All units

1828 units total · 1828 shown

ID	Title	Section
`00.01.01`	Real numbers, integers, rationals	Precalculus foundations
`00.01.02`	Absolute value and the triangle inequality	Precalculus foundations
`00.01.03`	Polynomials and rational expressions	Precalculus foundations
`00.01.E1`	Algebra and number-systems exercise pack (Lang Basic Mathematics Part I-II supplement)	Precalculus foundations
`00.02.05`	Function	Precalculus foundations
`00.03.01`	Linear equations and the line	Precalculus foundations
`00.03.02`	Quadratic equations and the quadratic formula	Precalculus foundations
`00.03.E1`	Functions, trigonometry, and coordinate-geometry exercise pack (Lang Basic Mathematics Part III-V supplement)	Precalculus foundations
`00.04.01`	Inequalities (linear and quadratic)	Precalculus foundations
`00.05.01`	Real exponents and exponential function	Precalculus foundations
`00.05.02`	Logarithms as inverses of exponentials	Precalculus foundations
`00.05.03`	Complex numbers (introductory)	Precalculus foundations
`00.06.01`	Right-triangle trigonometry	Precalculus foundations
`00.06.02`	Inverse trigonometric functions	Precalculus foundations
`00.06.03`	Law of sines and law of cosines	Precalculus foundations
`00.07.01`	Unit-circle trigonometry	Precalculus foundations
`00.07.02`	Trigonometric identities: sum, difference, and double-angle	Precalculus foundations
`00.08.01`	Trigonometric identities (addition formulas)	Precalculus foundations
`00.08.02`	Law of sines and law of cosines	Precalculus foundations
`00.09.01`	Cartesian coordinates and distance in the plane	Precalculus foundations
`00.10.01`	Conic sections (parabola, ellipse, hyperbola)	Precalculus foundations
`00.11.01`	Polar coordinates and parametric curves	Precalculus foundations
`00.11.02`	Conic-section parametrisations and intersections	Precalculus foundations
`00.12.01`	Mathematical induction	Precalculus foundations
`00.12.02`	Binomial theorem and Pascal's triangle	Precalculus foundations
`00.13.01`	Plane geometry (distance, area, pi)	Precalculus foundations
`00.13.02`	Solid geometry (volume)	Precalculus foundations
`01.01.01`	Field	Algebra & linear algebra
`01.01.02`	Dual space and double dual	Algebra & linear algebra
`01.01.03`	Vector space	Algebra & linear algebra
`01.01.04`	Subspace, basis, dimension	Algebra & linear algebra
`01.01.05`	Linear transformation: kernel, image, rank-nullity	Algebra & linear algebra
`01.01.06`	Systems of linear equations and the Kronecker-Capelli theorem	Algebra & linear algebra
`01.01.07`	Determinant: axiomatic + expansion + properties	Algebra & linear algebra
`01.01.08`	Eigenvalue, eigenvector, characteristic polynomial	Algebra & linear algebra
`01.01.09`	Gram-Schmidt orthonormalisation and finite-dim inner-product space	Algebra & linear algebra
`01.01.10`	Adjoint operator and isometry on a finite-dimensional inner-product space	Algebra & linear algebra
`01.01.11`	Jordan canonical form and minimal polynomial	Algebra & linear algebra
`01.01.12`	Singular value decomposition (finite-dim)	Algebra & linear algebra
`01.01.13`	Spectral theorem for normal operators on a finite-dim inner-product space (principal-axes theorem)	Algebra & linear algebra
`01.01.14`	Rayleigh quotient and the Courant-Fischer min-max characterisation of eigenvalues	Algebra & linear algebra
`01.01.15`	Bilinear form / quadratic form	Algebra & linear algebra
`01.01.16`	Invariant subspaces and the primary decomposition	Algebra & linear algebra
`01.01.17`	Change of basis and the transformation laws	Algebra & linear algebra
`01.01.18`	Linear manifolds, hyperplanes, and affine subspaces	Algebra & linear algebra
`01.01.19`	Simultaneous diagonalisation of two quadratic forms and the generalised eigenvalue problem	Algebra & linear algebra
`01.01.E1`	Linear algebra exercise pack (Apostol Vol. 2 Ch. 1-5 supplement)	Algebra & linear algebra
`01.02.01`	Group	Algebra & linear algebra
`01.02.02`	Subgroup, coset, quotient group, isomorphism theorems	Algebra & linear algebra
`01.02.03`	Group action, orbit-stabiliser, class equation	Algebra & linear algebra
`01.02.04`	Sylow theorems	Algebra & linear algebra
`01.02.05`	Solvable group, nilpotent group, Jordan-Holder theorem	Algebra & linear algebra
`01.02.06`	Ring, ring homomorphism, and ideal	Algebra & linear algebra
`01.02.07`	Polynomial rings, PIDs, UFDs, and Euclidean domains	Algebra & linear algebra
`01.02.08`	Localisation of a commutative ring	Algebra & linear algebra
`01.02.09`	Category, functor, natural transformation, the Yoneda lemma, and adjunction	Algebra & linear algebra
`01.02.10`	Tensor product of modules (commutative case)	Algebra & linear algebra
`01.02.11`	Exact sequence, short five lemma, snake lemma	Algebra & linear algebra
`01.02.12`	Algebraic field extension, degree, splitting field	Algebra & linear algebra
`01.02.13`	Fundamental Theorem of Galois Theory (finite case)	Algebra & linear algebra
`01.02.14`	Semisimple rings, the Artin-Wedderburn structure theorem, the Jacobson radical	Algebra & linear algebra
`01.02.15`	Galois cohomology, Hilbert's Theorem 90, and the Brauer group of a field	Algebra & linear algebra
`01.02.16`	Nakayama's lemma	Algebra & linear algebra
`01.02.17`	Hilbert basis theorem; Noetherian rings and modules	Algebra & linear algebra
`01.02.19`	Tensor algebra, exterior algebra, symmetric algebra	Algebra & linear algebra
`01.02.20`	Free group, free product, group presentation	Algebra & linear algebra
`01.02.22`	Krull dimension; Krull's principal ideal theorem	Algebra & linear algebra
`01.02.30`	Chain complex in an abelian category	Algebra & linear algebra
`01.02.31`	Chain homotopy and the homotopy category $K (A)$	Algebra & linear algebra
`01.02.32`	Mapping cone of a chain map and the distinguished triangle	Algebra & linear algebra
`01.02.33`	Abelian category and Grothendieck axioms AB1-AB5	Algebra & linear algebra
`01.02.35`	Dold-Kan correspondence	Algebra & linear algebra
`02.01.01`	Topological space	Analysis
`02.01.02`	Continuous map	Analysis
`02.01.05`	Metric space	Analysis
`02.01.06`	Quotient and identification topology	Analysis
`02.01.07`	Fibration (Hurewicz and Serre)	Analysis
`02.01.08`	Cofibration and homotopy extension property	Analysis
