1970 multiplicity-one at higher level ${\Gamma }_0(N)$: every newform is uniquely determined (up to scalar) by its Hecke eigenvalue sequence $\{a_p(f):p\nmid N\}$ plus Atkin-Lehner signs. Deligne-Serre 1974 attaches a $2$-dimensional $\ell$-adic Galois representation ${\rho }_{f,\ell }:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\mathrm{GL}}_2(\overline{{\mathbb{Q}}_{\ell }})$ to every normalised weight-$\ge 1$ cusp newform $f$ with $\mathrm{tr}{\rho }_{f,\ell }({\mathrm{Frob}}_p)=a_p(f)$ and $\det {\rho }_{f,\ell }({\mathrm{Frob}}_p)=p^{k-1}$ for $p\nmid N\ell$. Worked example: the discriminant cusp form $\Delta (z)=q\prod (1-q^n{)}^{24}$ of weight $12$ on ${\mathrm{SL}}_2(\mathbb{Z})$ is the unique normalised cusp form in a one-dimensional space, automatically a Hecke eigenform with eigenvalues $\tau (n)$ (Ramanujan tau), the multiplicativity $\tau (mn)=\tau (m)\tau (n)$ for coprime $(m,n)$ and the recursion $\tau (p)\tau (p^n)=\tau (p^{n+1})+p^{11}\tau (p^{n-1})$ being direct consequences. Deligne 1974 proved $|\tau (p)|\le 2p^{11/2}$ (Ramanujan-Petersson). Originator chain: Ramanujan 1916 (multiplicativity conjectures for $\tau$); Mordell 1917 Proc. Cambridge Phil. Soc. (direct proof via $L$-function); Hecke 1936-37 Math. Ann. 112 + 114 (operators $T_n$, Hecke algebra, Euler product); Petersson 1939 Math. Ann. 116 (Petersson inner product, self-adjointness); Eichler 1954 + Shimura 1958 (cohomological realisation); Atkin-Lehner 1970 (new/old decomposition, multiplicity-one); Deligne 1971 + Deligne-Serre 1974 (Galois representations); Deligne 1974 (Ramanujan-Petersson). Forward-connects to [21.04.03] Eichler-Shimura correspondence, [21.05.01] $\ell$-adic Galois representations, [21.06.01] modularity theorem and BSD, [21.03.01] Riemann zeta (sibling Dirichlet series), [21.03.02] Dirichlet $L$-functions (sibling ${\mathrm{GL}}_1$-case), [21.03.03] Dedekind / Hecke / Artin $L$ (sibling higher-rank generalisations). Lean status: partial, module Codex.NumberTheory.ModularForms.HeckeOperators declares def heckeOperator (p : ℕ) [Fact (Nat.Prime p)] : ModularForm k → ModularForm k, theorem hecke_multiplicative, theorem hecke_prime_power_recursion, theorem hecke_eigenform_fourier_coeff_eq_eigenvalue, and theorem hecke_petersson_self_adjoint with sorry-stubbed proof bodies pending Mathlib's coset-decomposition API for ${\mathrm{SL}}_2(\mathbb{Z})$ acting on determinant-$n$ integer matrices and the integration-on-fundamental-domain machinery for the Petersson inner product.

  requires: sibling `number-theory.modular-forms-sl2z` (21.04.01 — defines the ambient space $M_k(\mathrm{SL}_2(\mathbb{Z}))$ on which the Hecke operators act); `complex-analysis.complex-numbers-euler` (02.09.01 — Fourier-series / Dirichlet-series machinery)

• foundations.subgroup-coset-quotientopen unit 01.02.02 →

  Subgroup, coset, and quotient group with the isomorphism theorems

  Subgroup as closed-under-operation-and-inverse subset. Left and right cosets; Lagrange's theorem ($|G|=[G:H]\cdot |H|$). Normal subgroup $N◃G$ as kernel of a homomorphism, equivalently subgroup whose left and right cosets coincide. Quotient $G/N$ with the canonical projection $\pi :G\twoheadrightarrow G/N$. The four isomorphism theorems: first (kernel-image: $G/\ker f\cong \mathrm{im}f$), second (diamond: $(HN)/N\cong H/(H\cap N)$), third (lattice modular law: $(G/N)/(M/N)\cong G/M$ when $N\subseteq M◃G$), fourth (correspondence: subgroups of $G/N$ correspond bijectively to subgroups of $G$ containing $N$). Used everywhere downstream — character theory of finite groups, Galois theory, classifying spaces, deck transformation groups.