`02.01.09`	Compact-open topology and function spaces	Analysis
`02.01.10`	Fibre Homotopy Equivalence and Dold's Theorem	Analysis
`02.01.E1`	Point-set topology and the fundamental groupoid exercise pack (Brown, Topology and Groupoids supplement)	Analysis
`02.02.01`	Real-number axioms (ordered field)	Analysis
`02.03.02`	Cauchy sequences and Bolzano-Weierstrass	Analysis
`02.03.03`	Infinite series: convergence and the standard tests	Analysis
`02.04.01`	Step-function integral and the Darboux integral	Analysis
`02.04.03`	Integrability of continuous functions on [a,b]	Analysis
`02.04.04`	Fundamental theorems of calculus (FTC1 and FTC2)	Analysis
`02.04.06`	Improper integrals and the comparison test	Analysis
`02.05.01`	Multi-variable limit and continuity	Analysis
`02.05.02`	Mean value theorem (Rolle, Lagrange, Cauchy)	Analysis
`02.05.03`	Chain rule for multi-variable functions	Analysis
`02.05.04`	Implicit and inverse function theorems	Analysis
`02.05.05`	Taylor's theorem and extrema in several variables	Analysis
`02.05.E1`	Multivariable calculus exercise pack (Apostol Vol. 2 Ch. 8-9 supplement)	Analysis
`02.06.01`	Logarithm as an integral	Analysis
`02.06.02`	n-th-order linear ODE with constant coefficients	Analysis
`02.06.03`	Systems of linear ODEs and the matrix exponential	Analysis
`02.06.04`	Hyperbolic functions	Analysis
`02.06.E1`	Ordinary differential equations exercise pack (Apostol Vol. 2 Ch. 6-7 supplement)	Analysis
`02.07.01`	Sigma-algebra, Measurable Space, and the Borel Sigma-algebra	Analysis
`02.07.02`	Lebesgue Outer Measure and the Carathéodory Construction	Analysis
`02.07.03`	Measurable Functions, Simple Functions, Egorov's Theorem, and Lusin's Theorem	Analysis
`02.07.04`	Lebesgue Integral Construction and the Monotone Convergence Theorem	Analysis
`02.07.05`	Fatou's Lemma and the Dominated Convergence Theorem	Analysis
`02.07.06`	L^p Spaces: Hölder, Minkowski, and Riesz-Fischer Completeness	Analysis
`02.07.07`	Fubini-Tonelli Theorem and Product Measures	Analysis
`02.07.08`	Absolute Continuity and the Radon-Nikodym Theorem	Analysis
`02.07.09`	The Whitney Extension Theorem and the Whitney Cube Decomposition	Analysis
`02.07.10`	Rademacher's theorem	Analysis
`02.07.11`	The Area and Coarea Formulas	Analysis
`02.07.E1`	Geometric measure theory exercise pack (Whitney / Federer Ch. 2-3 supplement)	Analysis
`02.08.01`	First-order linear and separable ODEs	Analysis
`02.08.02`	Second-order linear ODEs with constant coefficients	Analysis
`02.09.01`	Complex numbers and Euler's formula	Analysis
`02.10.01`	Fourier Series and the Riemann-Lebesgue Lemma	Analysis
`02.10.04`	Fourier Transform on R^n and the Plancherel Theorem	Analysis
`02.10.05`	Surface integral and parametric surfaces	Analysis
`02.10.06`	The Bochner-Minlos theorem and characteristic functionals on nuclear spaces	Analysis
`02.10.07`	The Radon transform: inversion, Plancherel, and the range theorem	Analysis
`02.10.E1`	Vector calculus exercise pack (Apostol Vol. 2 Ch. 10-12 supplement)	Analysis
`02.11.01`	Bounded linear operators	Analysis
`02.11.02`	Hahn-Banach theorem (analytic and geometric forms)	Analysis
`02.11.03`	Unbounded self-adjoint operators	Analysis
`02.11.04`	Banach space fundamentals	Analysis
`02.11.05`	Compact operators	Analysis
`02.11.06`	Normed vector space	Analysis
`02.11.07`	Inner product space	Analysis
`02.11.08`	Hilbert space	Analysis
`02.11.09`	Open mapping and closed graph theorems	Analysis
`02.12.01`	Phase space, vector field, integral curve	Analysis
`02.12.02`	Phase flow / one-parameter group $g^{t}$	Analysis
`02.12.05`	Rectification (straightening) of a vector field	Analysis
`02.12.08`	Lyapunov stability (direct method)	Analysis
`02.12.10`	Poincaré-Bendixson theorem	Analysis
`02.12.12`	First integrals / conserved quantities	Analysis
`02.12.13`	Inhomogeneous linear ODE / variation of constants	Analysis
`02.12.14`	Limit cycle and Liénard / Van der Pol systems	Analysis
`02.12.17`	Bifurcation theory pointer	Analysis
`02.12.E1`	Qualitative theory of ODEs exercise pack (Arnold Ch. 2-3 supplement)	Analysis
`02.13.01`	Laplace Equation, Harmonic Functions, Mean-Value Property, and Maximum Principle	Analysis
`02.13.02`	Poisson Equation, Fundamental Solution, and Newtonian Potential	Analysis
`02.13.03`	Heat Equation, Heat Kernel, and Duhamel's Principle	Analysis
`02.13.04`	Wave Equation, d'Alembert Solution, Spherical Means, and Huygens Principle	Analysis
`02.13.05`	Whitney deformation theorem	Analysis
`02.13.06`	The Cauchy-Kovalevskaya Theorem and Holmgren Uniqueness	Analysis
`02.13.07`	Rectifiable currents	Analysis
`02.13.11`	Slicing of currents	Analysis
`02.13.E1`	Integration and currents exercise pack (Whitney Ch. I, IX, XI supplement)	Analysis
`02.14.01`	Wave-front set of a distribution	Analysis
`02.14.02`	Pseudo-differential operators on a manifold	Analysis
`02.14.03`	Propagation of singularities along Hamiltonian flow	Analysis
`02.14.04`	The theory of distributions and the Schwartz kernel theorem	Analysis
`02.14.05`	Rigged Hilbert space (Gel'fand triple) and the nuclear spectral theorem	Analysis
`02.15.01`	Brownian motion and the Wiener process	Analysis
`02.15.02`	The Itô integral and Itô's formula	Analysis
`02.15.03`	Stochastic differential equations, diffusions, and the infinitesimal generator	Analysis
`02.15.04`	The Feynman-Kac formula	Analysis
`02.15.05`	The Stratonovich Integral and Stratonovich Calculus	Analysis
`02.16.01`	Sobolev Inequalities: the Gagliardo-Nirenberg-Sobolev and Morrey Inequalities	Analysis
`02.16.02`	Trace and Extension Theorems for Sobolev Functions	Analysis
`02.16.03`	The Rellich-Kondrachov Compactness Theorem and the Poincaré Inequalities	Analysis
`02.16.04`	Lax-Milgram and Existence of Weak Solutions of Elliptic Boundary-Value Problems	Analysis
`02.16.05`	The Fredholm Alternative and Eigenvalues for Second-Order Elliptic Operators	Analysis
`02.17.01`	Interior and Boundary H^2 Regularity of Weak Elliptic Solutions	Analysis
`02.17.02`	Maximum Principles for General Second-Order Elliptic Operators	Analysis
`02.17.03`	The Alexandrov-Bakelman-Pucci Estimate	Analysis
`02.17.04`	Schauder Theory: Interior and Boundary C^{2,alpha} Estimates	Analysis
`02.17.05`	The Classical Dirichlet Problem via the Method of Continuity	Analysis
`02.17.06`	Lp (Calderón-Zygmund) W^{2,p} Estimates for Elliptic Equations	Analysis
`02.17.07`	De Giorgi-Nash-Moser Theory: Local Boundedness and Holder Continuity of Weak Solutions	Analysis
`02.17.08`	The Harnack Inequality for Elliptic Equations (Moser and Krylov-Safonov)	Analysis
`02.17.09`	Quasilinear Elliptic Equations: Gradient Estimates and Existence by Leray-Schauder	Analysis
`02.18.01`	Galerkin Existence and Energy Estimates for Second-Order Parabolic Equations	Analysis
`02.18.02`	Galerkin Existence and Finite Propagation Speed for Second-Order Hyperbolic Equations	Analysis
`02.18.03`	C0-Semigroups and the Hille-Yosida Theorem	Analysis
`02.18.04`	The Direct Method of the Calculus of Variations	Analysis
`02.18.05`	Viscosity Solutions of Hamilton-Jacobi Equations	Analysis
`02.18.06`	Scalar Conservation Laws: Shocks, Rankine-Hugoniot, and Entropy Solutions	Analysis
`02.19.01`	The Hardy-Littlewood Maximal Function and the Vitali Covering Lemma	Analysis
`02.19.02`	The Calderón-Zygmund Decomposition	Analysis
`02.19.03`	Calderón-Zygmund Singular Integral Operators: Lp Boundedness	Analysis
`02.19.04`	The Riesz Transforms	Analysis
`02.19.05`	Riesz and Bessel Potentials and the Hardy-Littlewood-Sobolev Inequality	Analysis
`02.20.01`	BMO and the John-Nirenberg Inequality	Analysis
`02.20.02`	Real-Variable Hardy Spaces H^p and the Atomic Decomposition	Analysis
`02.20.03`	Littlewood-Paley Theory and the Square Function	Analysis
`02.20.04`	Fourier Multipliers and the Hörmander-Mikhlin Theorem	Analysis
`02.20.05`	Oscillatory Integrals and the Method of Stationary Phase	Analysis
`02.21.01`	Dispersive Decay Estimates for the Schrödinger and Wave Propagators	Analysis
`02.21.02`	Strichartz Estimates via the TT* Method	Analysis
`02.21.03`	Local Well-Posedness for Semilinear NLS/NLW via Strichartz Contraction	Analysis
`02.21.04`	Conservation Laws, Global Well-Posedness, and the Energy-Critical Problem	Analysis
`02.21.05`	Bourgain X^{s,b} Spaces and Low-Regularity Well-Posedness	Analysis
`02.21.06`	Virial Identities, Blowup, and the Soliton-Stability Outlook	Analysis
`03.01.01`	Tensor product	Modern geometry
`03.01.02`	Associative algebra	Modern geometry
`03.01.03`	Ideal in an algebra	Modern geometry
`03.01.04`	Tensor algebra	Modern geometry
`03.01.05`	Quotient algebra	Modern geometry
`03.02.01`	Topological manifold	Differential geometry
`03.02.01`	Smooth manifold	Modern geometry
`03.02.02`	Smooth structure and atlases	Differential geometry
`03.02.03`	Smooth maps between manifolds	Differential geometry
`03.02.04`	Frobenius theorem	Differential geometry
`03.02.05`	Sectional curvature, Ricci tensor, scalar curvature	Differential geometry
`03.02.06`	Constant-curvature spaces and Killing-Hopf	Differential geometry
`03.02.07`	Killing fields and infinitesimal isometries	Differential geometry
`03.02.08`	Myers-Steenrod theorem	Differential geometry
`03.02.09`	Almost-complex structure (manifold-level)	Differential geometry
`03.02.10`	Complex manifold and the Dolbeault complex	Differential geometry
`03.02.11`	Hermitian manifold and the Kahler form	Differential geometry
`03.02.12`	Kahler identities and the Hodge decomposition (Kahler version)	Differential geometry
`03.02.13`	Isometric immersion and the second fundamental form	Differential geometry
`03.02.14`	Gauss, Codazzi, and Ricci equations	Differential geometry
`03.02.15`	Bochner technique and curvature vanishing theorems	Differential geometry
`03.02.16`	Weyl tensor and conformally flat metrics	Differential geometry
`03.02.17`	Lorentzian Hopf-Rinow and global hyperbolicity (introductory pseudo-Riemannian geometry)	Differential geometry
`03.02.18`	Petrov classification of Lorentzian 4-curvature	Differential geometry
`03.02.19`	Jacobi fields, conjugate points, and the Morse Index Theorem	Differential geometry
`03.02.20`	Handles, surgery, and the cobordism category	Differential geometry
`03.02.21`	Rearrangement and self-indexing Morse functions	Differential geometry
`03.02.22`	The Whitney trick and handle cancellation	Differential geometry
`03.02.23`	The h-cobordism theorem	Differential geometry
`03.02.24`	The generalised Poincaré conjecture in high dimensions	Differential geometry
`03.02.25`	The Lefschetz hyperplane theorem via Morse theory	Differential geometry
`03.02.26`	Harmonic maps: energy, tension field, and the harmonic-map equation	Differential geometry
`03.02.27`	Levi-Civita connection, exponential map, and gradient-like vector fields on a cobordism	Differential geometry
`03.02.28`	Pointer: surgery theory and the surgery exact sequence	Differential geometry
`03.02.29`	The harmonic-map heat flow and the Eells–Sampson theorem	Differential geometry
`03.02.30`	Morse functions, the Morse lemma, and the Morse index	Differential geometry
`03.02.31`	Handle attachment, CW homotopy type, and the Morse inequalities	Differential geometry
`03.02.32`	The Riemannian Hopf–Rinow theorem	Differential geometry
`03.02.33`	The Yamabe problem and the conformal Laplacian	Differential geometry
`03.02.34`	The Toponogov triangle comparison theorem	Differential geometry
`03.02.35`	Synge's theorem and the second variation of arc length	Differential geometry
`03.02.36`	Einstein metrics as critical points of the total scalar curvature	Differential geometry
`03.02.37`	Homogeneous Einstein metrics on G/H	Differential geometry
`03.02.38`	Infinitesimal Einstein deformations, the Lichnerowicz Laplacian on 2-tensors, and Koiso rigidity	Differential geometry
`03.02.39`	The Cartan–Ambrose–Hicks theorem (intrinsic rigidity)	Differential geometry
`03.02.40`	The path space as a CW complex: the fundamental theorem of Morse theory	Differential geometry
`03.02.41`	The two-spinor calculus: $ε$ -spinors, abstract indices, and the spinor–tensor dictionary	Differential geometry
`03.02.42`	Zero-rest-mass field equations and the spinor form of Maxwell, Weyl, and Dirac fields	Differential geometry
`03.02.43`	The Newman-Penrose spin-coefficient formalism	Differential geometry
`03.02.44`	Gauss-Bonnet and Chern-Gauss-Bonnet: from angle excess to the Euler class	Differential geometry