  requires: foundations.group

• number-theory.eichler-shimura-correspondenceopen unit 21.04.03 →

  Eichler-Shimura correspondence

  The Eichler-Shimura correspondence attaches to each normalised weight-$2$ cusp newform $f\in S_2({\Gamma }_0(N){)}^{\mathrm{new}}$ a continuous two-dimensional $\ell$-adic Galois representation ${\rho }_{f,\ell }:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\mathrm{GL}}_2(\overline{{\mathbb{Q}}_{\ell }})$, realised on the $f$-isotypic component of the rational $\ell$-adic Tate module $V_{\ell }=T_{\ell }{J}_0(N)\otimes {\mathbb{Q}}_{\ell }$ of the modular Jacobian $J_0(N)=\mathrm{Jac}(X_0(N))$. The representation is unramified outside $N\ell$ (Néron-Ogg-Shafarevich applied to $J_0(N)$ which has good reduction at $p\nmid N$), and satisfies the trace-of-Frobenius identity $\mathrm{tr}{\rho }_{f,\ell }({\mathrm{Frob}}_p)=a_p(f)$ and $\det {\rho }_{f,\ell }({\mathrm{Frob}}_p)=p$ for every $p\nmid N\ell$. Signature theorem (Eichler-Shimura congruence relation): on the special fibre $X_0(N)\otimes {\mathbb{F}}_p$ for $p\nmid N$, the Hecke correspondence $T_p$ satisfies $T_p\equiv {\mathrm{Frob}}_p+p⟨p⟩{\mathrm{Frob}}_p^{-1}(\mathrm{mod}\ell )$ on $T_{\ell }{J}_0(N)$ for $\ell \ne p$, derived from the moduli-theoretic split of the $p+1$ order-$p$ subgroups of an elliptic curve over $\overline{{\mathbb{F}}_p}$ into one Frobenius-kernel subgroup and $p$ Verschiebung-kernel subgroups. Eigenform decomposition: $V_{\ell }\otimes \overline{{\mathbb{Q}}_{\ell }}={⨁}_{[f]}{V}_{\ell ,f}$ ranges over Galois-conjugacy classes of weight-$2$ cusp newforms (modulo old-form contributions from ${\Gamma }_0(M)$, $M|N$), each $V_{\ell ,f}$ a free rank-$2$ $\overline{{\mathbb{Q}}_{\ell }}$-module on which $T_p$ acts by $a_p(f)$. Shimura's modular abelian variety: $A_f:=J_0(N)/I_f\cdot J_0(N)$ with $I_f=\ker ({\lambda }_f:{\mathbb{T}}_N\to K_f)$ is an abelian variety over $\mathbb{Q}$ of dimension $d=[K_f:\mathbb{Q}]$ where $K_f=\mathbb{Q}(a_p(f):p\nmid N)$ is the eigenvalue field; isogenous to ${\prod }_{\sigma }{E}_{f^{\sigma }}$ over Galois conjugates of $f$, with $L(A_f,s)={\prod }_{\sigma }L(f^{\sigma },s)$. Worked example: at level $N=11$, $\dim S_2({\Gamma }_0(11))=g(X_0(11))=1$, and the unique normalised weight-$2$ cusp newform $f=q-2q^2-q^3+2q^4+q^5+\cdots$ corresponds to the elliptic curve $E:y^2+y=x^3-x^2-10x-20$ of conductor $11$, with $a_p(f)=p+1-|E({\mathbb{F}}_p)|$ matching at every $p\ne 11$. Higher-weight extension (Deligne 1971): for $k\ge 2$, ${\rho }_{f,\ell }$ is built from the étale cohomology in degree $k-1$ of the Kuga-Sato variety ${\tilde{\mathcal{E}}}_N^{k-2}$ (smooth compactification of the $(k-2)$-fold fibre product of the universal elliptic curve over $Y_0(N)$), giving $\det {\rho }_{f,\ell }({\mathrm{Frob}}_p)=p^{k-1}$. Weight-1 case (Deligne-Serre 1974) via congruences with higher-weight forms; finite-image / Artin representations. Ramanujan-Petersson (Deligne 1974 Publ. Math. IHES 43): $|a_p(f)|\le 2p^{(k-1)/2}$ via Weil II applied to the étale cohomology of the modular / Kuga-Sato varieties — the modular-form Riemann hypothesis. Originator chain: Eichler 1954 Arch. Math. 5 (Brandt-matrix trace formula on $S_2({\Gamma }_0(N))$ identified with Frobenius trace on $H^1(X_0(N)\otimes {\mathbb{F}}_p)$); Shimura 1958 J. Math. Soc. Japan 10 (systematic construction of Galois representations from weight-2 eigenforms via the Jacobian $J_0(N)$); Shimura 1971 Introduction to the Arithmetic Theory of Automorphic Functions (canonical book-form treatment); Deligne 1971 (extension to weight $k\ge 2$ via Kuga-Sato); Deligne-Serre 1974 (weight-1 case); Deligne 1974 (Ramanujan-Petersson via Weil II); Carayol 1986 Ann. Sci. ENS 19 (local Langlands compatibility at $p|N$); Ribet 1990 Invent. Math. 100 (level-lowering / epsilon conjecture); Mazur-Wiles 1984 (Iwasawa Main Conjecture application); Wiles 1995 + BCDT 2001 (modularity theorem application). Forward-connects to [21.05.01] $\ell$-adic Galois representations (sibling, the systematic framework on which Eichler-Shimura is the originating example), [21.06.01] modularity theorem and BSD (every elliptic curve over $\mathbb{Q}$ arises as $A_f$ for some weight-$2$ cusp newform), [21.07.01] Iwasawa ${\mathbb{Z}}_p$-extensions (Mazur-Wiles Main Conjecture uses Eichler-Shimura Galois representations), [21.03.03] Dedekind / Hecke / Artin $L$ (sibling, $L(f,s)=L({\rho }_{f,\ell },s)$ identification); backward-connects to [21.04.01] modular forms on ${\mathrm{SL}}_2(\mathbb{Z})$, [21.04.02] Hecke operators and Hecke algebra, [06.06.03] Jacobian variety, [04.04.03] elliptic curves. Lean status: partial, module Codex.NumberTheory.ModularForms.EichlerShimura declares def Jacobian_X0N, def galoisRepOfEigenform, theorem trace_frob_eq_fourier_coeff, theorem eichler_shimura_congruence, theorem unramified_outside_N_ell, def modularAbelianVariety, theorem tate_module_eigenform_decomposition, def KugaSatoVariety, theorem deligne_higher_weight with sorry-stubbed proof bodies pending Mathlib's modular-curve / Jacobian over $\mathbb{Q}$ / étale cohomology / Néron-Ogg-Shafarevich / moduli-theoretic Hecke correspondence infrastructure.