`03.02.45`	Brownian motion on a Riemannian manifold	Differential geometry
`03.03.01`	Lie group	Modern geometry
`03.03.02`	Group action	Modern geometry
`03.03.03`	Orthogonal group	Modern geometry
`03.03.04`	Formal group law	Differential geometry
`03.03.05`	p-adic Lie group and the p-adic exponential	Differential geometry
`03.03.06`	Lie's third theorem (statement, simply-connected case)	Differential geometry
`03.03.07`	Invariant affine connections on a reductive homogeneous space and the canonical connection	Differential geometry
`03.03.10`	Lie groupoid: source, target, smooth composition	Modern geometry
`03.03.11`	Action Lie groupoid and action Lie algebroid	Modern geometry
`03.03.12`	Bisection group of a Lie groupoid; gauge transformations as bisections	Modern geometry
`03.03.13`	Groupoid as a small category with all morphisms invertible	Modern geometry
`03.04.01`	Lie algebra	Modern geometry
`03.04.02`	Differential forms	Modern geometry
`03.04.03`	Integration on manifolds	Modern geometry
`03.04.04`	Exterior derivative	Modern geometry
`03.04.05`	Stokes' theorem	Modern geometry
`03.04.06`	De Rham cohomology	Modern geometry
`03.04.07`	Mayer-Vietoris sequence for de Rham cohomology	Modern geometry
`03.04.08`	Variational calculus on manifolds	Modern geometry
`03.04.09`	Compactly-supported cohomology, integration along the fiber, and the de Rham Thom isomorphism	Modern geometry
`03.04.10`	Good covers, finite-dimensionality of de Rham cohomology, and the Mayer-Vietoris induction	Modern geometry
`03.04.11`	Čech-de Rham double complex and the tic-tac-toe principle	Modern geometry
`03.04.12`	Künneth formula for de Rham cohomology — two proofs	Modern geometry
`03.04.13`	Singular cohomology and the de Rham theorem (with $Z$ coefficients)	Modern geometry
`03.04.14`	Hypercohomology of a complex of sheaves	Modern geometry
`03.04.15`	Hodge Laplacian on a Riemannian manifold	Modern geometry
`03.04.16`	Lie algebroid: anchor, bracket, Leibniz law	Modern geometry
`03.04.17`	Lie functor: differentiating a Lie groupoid to its Lie algebroid	Modern geometry
`03.04.18`	Pradines integration theorem and Mackenzie transitive integrability	Modern geometry
`03.04.19`	Cotangent algebroid of a Poisson manifold; pointer to symplectic groupoids	Modern geometry
`03.04.20`	Surface integrals of 2-forms; flux of a vector field through an oriented surface	Modern geometry
`03.04.21`	Closed and exact forms; the Poincaré lemma; the angle 1-form	Modern geometry
`03.04.22`	Lie algebroid cohomology and the Chevalley-Eilenberg differential	Modern geometry
`03.04.23`	Representations of a Lie algebroid and flat A-connections	Modern geometry
`03.04.24`	Lie bialgebroids and Poisson groupoids: the Mackenzie-Xu duality	Differential geometry
`03.04.25`	Double Lie groupoids, double Lie algebroids, and VB-groupoids	Modern geometry
`03.04.E1`	Mayer-Vietoris and degree-theory exercise pack (Bott-Tu Ch. I supplement)	Modern geometry
`03.04.E2`	Differential forms and Stokes exercise pack (Shifrin / Arnold supplement)	Modern geometry
`03.05.00`	General fibre bundle	Differential geometry
`03.05.01`	Principal bundle	Modern geometry
`03.05.02`	Vector bundle	Modern geometry
`03.05.03`	Orthogonal frame bundle	Modern geometry
`03.05.04`	Connection on a vector bundle	Modern geometry
`03.05.05`	Double cover	Modern geometry
`03.05.06`	Vertical subbundle and fundamental vector fields	Differential geometry
`03.05.07`	Principal bundle with connection	Modern geometry
`03.05.08`	Complex vector bundle	Modern geometry
`03.05.09`	Curvature of a connection	Modern geometry
`03.05.10`	Sphere bundle, the global angular form, and the Hopf index theorem	Modern geometry
`03.05.11`	Horizontal lift and parallel transport	Differential geometry
`03.05.12`	Reduction of structure group; reduction of a connection	Differential geometry
`03.05.13`	Associated bundle and induced connection	Differential geometry
`03.05.14`	Torsion tensor and the two Cartan structural equations	Differential geometry
`03.05.15`	Linear connection via the frame bundle; soldering form	Differential geometry
`03.05.16`	Holonomy group and restricted holonomy	Differential geometry
`03.05.17`	Ambrose-Singer holonomy theorem	Differential geometry
`03.05.18`	Holonomy reduction theorem	Differential geometry
`03.05.19`	Holomorphic vector bundle	Differential geometry
`03.05.20`	Hermitian metric on a complex bundle; Chern connection	Differential geometry
`03.05.21`	Gauge groupoid of a principal bundle	Modern geometry
`03.05.22`	Atiyah algebroid of a principal bundle	Modern geometry
`03.05.23`	Connection on a principal bundle as splitting of the Atiyah algebroid	Modern geometry
`03.06.03`	Stiefel-Whitney classes	Modern geometry
`03.06.04`	Pontryagin and Chern classes	Modern geometry
`03.06.05`	Invariant polynomial on a Lie algebra	Modern geometry
`03.06.06`	Chern-Weil homomorphism	Modern geometry
`03.06.07`	Chern-Simons forms and transgression	Modern geometry
`03.06.08`	Kostant-Weil isomorphism and prequantum line bundle	Modern geometry
`03.06.09`	Dixmier-Douady class and $H^{3} (M, Z)$	Modern geometry
`03.06.10`	Stiefel-Whitney and Pontryagin numbers	Modern geometry
`03.06.11`	Hirzebruch signature theorem	Modern geometry
`03.06.12`	Unoriented bordism and Thom's theorem	Modern geometry
`03.06.13`	Oriented bordism and the Pontryagin-Thom construction	Modern geometry
`03.06.14`	Steenrod squares and the Wu formula	Modern geometry
`03.06.15`	Multiplicative sequences and the $L$ -, $\hat{A}$ -, Todd genera	Modern geometry
`03.06.16`	Whitney duality and immersion obstructions	Modern geometry
`03.06.17`	Combinatorial Pontryagin classes and exotic 7-spheres	Modern geometry
`03.06.18`	Chern character $ch (E)$ as a ring homomorphism	Modern geometry
`03.06.19`	Signature of a 4k-manifold and the intersection form	Modern geometry
`03.06.20`	Borel-Hirzebruch and the cohomology of $G / T$	Modern geometry
`03.06.21`	Godbillon-Vey class and secondary characteristic classes of foliations	Differential geometry
`03.06.23`	Modularity of the elliptic genus	Modern geometry
`03.06.24`	Bott-Taubes rigidity theorem	Modern geometry
`03.06.26`	Pointer: elliptic cohomology	Modern geometry