  requires: number-theory.modular-forms-sl2z, number-theory.hecke-operators, riemann-surfaces.jacobian-variety, algebraic-geometry.elliptic-curve

• number-theory.p-adic-l-functions-main-conjectureopen unit 21.07.02 →

  $p$-adic $L$-functions and the Iwasawa Main Conjecture

  The Kubota-Leopoldt $p$-adic L-function $L_p(s,\chi ):{\mathbb{Z}}_p\to {\mathbb{C}}_p$ attached to a Dirichlet character $\chi$ of conductor coprime to the odd prime $p$ is the unique continuous interpolation of the classical Dirichlet L-values at negative integers, $L_p(1-n,\chi )=(1-\chi {\omega }^{-n}(p)p^{n-1})L(1-n,\chi {\omega }^{-n})$ for positive integers $n$, with the Euler factor at $p$ removed and $\omega$ the Teichmuller character. Existence + uniqueness via Kummer congruences for generalised Bernoulli numbers $B_{n,\chi }$ — the refinement $n\equiv m(\mathrm{mod}{p}^{a-1}(p-1))$ implies the corrected ratios are congruent modulo $p^a$, giving uniform continuity on the dense subset $\{1-n:n\ge 1\}$ of ${\mathbb{Z}}_p$. Via the Amice transform (Amice 1964 Bull. Soc. Math. France 92), $L_p$ becomes an element ${\xi }_{\chi }$ of the Iwasawa algebra $\Lambda ={\mathbb{Z}}_p[[\Gamma ]]\cong {\mathbb{Z}}_p[[T]]$ where $\Gamma =\mathrm{Gal}(\mathbb{Q}({\mu }_{p^{\infty }})/\mathbb{Q})\cong {\mathbb{Z}}_p$. Iwasawa Main Conjecture (Iwasawa 1969 conjectured; Mazur-Wiles 1984 Invent. Math. 76 theorem): for an even non-principal Dirichlet character $\chi$ of conductor coprime to $p$, the characteristic ideal of the minus-part $\chi$-component of the cyclotomic Iwasawa module $X_{\chi }^{-}={\underleftarrow{\lim }}_n{A}_n^{(\chi ),-}$ (projective limit of $p$-Sylows of ideal class groups of $\mathbb{Q}({\mu }_{p^{n+1}})$, $\chi$-isotypic minus part) equals the ideal $({\xi }_{\chi })\subseteq \Lambda$. Mazur-Wiles strategy: cuspidal-Eisenstein congruences on ${\Gamma }_1(N{p}^r)$ via $\Lambda$-adic modular forms (Hida 1985 Ann. Sci. ENS 19), the Eisenstein ideal $I_E\subseteq \mathbb{T}$ with $\mathbb{T}/I_E\cong \Lambda$, the associated $\Lambda$-adic Galois representation $\rho :G_{\mathbb{Q}}\to {\mathrm{GL}}_2(\mathbb{T})$ with reduction to the diagonal Eisenstein representation, off-diagonal Galois cohomology classes lifting to $X_{\chi }^{-}$. Reverse divisibility via Iwasawa $p$-adic class-number formula + Ferrero-Washington 1979 Ann. Math. 109 (cyclotomic $\mu$-invariant vanishes for abelian fields). Generalisations: Wiles 1990 Ann. Math. 131 (totally real fields via Hilbert modular forms + Deligne-Ribet $p$-adic L); Rubin 1991 Invent. Math. 103 (Euler-system proof via cyclotomic units, imaginary quadratic fields via elliptic units); Greenberg 1989 Astérisque 165 (framework for arbitrary $p$-adic representations); Kato 2004 Astérisque 295 (Beilinson-Kato Euler system for modular Galois representations); Skinner-Urban 2014 Invent. Math. 195 (Main Conjecture for elliptic curves with good ordinary reduction, residually irreducible ${\bar{\rho }}_{E,p}$, characteristic ideal of dual Selmer = Mazur-Swinnerton-Dyer $p$-adic L); Bertolini-Darmon 2005 Ann. Math. 162 (anticyclotomic Main Conjecture via Heegner-point Euler system). BSD application: Skinner-Urban + Kato proves the $p$-part of Birch-Swinnerton-Dyer for elliptic curves of analytic rank zero; combined with Gross-Zagier 1986 Invent. Math. 84 and Bertolini-Darmon, addresses rank one over imaginary quadratic fields. Originator chain: Kummer 1851 (Bernoulli congruences); von Staudt-Clausen 1840 (Bernoulli denominators); Vandiver 1937 (irregular primes); Iwasawa 1959 Bull. Amer. Math. Soc. 65 (${\mathbb{Z}}_p$-extensions); Kubota-Leopoldt 1964 J. reine angew. Math. 214/215 ($p$-adic L); Iwasawa 1969 (Main Conjecture conjectured); Coates-Wiles 1977 Invent. Math. 39 (BSD-Iwasawa for CM elliptic curves); Mazur-Wiles 1984 (Main Conjecture for $\mathbb{Q}$); Wiles 1990 (totally real); Rubin 1991 (Euler systems); Greenberg 1989 (general framework); Kato 2004 (Beilinson-Kato); Bertolini-Darmon 2005 (anticyclotomic); Skinner-Urban 2014 (elliptic). Forward-connects to [21.06.01] modularity + BSD, [21.07.01] ${\mathbb{Z}}_p$-extensions + Iwasawa theory (sibling), [21.04.02] Hecke operators (Mazur-Wiles uses cuspidal-Eisenstein Hecke congruences), [21.05.01] $\ell$-adic Galois reps (input to Skinner-Urban), [21.03.01] Riemann zeta (Kubota-Leopoldt principal-character case), [21.03.02] Dirichlet L (general-character case). Lean status: partial, module Codex.NumberTheory.Iwasawa.PadicL declares def kubotaLeopoldtL, theorem KL_interpolation_property, def CyclotomicIwasawaModule, def characteristicIdeal, def iwasawaL, theorem main_conjecture_cyclotomic, plus theorem main_conjecture_elliptic (Skinner-Urban), all sorry-stubbed pending Mathlib infrastructure on $p$-adic measure theory, the Amice transform identifying continuous functions on ${\mathbb{Z}}_p$ with elements of $\Lambda ={\mathbb{Z}}_p[[T]]$, the Iwasawa algebra as a completed group ring, the structure theorem for finitely generated torsion $\Lambda$-modules, the cohomological apparatus of $\Lambda$-adic modular forms and the universal deformation Hecke algebra (Hida theory), and the Galois cohomology of Iwasawa modules.

  requires: number-theory.riemann-zeta` (21.03.01 — classical $\zeta$ at negative integers is the seed of $p$-adic interpolation via Bernoulli numbers); `number-theory.dirichlet-l-functions` (21.03.02 — Dirichlet $L$-values at negative integers are the interpolation targets)

• number-theory.modularity-bsd-conjectureopen unit 21.06.01 →

  Modularity Theorem (Statement) and BSD Conjecture

  The modularity theorem (Wiles 1995 + Taylor-Wiles 1995 for semistable $E$; Breuil-Conrad-Diamond-Taylor 2001 for the full statement): every elliptic curve $E$ over $\mathbb{Q}$ of conductor $N_E$ corresponds to a weight-$2$ cusp newform $f_E$ on ${\Gamma }_0(N_E)$ with rational Hecke eigenvalues such that $L(E,s)=L(f_E,s)$ — equivalently the $\ell$-adic Galois representations satisfy ${\rho }_{E,\ell }\cong {\rho }_{f_E,\ell }$ (Faltings 1983 isogeny theorem), equivalently $E$ is isogenous over $\mathbb{Q}$ to the modular abelian variety $A_{f_E}$ associated to $f_E$ by Eichler-Shimura. Local match at good primes: $a_p(f_E)=p+1-\#E({\mathbb{F}}_p)$ via the Eichler-Shimura congruence $T_p={\mathrm{Frob}}_p+p\cdot {\mathrm{Ver}}_p$ on $J_0(N_E)\mathrm{mod}p$ + the Hasse-Weil local zeta function for $E$. BSD conjecture, rank part (Birch-Swinnerton-Dyer 1965): ${\mathrm{ord}}_{s=1}L(E,s)=\mathrm{rank}E(\mathbb{Q})$. BSD refined leading-coefficient formula (Tate 1974): ${\lim }_{s\to 1}(s-1{)}^{-r}L(E,s)={\Omega }_E{R}_E\#\mathrm{Sha}(E/\mathbb{Q}){\prod }_p{c}_p/(\#E(\mathbb{Q}{)}_{\mathrm{tors}}{)}^2$ with ${\Omega }_E$ the real period ${\int }_{E(\mathbb{R})}|\omega |$, $R_E$ the Néron-Tate regulator, $\mathrm{Sha}(E/\mathbb{Q})$ the Tate-Shafarevich group, $c_p$ the Tamagawa numbers via Kodaira's local classification, $E(\mathbb{Q}{)}_{\mathrm{tors}}$ classified by Mazur 1977. Clay Millennium Open Problem since 2000, prize unclaimed as of 2026. Frey-Ribet-Serre-Wiles chain to Fermat: a non-zero $a^{\ell }+b^{\ell }=c^{\ell }$ at prime $\ell \ge 5$ gives the Frey curve $E_{a,b,c}:y^2=x(x-a^{\ell })(x+b^{\ell })$ of conductor $\mathrm{rad}(abc)$; Serre's 1987 $\epsilon$-conjecture proved by Ribet 1990 forces the mod-$\ell$ rep to come from a weight-$2$ newform of level $2$; modularity proved by Wiles 1995 + BCDT 2001 puts the rep at level $\mathrm{rad}(abc)$; but $\dim S_2({\Gamma }_0(2))=0$, contradiction. Partial results towards BSD: Coates-Wiles 1977 (CM rank-$0$ via elliptic-unit Euler system); Gross-Zagier 1986 (Heegner-point height = $c_E{L}^{\prime }(E,1)L(E^K,1)$, lower bound in analytic rank $1$); Kolyvagin 1989 (Euler system of Heegner points, upper bound + Sha finiteness in analytic rank $\le 1$); Kato 2004 (Beilinson elements, one inclusion of Iwasawa main conjecture); Skinner-Urban 2014 (reverse inclusion via Eisenstein congruences on $\mathrm{GU}(2,2)$, full main conjecture in many cases, $p$-part of BSD in analytic rank $\le 1$); Bhargava-Skinner-Zhang programme (BSD holds in a positive proportion when ordered by height). Wiles 1995 strategy: deformation theory of mod-$\ell$ Galois reps (Mazur 1989), the universal deformation ring $R_{\overline{\rho }}^{\square }$, the Hecke algebra ${\mathbb{T}}^{\square }$ with the same residual representation, the natural surjection $R\twoheadrightarrow \mathbb{T}$, the $R=\mathbb{T}$ theorem proved via Taylor-Wiles patching with auxiliary primes; mod-$\ell$ modularity at $\ell =3$ via Langlands-Tunnell, the $3$-$5$ switch when ${\overline{\rho }}_{E,3}$ reducible; BCDT 2001 extends to wild $3$-adic deformation at additive-reduction primes. Originator chain: Mordell 1922 + Weil 1929 (Mordell-Weil); Hasse 1933 (Hasse bound); Taniyama 1955 / Shimura 1960s (Taniyama-Shimura-Weil conjecture); Weil 1967 (converse theorem); Birch-Swinnerton-Dyer 1965 (BSD); Mazur 1977 (torsion classification); Coates-Wiles 1977 (CM BSD); Tate 1974 (BSD refined); Faltings 1983 (isogeny theorem); Frey 1986 + Serre 1987 + Ribet 1990 (chain to Fermat); Gross-Zagier 1986 + Kolyvagin 1989 (analytic rank $\le 1$); Wiles 1995 + Taylor-Wiles 1995 (modularity for semistable); BCDT 2001 (full modularity); Kato 2004 + Skinner-Urban 2014 (Iwasawa main conjecture). Forward-connects to [21.04.03] Eichler-Shimura correspondence (sibling-in-flight, technical bridge), [21.05.01] $\ell$-adic Galois representations (sibling-in-flight, Galois-rep modularity formulation), [21.07.01] ${\mathbb{Z}}_p$-extensions (sibling-in-flight, Iwasawa-theoretic refinement), [21.07.02] $p$-adic $L$-functions and main conjecture (sibling-in-flight, $p$-part of BSD), [21.06.02] Sato-Tate (successor, equidistribution refinement), [21.10.01] Langlands programme (unifying frame), [21.04.02] Hecke operators (operator-theoretic substrate), [21.04.01] modular forms (ambient analytic theory), [04.04.03] elliptic curves (algebraic-geometric foundation). Lateral connections to 04-algebraic-geometry (modular curve $X_0(N)$ and its Jacobian $J_0(N)$, Eichler-Shimura construction of $A_{f_E}$), 07-representation-theory (Galois representations as continuous reps of profinite groups). Lean status: partial, module Codex.NumberTheory.Modularity.ModularityBSD declares the elliptic-curve placeholder opaque EllipticCurveQ, def conductor, def lFunction, theorem modularity_theorem (Wiles + BCDT), theorem hasseWeilConjecture (corollary), opaque shaftaGroup, def regulator, def realPeriod, def tamagawaNumber, structure BirchSwinnertonDyerConjecture packaging the rank and refined-leading-coefficient statements, plus partial-result placeholders theorem coatesWiles_cmRankZero, theorem grossZagier_rankOneLowerBound, theorem kolyvagin_rankAndShaFinite — all sorry-stubbed pending the most ambitious formalisation target in the arithmetic-geometry corner of the Mathlib roadmap (elliptic-curve conductor, modular Galois reps, Eichler-Shimura, Faltings isogeny, Néron-Tate heights, Tate-Shafarevich finiteness, Tamagawa numbers via Kodaira classification, and the entire Wiles-Taylor-Wiles + BCDT deformation-theoretic proof of modularity).