`03.07.05`	Yang-Mills action	Modern geometry
`03.07.06`	Anti-self-dual (ASD) equation on a 4-manifold	Modern geometry
`03.07.07`	BPST instanton and the Bogomolny bound	Modern geometry
`03.07.08`	Conformal compactification and finite-action instantons	Modern geometry
`03.07.09`	Moduli space of ASD connections $M_{k} (S^{4})$	Modern geometry
`03.07.10`	ADHM construction (Atiyah-Drinfeld-Hitchin-Manin)	Modern geometry
`03.07.11`	Penrose twistor space and the Ward correspondence	Modern geometry
`03.07.12`	The Geometry of Twistors: Null Planes, the Twistor Norm, and the Robinson Congruence	Modern geometry
`03.07.14`	Penrose transform at linear level	Modern geometry
`03.07.16`	$B$ -field as a gerbe connection	Modern geometry
`03.07.17`	Chern-Simons functional on a 3-manifold	Modern geometry
`03.07.18`	Configuration space and slice theorem on $B^{*} (Y)$	Modern geometry
`03.07.19`	Spectral flow and the Floer grading mod 8	Modern geometry
`03.07.20`	Uhlenbeck compactness for ASD equations on cylinders	Modern geometry
`03.07.21`	Gluing theorem for instanton trajectories	Modern geometry
`03.07.22`	Orientations on instanton trajectory moduli	Modern geometry
`03.07.23`	Instanton Floer homology $H F_{*} (Y)$	Modern geometry
`03.07.24`	Relative Donaldson invariants for 4-manifolds with boundary	Modern geometry
`03.07.25`	Donaldson-Floer surgery exact triangle	Modern geometry
`03.07.26`	Atiyah-Floer conjecture	Modern geometry
`03.07.27`	Polyfolds (Hofer-Wysocki-Zehnder)	Modern geometry
`03.07.28`	Monopole-instanton Floer equivalence (Kronheimer-Mrowka)	Modern geometry
`03.07.29`	Electromagnetism as a U(1) Yang-Mills theory — the geometric dictionary	Modern geometry
`03.07.30`	Aharonov-Bohm effect and holonomy of U(1) connections	Modern geometry
`03.07.31`	BRST cohomology and Faddeev-Popov-ghost quantisation of gauge theories	Modern geometry
`03.07.32`	Anomalies via descent equations and the Atiyah-Singer index theorem	Modern geometry
`03.07.33`	Casson's invariant and the Euler characteristic of instanton Floer homology	Modern geometry
`03.07.34`	Simple type, basic classes, and the structure of Donaldson invariants	Modern geometry
`03.08.01`	Topological K-theory	Modern geometry
`03.08.02`	Adams operations $ψ^{k}$	Modern geometry
`03.08.03`	Thom isomorphism in K-theory	Modern geometry
`03.08.04`	Classifying space	Modern geometry
`03.08.05`	Universal bundle, $H^{*} (B U (k))$ , and the Borel presentation of flag-manifold cohomology	Modern geometry
`03.08.06`	Stable homotopy	Modern geometry
`03.08.07`	Bott periodicity	Modern geometry
`03.08.08`	Bott periodicity for U via Morse theory	Modern geometry
`03.08.09`	Worked K-theory computations: spheres, projective spaces, and tori	Modern geometry
`03.08.10`	Equivariant K-theory $K_{G} (X)$ and $R (G)$	Modern geometry
`03.08.11`	The group $J (X)$ and the $J$ -homomorphism	Modern geometry
`03.08.12`	KR-theory (K-theory with reality)	Modern geometry
`03.08.13`	Bott periodicity for O via iterated minimal geodesics	Modern geometry
`03.08.20`	Whitehead torsion and the s-cobordism theorem	Modern geometry
`03.09.02`	Clifford algebra	Modern geometry
`03.09.03`	Spin group	Modern geometry
`03.09.04`	Spin structure on an oriented Riemannian manifold	Modern geometry
`03.09.05`	Spinor bundle	Modern geometry
`03.09.06`	Fredholm operators	Modern geometry
`03.09.07`	Symbol of a differential operator	Modern geometry
`03.09.08`	Dirac operator	Modern geometry
`03.09.09`	Elliptic operators on a manifold	Modern geometry
`03.09.10`	Atiyah-Singer index theorem	Modern geometry
`03.09.11`	Clifford algebra classification — the 8×8 chessboard	Modern geometry
`03.09.12`	KR-theory and the (1,1)-periodicity theorem	Modern geometry
`03.09.13`	Triality on Spin(8) and exceptional Lie groups via spinors	Modern geometry
`03.09.14`	Generalised Dirac bundles and the Bochner-Weitzenböck identity	Modern geometry
`03.09.15`	Cl_k-linear Dirac operators and the KO-valued index	Modern geometry
`03.09.16`	Positive scalar curvature obstruction theory	Modern geometry
`03.09.17`	Witten positive-mass theorem via spinors	Modern geometry
`03.09.18`	Berger holonomy classification and parallel spinors	Modern geometry
`03.09.19`	Calibrated geometries — Special Lagrangian, associative, coassociative, Cayley	Modern geometry
`03.09.20`	Heat-kernel proof of the Atiyah-Singer index theorem	Modern geometry
`03.09.21`	Family, equivariant, and Lefschetz fixed-point index theorems	Modern geometry
`03.09.22`	Sobolev spaces, pseudodifferential operators, and elliptic parametrices	Modern geometry
`03.09.23`	Bismut superconnection	Modern geometry
`03.09.24`	Eta invariant and Atiyah-Patodi-Singer index theorem	Modern geometry
`03.09.25`	Kirillov character formula via the equivariant index	Modern geometry
`03.09.26`	Mathai-Quillen formalism and universal Thom forms	Modern geometry
`03.09.27`	Pure spinors and the spinor variety	Modern geometry
`03.09.28`	The Cartan Model of Equivariant de Rham Cohomology	Modern geometry
`03.09.29`	The probabilistic heat kernel and Bismut's formula	Modern geometry
`03.09.E1`	Clifford and spin algebra exercise pack (Lawson-Michelsohn Ch. I supplement)	Modern geometry
`03.09.E2`	Chapter IV applications exercise pack (Lawson-Michelsohn Ch. IV supplement)	Modern geometry
`03.10.02`	CFT basics	Modern geometry
`03.10.03`	The Wess-Zumino-Witten action and the level-k extension	Modern geometry
`03.10.04`	Minimal Models, the Kac Formula, and Null Vectors	Modern geometry
`03.10.05`	The Coulomb gas, screening charges, and the conformal bootstrap	Modern geometry
`03.11.01`	Central extension of a Lie algebra	Modern geometry
`03.11.02`	Infinite-dimensional Lie algebra representations	Modern geometry
`03.11.03`	Virasoro algebra	Modern geometry
`03.11.04`	The free loop space LM and transgression	Modern geometry
`03.11.05`	The geometric central extension of the loop group LG	Modern geometry
`03.12.00`	Fundamental group	Modern geometry
`03.12.01`	Homotopy and homotopy group	Modern geometry
`03.12.02`	Covering space	Modern geometry
`03.12.03`	Suspension	Modern geometry
`03.12.04`	Spectrum	Modern geometry
`03.12.05`	Eilenberg-MacLane space	Modern geometry
`03.12.06`	Sullivan minimal models and rational homotopy theory	Modern geometry
`03.12.07`	Whitehead tower, rational Hurewicz theorem, and Serre's finiteness	Modern geometry
`03.12.08`	Fundamental groupoid	Modern geometry
`03.12.09`	Seifert-van Kampen theorem	Modern geometry
`03.12.10`	CW complex	Modern geometry
`03.12.11`	Singular homology	Modern geometry
`03.12.12`	Simplicial and $Δ$ -complex homology	Modern geometry
`03.12.13`	Cellular homology and cellular approximation	Modern geometry
`03.12.14`	Excision theorem	Modern geometry
`03.12.15`	Eilenberg-Steenrod axioms	Modern geometry
`03.12.16`	Poincaré duality	Modern geometry
`03.12.17`	Cap product	Modern geometry
`03.12.18`	Universal coefficient theorem	Modern geometry
`03.12.19`	Hurewicz theorem	Modern geometry
`03.12.20`	Whitehead's theorem	Modern geometry
`03.12.21`	Blakers-Massey theorem	Modern geometry
`03.12.22`	$Δ$ -complex / semi-simplicial set	Modern geometry
`03.12.23`	Euler characteristic	Modern geometry
`03.12.24`	Simplicial set and the simplicial category Delta	Differential geometry
`03.12.25`	Simplicial sets and geometric realization	Modern geometry
`03.12.26`	Functorial CW approximation Gamma X = \|S_*X\|	Differential geometry
`03.12.27`	Puppe cofiber sequence	Differential geometry
`03.12.28`	Puppe fiber sequence	Differential geometry
`03.12.29`	Thom space and Thom isomorphism	Differential geometry
`03.12.30`	Minimal complex and minimal fibration	Differential geometry
`03.12.31`	Quillen model category	Modern geometry
`03.12.32`	Quillen functor and Quillen equivalence	Modern geometry
`03.12.33`	Kan-Quillen model structure on sSet	Modern geometry
`03.12.34`	Acyclic models and the Eilenberg-Zilber theorem	Modern geometry
`03.12.35`	Simplicial model category and the function complex	Modern geometry
`03.12.36`	Bisimplicial set, diagonal, and the realisation lemma	Differential geometry
`03.12.37`	Homotopy colimit via the Bousfield-Kan construction	Modern geometry
`03.12.38`	Bousfield-Kan spectral sequence	Modern geometry
`03.12.39`	Simplicial group and the W-bar classifying functor	Differential geometry
`03.12.40`	Postnikov tower of a Kan complex	Modern geometry
`03.12.41`	Twisted cartesian products and simplicial fibre bundles	Modern geometry
`03.12.42`	Combinatorial simplicial homotopy groups and the Kan-fibration long exact sequence	Modern geometry
`03.12.43`	Quasi-categories and the Joyal model structure	Modern geometry
`03.12.44`	Mixed Hodge structures on rational homotopy theory (Morgan's theorem)	Modern geometry
`03.12.45`	Arithmetic square and integral fracture theorems	Modern geometry
`03.12.46`	The periodicity and thick subcategory theorems	Modern geometry
`03.12.47`	HELP and the unified Whitehead / cellular approximation theorem	Modern geometry
`03.12.48`	Bousfield localisation of a model category	Modern geometry
`03.12.49`	Bialgebra, Hopf algebra, and the Milnor-Moore theorem	Differential geometry
`03.12.50`	The Cartan model for the minimal model of a homogeneous space	Modern geometry
`03.12.51`	Massey products and the formality condition	Modern geometry
`03.12.52`	Relative homotopy group $π_{n} (X, A, x_{0})$	Modern geometry
`03.12.53`	Whitehead's crossed module of a pair	Modern geometry
`03.12.54`	Filtered space	Modern geometry
`03.12.55`	Crossed complex of a filtered space	Modern geometry
`03.12.56`	Higher Homotopy Seifert-van Kampen theorem	Modern geometry
`03.12.57`	Cubical $ω$ -groupoid $ρ (X_{*})$	Modern geometry
`03.12.58`	Free crossed resolution of a group	Modern geometry
`03.12.59`	Classifying space of a crossed complex	Modern geometry
`03.12.60`	Localisation of nilpotent spaces at a set of primes	Modern geometry
`03.12.61`	Nilpotent groups and nilpotent spaces	Modern geometry
`03.12.E1`	Rational homotopy and Sullivan minimal-model exercise pack (Bott-Tu Ch. III §19 supplement)	Modern geometry
`03.12.E2`	Singular and cellular homology exercise pack (Hatcher Ch. 2 supplement)	Modern geometry
`03.12.E3`	Simplicial homotopy theory exercise pack (Goerss-Jardine supplement)	Modern geometry
`03.12.E4`	Localization and completion exercise pack (May-Ponto supplement)	Modern geometry
`03.12.E5`	Simplicial objects and Dold-Kan exercise pack (May supplement)	Modern geometry
`03.13.01`	Spectral sequences — exact couples, filtered complexes, double complexes	Modern geometry
`03.13.02`	Leray-Serre spectral sequence and the Gysin sequence	Modern geometry
`03.13.03`	Leray-Hirsch theorem and the splitting principle for vector bundles	Modern geometry
`03.13.04`	Atiyah-Hirzebruch spectral sequence	Modern geometry
`03.13.05`	The Brown-Peterson spectrum BP and its Hopf algebroid	Modern geometry
`03.13.06`	The chromatic spectral sequence	Modern geometry
`03.13.07`	Greek-letter elements in the stable homotopy of spheres	Modern geometry
`03.13.08`	The telescope conjecture and its disproof	Modern geometry
`03.13.E1`	Spectral-sequence computation exercise pack (Bott-Tu Ch. III supplement)	Modern geometry
`03.14.01`	Quantum free particle as a representation of E(3)	Modern geometry
`03.14.02`	Complex structures and quantization: squeezed states	Modern geometry
`03.15.01`	Gradient flow, stable/unstable manifolds, and the Morse-Smale condition	Modern geometry
`03.15.02`	Trajectory spaces, the Fredholm setup, and transversality	Modern geometry
`03.15.03`	Compactness: broken trajectories	Modern geometry
`03.15.04`	Gluing of trajectories	Modern geometry
`03.15.05`	Coherent orientations and characteristic signs	Modern geometry
`03.15.06`	The Morse complex and $\partial^{2} = 0$	Modern geometry
`03.15.07`	Continuation maps and invariance of $H M_{*}$	Modern geometry
`03.15.08`	The Morse Homology Theorem	Modern geometry
`03.15.09`	Morse cohomology, cup product, and the ring structure	Modern geometry
`03.15.10`	Poincaré duality via flow reversal and the filtered Morse spectral sequence	Modern geometry
`03.15.11`	From finite-dimensional Morse homology to Floer homology	Modern geometry
`03.15.12`	Witten's supersymmetric Morse theory (survey/pointer)	Modern geometry
`03.15.E1`	Morse homology exercise pack (Schwarz Morse Homology supplement)	Modern geometry