  requires: algebraic-geometry.elliptic-curves` (04.04.03 — the elliptic curve $E/\mathbb{Q}$ on which both modularity and BSD are stated); sibling `number-theory.modular-forms-sl2-z` (21.04.01 — weight-$k$ modular forms; the modularity bridge identifies $E$ with a weight-$2$ cusp newform); sibling `number-theory.hecke-operators` (21.04.02 — Hecke eigenvalues $a_p(f) = p + 1 - N_p(E)$ at primes of good reduction); sibling `number-theory.riemann-zeta` (21.03.01 — prototype $L$-function with analytic continuation, functional equation, and Euler product); sibling `number-theory.dirichlet-l-functions` (21.03.02 — $\mathrm{GL}_1$-case of the $L$-function / Galois-representation correspondence)

• number-theory.ell-adic-galois-representationsopen unit 21.05.01 →

  $\ell$-adic Galois representations

  An $\ell$-adic Galois representation is a continuous group homomorphism $\rho :G_K\to {\mathrm{GL}}_n({\mathbb{Q}}_{\ell })$ for $K$ a number field, $\ell$ a rational prime, and $n\ge 1$ the dimension; continuity is with respect to the profinite topology on the absolute Galois group $G_K=\mathrm{Gal}(\overline{K}/K)$ and the $\ell$-adic topology on ${\mathrm{GL}}_n({\mathbb{Q}}_{\ell })$. Equivalently, $\rho$ stabilises a ${\mathbb{Z}}_{\ell }$-lattice and is described by a continuous map into ${\mathrm{GL}}_n({\mathbb{Z}}_{\ell })$. Three foundational examples: (1) the cyclotomic character ${\chi }_{\mathrm{cyc}}:G_{\mathbb{Q}}\to {\mathbb{Z}}_{\ell }^{\times }$ determined by $\sigma (\zeta )={\zeta }^{{\chi }_{\mathrm{cyc}}(\sigma )}$ on primitive ${\ell }^n$-th roots of unity, with ${\chi }_{\mathrm{cyc}}({\mathrm{Frob}}_p)=p$ for $p\ne \ell$; (2) the Tate module $T_{\ell }E={\underleftarrow{\lim }}_{n}E[{\ell }^n]$ of an elliptic curve $E/K$, a free ${\mathbb{Z}}_{\ell }$-module of rank $2$ on which $G_K$ acts continuously via ${\rho }_{E,\ell }:G_K\to {\mathrm{GL}}_2({\mathbb{Z}}_{\ell })$, with $\det {\rho }_{E,\ell }={\chi }_{\mathrm{cyc}}$ by the Weil pairing and $\mathrm{tr}{\rho }_{E,\ell }({\mathrm{Frob}}_p)=a_p(E)=p+1-\#E({\mathbb{F}}_p)$ at primes of good reduction (Hasse bound $|a_p(E)|\le 2\sqrt{p}$); (3) the modular Galois representation ${\rho }_{f,\ell }:G_{\mathbb{Q}}\to {\mathrm{GL}}_2(\overline{{\mathbb{Q}}_{\ell }})$ attached to a normalised cuspidal Hecke eigenform $f$ of weight $k\ge 2$ and level $N$ (Deligne 1971 via étale cohomology of Kuga-Sato varieties; Deligne-Serre 1974 for $k=1$), with $\mathrm{tr}{\rho }_{f,\ell }({\mathrm{Frob}}_p)=a_p(f)$ and $\det {\rho }_{f,\ell }({\mathrm{Frob}}_p)=p^{k-1}\epsilon (p)$ for $p\nmid N\ell$. Signature theorem (Néron-Ogg-Shafarevich 1968): ${\rho }_{E,\ell }$ is unramified at every prime $v\nmid \ell$ where $E$ has good reduction, proved via the formal-group argument that the reduction map $E[{\ell }^n]({\bar{K}}_v)\to \bar{E}[{\ell }^n](\overline{k(v)})$ is an isomorphism when the residue characteristic is coprime to $\ell$. Consequently every geometric $\ell$-adic Galois representation is unramified outside a finite set of places of $K$ (smooth-proper base change in SGA 4 for the general case). Geometric origin via étale cohomology: $H_{\text{eˊt}}^i(X_{\bar{K}},{\mathbb{Q}}_{\ell })$ for $X/K$ smooth proper is the canonical source of geometric $\ell$-adic Galois representations, with Frobenius eigenvalues at good-reduction primes being Weil numbers of weight $i$ (absolute value $q^{i/2}$ for $q=N(v)$) by the Weil conjectures (Deligne 1974 Publ. Math. IHÉS 43 = Weil I; Deligne 1980 = Weil II). For elliptic curves: $H_{\text{eˊt}}^1(E_{\bar{K}},{\mathbb{Z}}_{\ell })\cong (T_{\ell }E{)}^{\vee }$. Serre's open-image theorem (1972): for non-CM $E/\mathbb{Q}$, ${\rho }_{E,\ell }(G_{\mathbb{Q}})={\mathrm{GL}}_2({\mathbb{Z}}_{\ell })$ for almost every $\ell$. Tate conjecture (Tate 1966 conjecture; Faltings 1983 theorem): $\mathrm{Hom}(A,B)\otimes {\mathbb{Z}}_{\ell }\to {\mathrm{Hom}}_{G_K}(T_{\ell }A,T_{\ell }B)$ is an isomorphism for abelian varieties $A,B$ over a number field. Fontaine's $p$-adic Hodge theory: at the prime $p=\ell$, the period rings $B_{\mathrm{cris}}\subset B_{\mathrm{st}}\subset B_{\mathrm{dR}}$ stratify $p$-adic representations of local Galois groups into crystalline / semistable / de Rham / Hodge-Tate; geometric representations from smooth proper varieties with good reduction are crystalline. Deformation theory and modularity: Mazur 1989 deformation rings $R_{\bar{\rho }}^{\mathrm{univ}}$; Wiles 1995 $R=T$ theorem proves modularity of semistable $E/\mathbb{Q}$ and Fermat's Last Theorem; Khare-Wintenberger 2009 prove Serre's modularity conjecture for odd irreducible mod-$\ell$ representations of $G_{\mathbb{Q}}$. Originator chain: Tate 1966 (Tate module); Serre 1968 (formalism, monograph); Néron-Ogg-Shafarevich 1968 (good-reduction criterion); Deligne 1971 (modular Galois reps via Kuga-Sato); Serre 1972 (open image); Grothendieck 1972-77 SGA 4 / 4.5 / 5 (étale cohomology); Deligne 1974 (Weil I); Deligne-Serre 1974 (weight-1 case); Deligne 1980 (Weil II); Fontaine 1982 (Hodge-Tate); Faltings 1983 (Mordell + Shafarevich + Tate); Mazur 1989 (deformation theory); Fontaine 1994 (period rings); Wiles 1995, Taylor-Wiles 1995, BCDT 2001 (modularity); Khare-Wintenberger 2009 (Serre's conjecture). Forward-connects to [21.04.03] Eichler-Shimura correspondence (sibling-in-flight, modular Galois rep construction for $k=2$), [21.06.01] modularity + BSD (sibling-in-flight, ${\rho }_{E,\ell }\cong {\rho }_{f_E,\ell }$), [21.07.01] Iwasawa theory (sibling-in-flight, cyclotomic-tower deformation), [21.04.02] Hecke operators (sibling, Hecke-eigenvalue / Frobenius-trace dictionary), [21.04.01] modular forms (sibling, source space for modular Galois reps), [21.03.01] Riemann zeta (sibling, the simplest motivic $L$-function), [21.03.02] Dirichlet $L$ (sibling, ${\mathrm{GL}}_1$-case of the Galois-representation $L$-function). Lateral connections to 04-algebraic-geometry (étale cohomology of smooth proper varieties), 07-rep-theory (continuous representations of profinite groups, Tannakian categories). Lean status: partial, module Codex.NumberTheory.GaloisReps.EllAdic declares structure EllAdicGaloisRep (number field, prime, dimension, continuous representation), def cyclotomicCharacter (${\chi }_{\mathrm{cyc}}:G_{\mathbb{Q}}\to {\mathbb{Z}}_{\ell }^{\times }$), def tateModuleEllipticCurve ($T_{\ell }E$ as the inverse limit of ${\ell }^n$-torsion), def galoisRepEllipticCurve (${\rho }_{E,\ell }:G_K\to {\mathrm{GL}}_2({\mathbb{Z}}_{\ell })$), def modularGaloisRep (${\rho }_{f,\ell }$ attached to a Hecke eigenform), theorem unramified_outside_finite_set (Néron-Ogg-Shafarevich + smooth-proper base change), and the étale-cohomology placeholder def etaleCohomologyGaloisRep, all sorry-stubbed pending Mathlib infrastructure on the absolute Galois group as a profinite group with explicit continuity machinery, the cyclotomic-tower construction, the inverse-limit machinery for ${\ell }^n$-torsion of abelian varieties, the good-reduction integral model and Néron-Ogg-Shafarevich criterion, the Kuga-Sato variety construction and Deligne-Serre attachment theorem for modular Galois representations, and the étale-cohomology API for smooth proper varieties over a number field with smooth-proper base change.

  requires: sibling `number-theory.hecke-operators` (21.04.02 — modular Galois representation $\rho_{f, \ell}$ is attached via Hecke-eigenvalue data, with $\mathrm{tr}\, \rho_{f, \ell}(\mathrm{Frob}_p) = a_p(f)$); `foundations.algebraic-field-extension` (01.02.12 — Galois-theoretic machinery, finite extensions, splitting fields underlying the absolute Galois group $G_K = \mathrm{Gal}(\overline{K}/K)$)

• number-theory.zp-extensions-iwasawaopen unit 21.07.01 →

  ${\mathbb{Z}}_p$-extensions and Iwasawa Theory

  A ${\mathbb{Z}}_p$-extension of a number field $K$ is a Galois extension $K_{\infty }/K$ with $\mathrm{Gal}(K_{\infty }/K)\cong {\mathbb{Z}}_p$ as topological group; equivalently the directed union of a tower $K=K_0\subset K_1\subset \cdots$ with $[K_n:K]=p^n$ and $\mathrm{Gal}(K_n/K)\cong \mathbb{Z}/p^n$. The cyclotomic ${\mathbb{Z}}_p$-extension $K_{\infty }^{\mathrm{cyc}}=K\cdot {\mathbb{Q}}_{\infty }^{\mathrm{cyc}}$ is the canonical example; for $K=\mathbb{Q}$ it is the unique ${\mathbb{Z}}_p$-extension. The number of independent ${\mathbb{Z}}_p$-extensions is $r_2(K)+1+{\delta }_K$ with ${\delta }_K\ge 0$ controlled by the Leopoldt conjecture. The Iwasawa algebra $\Lambda ={\mathbb{Z}}_p[[\Gamma ]]:={\underleftarrow{\lim }}_n{\mathbb{Z}}_p[\Gamma /{\Gamma }^{p^n}]$ is a complete Noetherian regular local UFD of Krull dimension $2$, with maximal ideal $\mathfrak{m}=(p,T)$ and residue field ${\mathbb{F}}_p$; the choice of a topological generator $\gamma \in \Gamma$ gives a topological-ring isomorphism $\Lambda \cong {\mathbb{Z}}_p[[T]]$ sending $\gamma -1\mapsto T$. The kernel of $\Lambda \twoheadrightarrow {\mathbb{Z}}_p[\Gamma /{\Gamma }^{p^n}]$ is generated by ${\omega }_n(T)=(1+T{)}^{p^n}-1$. The Weierstrass preparation theorem factors each non-zero $f\in \Lambda$ as $f=p^{\mu }\cdot u(T)\cdot P(T)$ with $u\in {\Lambda }^{\times }$ and $P\in {\mathbb{Z}}_p[T]$ a distinguished polynomial (monic, non-leading coefficients in $p{\mathbb{Z}}_p$); the invariants $\mu =v_p(f)$ and $\lambda =\mathrm{deg}P$ are intrinsic to $f$. Signature theorem (Iwasawa 1959 growth formula): for a ${\mathbb{Z}}_p$-extension $K_{\infty }/K$, the Iwasawa module $X_{\infty }={\underleftarrow{\lim }}_n{\mathrm{Cl}}_p(K_n)$ (inverse limit of $p$-Sylows of class groups under norm maps) is a finitely generated $\Lambda$-module, $\Lambda$-torsion when only finitely many primes ramify (always true for the cyclotomic ${\mathbb{Z}}_p$-extension), and there exist integers $\lambda \ge 0$, $\mu \ge 0$, $\nu \in \mathbb{Z}$ such that $|{\mathrm{Cl}}_p(K_n)|=p^{\lambda n+\mu p^n+\nu }$ for $n\gg 0$. Proof: the structure theorem for finitely generated torsion $\Lambda$-modules gives a pseudo-isomorphism $X_{\infty }\sim {⨁}_i\Lambda /(f_i^{n_i})\oplus {⨁}_j\Lambda /(p^{m_j})$ with $f_i$ distinguished; the control theorem identifies ${\mathrm{Cl}}_p(K_n)\cong X_{\infty }/{\omega }_n{X}_{\infty }$ for $n\gg 0$; the size of the ${\omega }_n$-quotient is $p^{\lambda n}$ from the cyclic-polynomial factors ($\lambda =\sum n_i\mathrm{deg}{f}_i$) and $p^{\mu p^n}$ from the $p$-power factors ($\mu =\sum m_j$), with the stabilising correction $\nu$. The characteristic ideal ${\mathrm{char}}_{\Lambda }(M)=(p^{\mu }\prod f_i^{n_i})\subseteq \Lambda$ is the principal-ideal Iwasawa invariant of a finitely generated torsion $\Lambda$-module, the analogue of the order of a finite abelian group. Ferrero-Washington 1979: $\mu =0$ for the cyclotomic ${\mathbb{Z}}_p$-extension of any abelian extension of $\mathbb{Q}$, proved by direct analytic computation on Stickelberger digits of Bernoulli numbers. Greenberg conjecture (1976): $\lambda =\mu =0$ for the cyclotomic ${\mathbb{Z}}_p$-extension of a totally real field; open in general, extensively verified numerically. Generalises through Mazur 1972 (elliptic-curve Iwasawa theory, $X_{\infty }(E)={\underleftarrow{\lim }}_n{\mathrm{Sel}}_{p^{\infty }}(E/K_n{)}^{\vee }$), Greenberg 1989 (Iwasawa theory for arbitrary $p$-adic Galois representations), Coates-Fukaya-Kato-Sujatha-Venjakob 2005 (non-commutative Iwasawa theory). Originator chain: Kummer 1850s (irregular primes, $p$-parts of $\mathrm{Cl}(\mathbb{Q}({\zeta }_p))$); Vandiver 1929 (Vandiver conjecture on ${\mathrm{Cl}}_p(\mathbb{Q}({\zeta }_p{)}^{+})$); Iwasawa 1959 Bull. AMS 65 (originator paper); Iwasawa 1969 J. Math. Soc. Japan 20 (general ${\mathbb{Z}}_p$-extensions, Main Conjecture conjectured); Iwasawa 1973 Ann. Math. 98 (analytic side); Mazur 1972 Invent. Math. 18 (elliptic curves); Greenberg 1976 Amer. J. Math. 98 (Greenberg conjecture, Iwasawa BSD); Ferrero-Washington 1979 Ann. Math. 109 ($\mu =0$); Mazur-Wiles 1984 Invent. Math. 76 (Main Conjecture for $\mathbb{Q}$); Wiles 1990 Ann. Math. 131 (totally real fields); Greenberg 1989 (general $p$-adic representations); Coates-Fukaya-Kato-Sujatha-Venjakob 2005 (non-commutative). Forward-connects to [21.07.02] $p$-adic $L$-functions and the Iwasawa Main Conjecture (sibling, analytic side), [21.06.01] modularity + BSD (Iwasawa formulation of BSD), [21.05.01] $\ell$-adic Galois representations (Selmer-group Iwasawa theory), [21.03.01]/[21.03.02] Riemann zeta and Dirichlet $L$-functions (interpolation targets of $p$-adic $L$). Lean status: partial, module Codex.NumberTheory.Iwasawa.ZpExtensions declares structure ZpExtension, def iwasawaAlgebra (as PowerSeries ℤ_[p]), noncomputable def omegaPoly, noncomputable def iwasawaModule, def IsDistinguished, theorem structure_theorem_iwasawa_modules, noncomputable def charIdealGen, structure IwasawaInvariants, theorem iwasawa_growth_formula, theorem ferrero_washington_mu_zero, theorem iwasawa_main_conjecture_Q (statement only) — all sorry-stubbed pending Mathlib infrastructure on the infinite Galois-extension API at the ${\mathbb{Z}}_p$-pro-cyclic level, the pro-finite inverse-limit module theory over a complete topological ring, the Weierstrass preparation theorem for ${\mathbb{Z}}_p[[T]]$, the structure theorem for finitely generated torsion modules over a two-dimensional regular local UFD, and the class-field-theoretic control theorem identifying ${\mathrm{Cl}}_p(K_n)\cong X_{\infty }/{\omega }_n{X}_{\infty }$.

  requires: foundations.field-extension-splitting-field` (01.02.12 — Galois-theoretic infinite extension foundations); `lie-groups.padic-lie-group-and-the-padic-exponential` (03.03.05 — topology of $\mathbb{Z}_p$ as the Galois group of the tower)