`03.16.01`	The Atiyah–Segal axioms for topological quantum field theory	Modern geometry
`03.16.02`	Classification of 2d oriented TQFTs: the Frobenius-algebra theorem	Modern geometry
`03.16.03`	Extended TQFT and the cobordism hypothesis	Modern geometry
`03.16.04`	Invertible field theories and the Freed–Hopkins classification	Modern geometry
`03.16.05`	Anomalies as invertible field theories in one dimension higher	Modern geometry
`03.16.06`	Chern-Simons theory as a quantum TQFT, the Jones polynomial, and Reshetikhin-Turaev	Modern geometry
`04.01.01`	Sheaf	Algebraic geometry
`04.01.02`	Stalk of a sheaf	Algebraic geometry
`04.01.03`	Sheafification	Algebraic geometry
`04.01.04`	Direct and inverse image of sheaves	Algebraic geometry
`04.02.01`	Scheme	Algebraic geometry
`04.02.02`	Affine scheme	Algebraic geometry
`04.02.03`	Projective scheme	Algebraic geometry
`04.02.04`	Morphism of schemes	Algebraic geometry
`04.02.05`	Smooth, étale, and unramified morphisms	Algebraic geometry
`04.02.07`	Nullstellensatz and dimension theory	Algebraic geometry
`04.03.01`	Sheaf cohomology	Algebraic geometry
`04.03.02`	Local systems, monodromy, and twisted cohomology	Algebraic geometry
`04.03.03`	Čech cohomology of sheaves on schemes	Algebraic geometry
`04.03.04`	Cohomology of line bundles on projective space	Algebraic geometry
`04.03.05`	Serre's vanishing and finiteness theorems	Algebraic geometry
`04.03.06`	Derived functors and Ext	Algebraic geometry
`04.03.07`	Higher direct images and base change	Algebraic geometry
`04.03.08`	Étale cohomology and $ℓ$ -adic cohomology of varieties	Algebraic geometry
`04.03.10`	Triangulated category — Verdier axioms TR1-TR4 and the octahedral axiom	Algebraic geometry
`04.03.11`	Derived category $D (A)$ — localisation at quasi-isomorphisms	Algebraic geometry
`04.03.12`	Derived functors $R F$ and $L F$ via derived categories	Algebraic geometry
`04.03.13`	Grothendieck spectral sequence	Algebraic geometry
`04.03.14`	Spectral sequence of a filtered complex	Algebraic geometry
`04.03.15`	Sheaf cohomology - Leray spectral sequence (general form)	Algebraic geometry
`04.03.16`	Six-functor formalism — adjunctions and base change	Algebraic geometry
`04.03.17`	Derived tensor product ⊗^L and Tor in derived categories	Algebraic geometry
`04.03.18`	t-Structure on a triangulated category — heart and truncations	Algebraic geometry
`04.03.19`	Perverse sheaves Perv(X) — pointer + foundations	Algebraic geometry
`04.03.20`	Hochschild homology and cohomology	Algebraic geometry
`04.03.21`	Hochschild-Kostant-Rosenberg theorem	Algebraic geometry
`04.03.22`	Cyclic homology and Connes' long exact sequence	Algebraic geometry
`04.03.23`	The Verdier quotient of a triangulated category	Algebraic geometry
`04.03.E1`	Cohomology of schemes exercise pack (Hartshorne Ch. III supplement)	Algebraic geometry
`04.04.01`	Riemann-Roch theorem for curves	Algebraic geometry
`04.04.02`	Hurwitz formula	Algebraic geometry
`04.04.03`	Elliptic curves	Algebraic geometry
`04.04.04`	Castelnuovo's Genus Bound for Space Curves and Extremal Curves	Algebraic geometry
`04.04.08`	Petri map mu_0 and Gieseker-Petri theorem	Algebraic geometry
`04.04.09`	Clifford's theorem with equality	Algebraic geometry
`04.04.10`	Martens' theorem and Mumford's strengthening	Algebraic geometry
`04.04.11`	Gonality of a curve	Algebraic geometry
`04.04.13`	Determinantal varieties and the Porteous formula	Algebraic geometry
`04.04.14`	The Enriques-Babbage-Petri Theorem on Canonical Curves	Algebraic geometry
`04.04.15`	Fulton-Lazarsfeld connectedness theorem	Algebraic geometry
`04.04.16`	Lazarsfeld's K3-vector-bundle proof of Petri	Algebraic geometry
`04.04.E1`	Curves exercise pack (Hartshorne Ch. IV supplement)	Algebraic geometry
`04.05.01`	Weil divisor	Algebraic geometry
`04.05.02`	Picard group	Algebraic geometry
`04.05.03`	Line bundle on a scheme	Algebraic geometry
`04.05.04`	Cartier divisor	Algebraic geometry
`04.05.05`	Ample and very ample line bundle	Algebraic geometry
`04.05.06`	Intersection pairing on a surface	Algebraic geometry
`04.05.07`	Adjunction formula on a surface	Algebraic geometry
`04.05.08`	Riemann-Roch theorem for surfaces	Algebraic geometry
`04.05.09`	Hodge index theorem	Algebraic geometry
`04.05.10`	Hirzebruch-Riemann-Roch theorem (general dimension)	Algebraic geometry
`04.05.11`	Worked Hirzebruch-Riemann-Roch computations	Algebraic geometry
`04.05.12`	Pointer: Grothendieck-Riemann-Roch (GRR)	Algebraic geometry
`04.05.E1`	Surfaces exercise pack (Hartshorne Ch. V supplement)	Algebraic geometry
`04.06.01`	Quasi-coherent sheaf	Algebraic geometry
`04.06.02`	Coherent sheaf	Algebraic geometry
`04.07.01`	Projective space	Algebraic geometry
`04.07.02`	Blowup	Algebraic geometry
`04.07.03`	Monads on projective space and the Beilinson resolution	Algebraic geometry
`04.07.04`	Stable rank-2 bundles on projective space and Barth's theorem	Algebraic geometry
`04.08.01`	Sheaf of differentials	Algebraic geometry
`04.08.02`	Canonical sheaf	Algebraic geometry
`04.08.03`	Serre duality	Algebraic geometry
`04.09.01`	Hodge decomposition	Algebraic geometry
`04.09.02`	Kodaira vanishing theorem	Algebraic geometry
`04.09.03`	Serre's GAGA comparison theorem	Algebraic geometry
`04.09.05`	The ddbar-lemma	Algebraic geometry
`04.09.07`	Hard Lefschetz theorem	Algebraic geometry
`04.09.08`	Hodge-Riemann bilinear relations	Algebraic geometry
`04.09.09`	Lefschetz (1,1)-theorem	Algebraic geometry
`04.09.10`	Akizuki-Nakano vanishing theorem	Algebraic geometry
`04.09.11`	Kodaira embedding theorem	Algebraic geometry
`04.10.01`	Moduli of curves	Algebraic geometry
`04.10.02`	Geometric invariant theory	Algebraic geometry
`04.10.03`	Hilbert-Mumford numerical criterion	Algebraic geometry
`04.10.04`	Kempf-Ness theorem and the GIT-symplectic dictionary	Algebraic geometry
`04.10.05`	Hilbert scheme Hilb^P(X)	Algebraic geometry
`04.10.06`	Moduli of vector bundles on a curve and slope stability	Algebraic geometry
`04.10.07`	Linear algebraic groups, reductivity, and finite generation of invariants	Algebraic geometry
`04.10.08`	Kirwan stratification of the unstable locus	Algebraic geometry
`04.10.09`	Variation of GIT (VGIT)	Algebraic geometry
`04.10.11`	Gieseker stability and moduli of sheaves	Algebraic geometry