• number-theory.dedekind-hecke-artin-l-functionsopen unit 21.03.03 →

  Dedekind Zeta Function, Hecke $L$-Functions, Artin $L$-Functions

  Survey unit consolidating the $L$-function family ${\zeta }_K$, Hecke $L$, Artin $L$ as the spine of modern number theory. Dedekind 1879: introduced ${\zeta }_K(s)={\sum }_{\mathfrak{a}\subset {\mathcal{O}}_K}N(\mathfrak{a}{)}^{-s}$ on $\mathrm{Re}(s)>1$ for any number field $K$, with Euler product ${\prod }_{\mathfrak{p}}(1-N(\mathfrak{p}{)}^{-s}{)}^{-1}$ over prime ideals encoding unique factorisation of ideals in the Dedekind domain ${\mathcal{O}}_K$. Signature theorem (analytic class-number formula): ${\mathrm{res}}_{s=1}{\zeta }_K(s)=2^{r_1}(2\pi {)}^{r_2}{h}_K{R}_K/(w_K\sqrt{|d_K|})$, linking the residue to the five invariants $(h_K,R_K,w_K,|d_K|,(r_1,r_2))$ by Minkowski-lattice-point counting of ideals of bounded norm. Hecke 1918, 1920: extended to Hecke characters $\chi :I_K(\mathfrak{m})\to {\mathbb{C}}^{\times }$ on the group of fractional ideals coprime to a modulus $\mathfrak{m}$, factoring through the ray class group on the finite part and prescribed by an algebraic Grössencharacter on the archimedean part. Hecke $L$-function: $L(s,\chi )={\sum }_{\mathfrak{a}}\chi (\mathfrak{a})N(\mathfrak{a}{)}^{-s}={\prod }_{\mathfrak{p}}(1-\chi (\mathfrak{p})N(\mathfrak{p}{)}^{-s}{)}^{-1}$. Functional equation (Hecke 1920): completed $\Lambda (s,\chi )=N(\mathfrak{f}(\chi ){)}^{s/2}{\prod }_v{L}_v(s,\chi )L(s,\chi )$ with archimedean Gamma factors satisfies $\Lambda (s,\chi )=W(\chi )\Lambda (1-s,\overline{\chi })$, root number $|W(\chi )|=1$, Gauss-sum-built. Proved via higher-dimensional theta inversion on the lattice ${\mathcal{O}}_K\subset K{\otimes }_{\mathbb{Q}}\mathbb{R}$. Artin 1923: extended to Artin $L$-functions $L(s,\rho ,L/K)={\prod }_{\mathfrak{p}}\det (1-\rho ({\mathrm{Frob}}_{\mathfrak{p}})N(\mathfrak{p}{)}^{-s}\mid V^{I_{\mathfrak{p}}}{)}^{-1}$ attached to a finite-dimensional Galois representation $\rho :\mathrm{Gal}(L/K)\to {\mathrm{GL}}_n(\mathbb{C})$, with inertia-fixed restriction at ramified primes. Signature theorem (Artin factorisation): ${\zeta }_L(s)={\prod }_{\rho \in \hat{G}}L(s,\rho ,L/K{)}^{\dim \rho }$ over irreducible representations of $\mathrm{Gal}(L/K)$, proved by Euler-factor matching using the regular-representation decomposition ${\mathrm{Reg}}_G={⨁}_{\rho }{\rho }^{\dim \rho }$ and Artin additivity / inductivity. Artin holomorphy conjecture (1923, open in general): $L(s,\rho )$ entire for $\rho$ non-trivial irreducible; known for $\dim \rho =1$ via Artin reciprocity (Artin 1927) identifying one-dimensional Artin $L$ with Hecke $L$ of a Grössencharacter; known for solvable two-dimensional $\rho$ (Langlands-Tunnell base change); known for odd two-dimensional $\rho /\mathbb{Q}$ (Khare-Wintenberger 2009, Serre's modularity conjecture). Brauer 1947: every Artin $L$-function meromorphic on $\mathbb{C}$, via the Brauer induction theorem (every character of a finite group is an integral combination of induced characters from one-dimensional characters of elementary subgroups) combined with Artin inductivity and one-dimensional-equals-Hecke. Aramata-Brauer corollary: ${\zeta }_L(s)/{\zeta }_K(s)$ is entire for every finite Galois $L/K$. Tate 1950 (PhD thesis): adelic-and-Fourier-analytic reformulation casting Hecke $L$ as ${\mathrm{GL}}_1$ zeta integrals ${\int }_{{\mathbb{A}}_K^{\times }}f(x)\chi (x)|x{|}^s{d}^{\times }x$ against Schwartz-Bruhat functions on the idele group, with the functional equation derived from Pontryagin self-duality of ${\mathbb{A}}_K$. Weil 1952: classification of Hecke characters by type at infinity, isolating Type $A_0$ algebraic Grössencharacters appearing in CM elliptic-curve $L$-functions (Deuring 1953-57, Shimura 1971). Langlands 1970: functoriality conjectures placing every Artin $L$ inside the ${\mathrm{GL}}_n$ automorphic $L$-function framework, implying Artin holomorphy as a corollary of functoriality. Generalises through [21.04.01] modular forms (where weight-1 cusp forms attach two-dimensional Artin representations via Deligne-Serre 1974), [21.05.01] $\ell$-adic Galois representations (where the Artin formalism extends to $\ell$-adic coefficients via Grothendieck-Deligne étale cohomology), [21.06.01] modularity + BSD (the modular-Galois reciprocity $L(s,{\rho }_E)=L(s,f_E)$ for elliptic curves over $\mathbb{Q}$), [21.10.01] Langlands programme (the chapter-closing unification). Originator chain: Dirichlet 1837 (Dirichlet $L$ and class-number formula for quadratic fields); Dedekind 1879 (general ${\zeta }_K$, ideal-theoretic algebraic number theory, analytic class-number formula); Hecke 1918, 1920 (Hecke characters and Hecke $L$, theta-symmetry functional equation); Artin 1923 (Artin $L$-functions, factorisation, holomorphy conjecture); Artin 1927/1930 (Artin reciprocity, one-dimensional Artin = Hecke); Brauer 1947 (Brauer induction, meromorphic continuation); Tate 1950 (adelic Tate thesis); Weil 1952 (Type $A_0$); Deuring 1953-57 (CM elliptic-curve $L$); Langlands 1970 (functoriality conjectures); Langlands-Tunnell 1980-81 (two-dimensional solvable Artin); Wiles 1995 + BCDT 2001 (modularity for elliptic curves over $\mathbb{Q}$); Khare-Wintenberger 2009 (Serre's modularity for odd two-dimensional Artin over $\mathbb{Q}$). Lean status: partial, module Codex.NumberTheory.LFunctions.DedekindHeckeArtin declares noncomputable def dedekindZeta, structure HeckeCharacter, noncomputable def heckeL, noncomputable def artinL, theorem artin_factorisation, theorem artinL_additive, theorem artinL_inductive, theorem artinL_meromorphic, def ArtinHolomorphy, theorem aramata_brauer, plus the Dedekind functional equation and class-number-formula residue statements — all sorry-stubbed pending Mathlib infrastructure on ideal-norm Dirichlet-series meromorphic continuation, the idele-class-group Hecke-character formalism, the Artin-formalism connecting finite-group representation theory to $L$-functions, and the Brauer induction theorem with its Hecke-reduction corollary.

  requires: sibling `number-theory.riemann-zeta` (21.03.01 — degenerate $K = \mathbb{Q}$ case of $\zeta_K$); sibling `number-theory.dirichlet-l-functions` (21.03.02 — one-dimensional Hecke character on $\mathbb{Q}$ is a Dirichlet character; rank-1 case of Hecke $L$); `algebra.algebraic-field-extension` (01.02.12 — number field $K/\mathbb{Q}$ and Galois extensions); `representation-theory.group-representation` (07.01.01 — codomain of an Artin representation $\rho : \mathrm{Gal}(L/K) \to \mathrm{GL}_n(\mathbb{C})$)

• diff-geom.padics-lie-groupopen unit 03.03.05 →

  $p$-adic Lie groups and the $p$-adic exponential

  $p$-adic Lie group = Lie group object in the category of $p$-adic analytic manifolds. The $p$-adic exponential ${\exp }_p:p{\mathbb{Z}}_p\to 1+p{\mathbb{Z}}_p$ converges on the open ball of radius $p^{-1/(p-1)}$ around 0 (for $p$ odd; smaller radius for $p=2$). ${\mathbb{Z}}_p$ itself is the prototype $p$-adic Lie group, and its profinite topology is the source of the Iwasawa-algebra structure $\Lambda ={\mathbb{Z}}_p[[\Gamma ]]$. Standard examples include ${\mathrm{GL}}_n({\mathbb{Z}}_p)$, ${\mathrm{SL}}_n({\mathbb{Z}}_p)$, and the unit group of a $p$-adic division algebra. The Lazard correspondence equates $p$-saturable $p$-adic Lie groups with ${\mathbb{Z}}_p$-Lie algebras of finite rank. Used in: Galois representations of number fields (via $\mathrm{Gal}(K_{\infty }/K)\cong {\mathbb{Z}}_p$ for cyclotomic and other towers), $p$-adic Hodge theory (Fontaine's rings $B_{\mathrm{dR}},B_{\mathrm{cris}}$), Iwasawa theory, $p$-adic representation theory.

  requires: lie-groups.lie-group-and-the-lie-algebra, topology.metric-space

• biology.metacommunity-theoryno unit yet

  Metacommunity dynamics and spatial ecology

  Metacommunity theory extends community ecology to spatially structured landscapes. Four paradigms (patch dynamics, species sorting, mass effects, neutral theory) organise the field. Key results include the competition-colonisation trade-off (Tilman 1994), MacArthur-Wilson island biogeography equilibrium, Hubbell's neutral theory species-abundance predictions, and beta diversity partitioning into turnover and nestedness components. Builds toward landscape ecology and conservation biogeography.

  requires: biology.community-ecology

• biology.food-web-network-structureno unit yet

  Food web assembly, network structure, and interaction strength

  Food web assembly theory (Drake 1990, Law & Morton 1996) addresses how communities self-organise through sequential colonisation. Network motifs reveal structural building blocks: nested architecture in mutualistic webs, modular structure in antagonistic webs. The niche model (Williams & Martinez 2000) generates realistic food webs from simple allometric and niche-overlap rules. Interaction strength distributions are strongly skewed (many weak, few strong), and this skew stabilises food web dynamics (McCann et al. 1998). Trophic cascades in complex webs differ from linear-chain cascades due to omnivory and detrital pathways.

  requires: biology.community-ecology

• biology.complexity-stabilityno unit yet

  Complexity-stability analysis in ecological communities

  May's (1972) random-matrix analysis showed that increasing species richness and connectance destabilises randomly assembled communities when the stability criterion sigma*sqrt(SC) > 1. Real food webs resolve this paradox through compartmentalisation, interaction-strength skew, and allometric constraints on body-size ratios. Weak interactions provide stabilising feedback (McCann et al. 1998). Omnivory dampens population oscillations. The interplay between network structure and dynamical stability is the central problem of theoretical community ecology.

  requires: biology.community-ecology

• biology.hormone-receptor-signallingno unit yet

  Hormone receptor mechanisms and intracellular signalling cascades

  G-protein-coupled receptors (GPCRs) mediate peptide hormone signalling via cAMP and IP3/DAG second-messenger cascades. Receptor tyrosine kinases (RTKs) mediate insulin and growth factor signalling via PI3K/Akt and MAPK pathways. Nuclear hormone receptors mediate steroid and thyroid hormone action as ligand-activated transcription factors. Receptor desensitisation (GRK phosphorylation, beta-arrestin internalisation) and downregulation modulate cellular responsiveness to sustained hormone exposure. Covered as a Master section within 18.07.01.

  requires: biology.endocrine-hormones

• biology.endocrine-dynamicsno unit yet

  Mathematical models of endocrine feedback dynamics

  The Goodwin oscillator model of hormone cascades with Hill-type negative feedback. Delay differential equations capturing transport and action delays in endocrine axes. Hopf bifurcation from steady-state to oscillatory hormone release when feedback cooperativity or delays exceed critical thresholds. Pulsatile hormone release and ultradian rhythms as emergent properties of feedback dynamics. Frequency-dependent decoding of GnRH pulse trains by gonadotroph cells. Covered as a Master section within 18.07.01.

  requires: biology.endocrine-hormones

• biology.parathyroid-calcium-regulationno unit yet

  Parathyroid-calcium-vitamin D axis and mineral homeostasis

  Three-hormone system (PTH, calcitriol, calcitonin) regulating ionised calcium concentration. The calcium-sensing receptor (CaSR) on parathyroid chief cells as the molecular set point. PTH actions on bone (osteoclast activation via RANKL), kidney (calcium reabsorption, phosphate excretion, 1-alpha-hydroxylase stimulation), and intestine (indirect via vitamin D activation). Vitamin D activation cascade: skin synthesis, hepatic 25-hydroxylation, renal 1-alpha-hydroxylation regulated by PTH and FGF-23. Calcitonin as physiological antagonist from thyroid C cells. Covered as a Master section within 18.07.01.

  requires: biology.endocrine-hormones

• biology.metabolism-first-originno unit yet

  Metabolism-first scenario and the iron-sulfur world

  Autotrophic-origin models proposing that self-sustaining chemical reaction networks preceded self-replicating molecules. Wachtershauser's iron-sulfur world (1988): pyrite precipitation provides exergonic driving force and positively charged surface for concentrating anions; surface metabolism builds complexity through successive extension without enzymatic machinery. Russell-Lane alkaline hydrothermal vent model: Lost City-type vents produce natural proton gradients across iron-sulfur mineral pore walls, providing continuous free energy (proton motive force), mineral catalysis, and physical compartmentalisation. The Wood-Ljungdahl pathway (acetogens and methanogens) is the strongest candidate for a primordial carbon-fixation route. Universal chemiosmotic coupling (Mitchell 1961, Nobel 1978) across all three domains of life is interpreted as a relic of the vent environment. Covered as a Master section within 19.15.01.

  requires: biology.origin-of-life

• biology.quasispecies-error-thresholdno unit yet

  Quasispecies theory and the error threshold

  Eigen's quasispecies theory formalises the tension between replication fidelity and genome length. The error threshold relation constrains maximum genome length to approximately ln(sigma)/(1-q) where sigma is the superiority parameter and q is per-base copying fidelity. For RNA without proofreading (q ~ 0.99), the maximum stable genome is ~230 nucleotides — the information catastrophe. Spiegelman's monster (serial transfer experiment, 1967) empirically demonstrates information reduction under selection for replication speed alone. Hypercycles (Eigen & Schuster 1977) distribute information across multiple cooperating replicators to exceed the single-molecule threshold, but require compartmentalisation for parasite resistance (Szathmary & Demeter 1987). The transition to DNA with proofreading enzymes lowers the error rate to 10^-6 to 10^-9, permitting million-base-pair genomes. Covered as a Master section within 19.15.01.

  requires: biology.origin-of-life

• biology.homochirality-originno unit yet

  Homochirality and the origin of biological asymmetry

  Terrestrial life uses exclusively L-amino acids and D-sugars (homochirality), despite prebiotic synthesis producing racemic mixtures. Frank's model (1953) shows that coupled autocatalysis and mutual antagonism amplify any tiny initial enantiomeric excess to homochirality. The Soai reaction (1995) demonstrates asymmetric autocatalytic amplification from 10^-6 enantiomeric excess to greater than 99.5%. Viedma ripening (2005) achieves complete chiral conversion via grinding-induced Ostwald ripening with solution-phase racemisation. Proposed initial bias sources include parity violation in the weak nuclear force (Vester-Ulbricht hypothesis, enantiomeric excess ~10^-11) and circularly polarised UV in star-forming regions. Carbonaceous chondrite meteorites show L-amino acid excesses of 2-18%, consistent with a systematic physical cause. Covered as a Master section within 19.15.01.

  requires: biology.origin-of-life