`04.10.12`	Bridgeland stability conditions	Algebraic geometry
`04.10.13`	K-stability and the Yau-Tian-Donaldson conjecture	Algebraic geometry
`04.10.14`	Non-reductive GIT	Algebraic geometry
`04.10.15`	Derived GIT and magic windows	Algebraic geometry
`04.10.16`	Abelian varieties: group law, polarizations, and the dual	Algebraic geometry
`04.10.20`	Deformation theory of smooth curves	Algebraic geometry
`04.10.22`	Stable curve and Deligne-Mumford stability	Algebraic geometry
`04.10.26`	Forgetful and gluing morphisms on $\overline{M}_{g, n}$	Algebraic geometry
`04.10.29`	Limit linear series (Eisenbud-Harris)	Algebraic geometry
`04.10.30`	Hurwitz numbers and the Hurwitz scheme	Algebraic geometry
`04.10.31`	Severi varieties of nodal plane curves and the Harris irreducibility theorem	Algebraic geometry
`04.10.32`	ELSV formula: Hurwitz numbers as Hodge integrals	Algebraic geometry
`04.10.33`	Explicit low-genus moduli: Igusa-Clebsch invariants for M_2 and the plane-quartic model of M_3	Algebraic geometry
`04.10.34`	Torelli morphism and Torelli theorem	Algebraic geometry
`04.10.35`	Moduli of stable maps and Gromov-Witten invariants	Algebraic geometry
`04.11.01`	Algebraic torus and character/cocharacter lattices	Algebraic geometry
`04.11.02`	Rational polyhedral cone and dual cone	Algebraic geometry
`04.11.03`	Affine toric variety $U_{σ}$	Algebraic geometry
`04.11.04`	Fan and the toric variety $X_{Σ}$	Algebraic geometry
`04.11.05`	Smoothness and completeness via fans	Algebraic geometry
`04.11.06`	Orbit-cone correspondence	Algebraic geometry
`04.11.07`	Toric resolution of singularities	Algebraic geometry
`04.11.08`	Toric divisor and support function	Algebraic geometry
`04.11.09`	Toric Picard group	Algebraic geometry
`04.11.10`	Polytope-fan dictionary; the line bundle $L_{P}$	Algebraic geometry
`04.11.11`	Algebraic moment map and the polytope	Algebraic geometry
`04.11.12`	Cohomology of a smooth complete toric variety	Algebraic geometry
`04.11.13`	Toric intersection theory and mixed volumes	Algebraic geometry
`04.11.14`	Bernstein-Kushnirenko theorem	Algebraic geometry
`04.11.15`	The Cox homogeneous coordinate ring and the toric GIT quotient	Algebraic geometry
`04.11.16`	Reflexive polytope and Batyrev mirror duality (pointer)	Algebraic geometry
`04.12.01`	Tropical semiring and tropical polynomial	Algebraic geometry
`04.12.02`	Tropical curve as balanced rational metric graph	Algebraic geometry
`04.12.03`	Kapranov's theorem (fundamental theorem of tropical geometry)	Algebraic geometry
`04.12.04`	Newton polytope and non-archimedean amoeba	Algebraic geometry
`04.12.05`	Mikhalkin's correspondence theorem	Algebraic geometry
`04.12.06`	Nishinou-Siebert correspondence theorem	Algebraic geometry
`04.12.07`	Toric degeneration of a Calabi-Yau variety	Algebraic geometry
`04.12.08`	Dual Intersection Complex; Tropical Manifold B	Algebraic geometry
`04.12.09`	Gross-Siebert Reconstruction Theorem (Statement)	Algebraic geometry
`04.12.10`	Strominger-Yau-Zaslow (SYZ) Conjecture	Algebraic geometry
`04.12.11`	Slab function and structure of a tropical manifold	Algebraic geometry
`04.12.12`	Theta function of a polarised tropical manifold	Algebraic geometry
`04.12.13`	Period integral and the mirror map (pointer)	Algebraic geometry
`04.12.14`	Logarithmic structures and log smooth morphisms	Algebraic geometry
`04.12.15`	Log Gromov-Witten Invariants (pointer)	Algebraic geometry
`04.12.16`	The A-model, the B-model, and the mirror symmetry conjecture	Algebraic geometry
`04.12.17`	Special Lagrangian fibrations and McLean's theorem	Algebraic geometry
`05.00.01`	Lagrangian mechanics on the tangent bundle	Symplectic geometry
`05.00.02`	Hamilton's principle of least action	Symplectic geometry
`05.00.03`	Legendre transform	Symplectic geometry
`05.00.04`	Noether's theorem	Symplectic geometry
`05.00.05`	The charged particle and the twisted symplectic form	Symplectic geometry
`05.00.06`	Galilean group and Newtonian mechanics	Symplectic geometry
`05.00.07`	Galilei group and Bargmann central extension	Symplectic geometry
`05.00.08`	Mechanical similarity / virial theorem	Symplectic geometry
`05.00.09`	Worked Lagrangian examples	Symplectic geometry
`05.00.10`	Scattering and Rutherford formula	Symplectic geometry
`05.00.11`	Small oscillations and normal modes	Symplectic geometry
`05.00.14`	Motion in a non-inertial frame / Coriolis force	Symplectic geometry
`05.00.E1`	Lagrangian and variational mechanics exercise pack (Arnold Part II supplement)	Symplectic geometry
`05.01.01`	Symplectic vector space	Symplectic geometry
`05.01.02`	Symplectic manifold	Symplectic geometry
`05.01.03`	Symplectic group	Symplectic geometry
`05.01.04`	Darboux's theorem	Symplectic geometry
`05.01.05`	Moser's trick	Symplectic geometry
`05.02.01`	Hamiltonian vector field	Symplectic geometry
`05.02.02`	Poisson bracket and Poisson manifold	Symplectic geometry
`05.02.03`	Integrable system	Symplectic geometry
`05.02.04`	Action-angle coordinates	Symplectic geometry
`05.02.05`	Cotangent bundle as canonical symplectic manifold	Symplectic geometry
`05.02.06`	Geodesic flow as a Hamiltonian flow	Symplectic geometry
`05.02.07`	Liouville's volume theorem	Symplectic geometry
`05.02.08`	Poincaré recurrence theorem	Symplectic geometry
`05.02.09`	Poincaré-Cartan integral invariants	Symplectic geometry
`05.02.10`	The Routhian	Symplectic geometry
`05.02.11`	Maupertuis' principle and abbreviated action	Symplectic geometry
`05.02.12`	Hamiltonian monodromy and the spherical pendulum	Symplectic geometry
`05.02.E1`	Hamiltonian mechanics and canonical transformations exercise pack (Arnold Part III supplement)	Symplectic geometry
`05.03.01`	Coadjoint orbit	Symplectic geometry
`05.03.02`	Souriau Gibbs state on a symplectic G-space	Symplectic geometry
`05.03.03`	Classification of homogeneous symplectic manifolds (Kirillov-Kostant-Souriau)	Symplectic geometry
`05.04.01`	Moment map	Symplectic geometry
`05.04.02`	Marsden-Weinstein symplectic reduction	Symplectic geometry
`05.04.03`	Atiyah-Guillemin-Sternberg convexity theorem	Symplectic geometry
`05.04.04`	Delzant theorem (symplectic toric classification)	Symplectic geometry
`05.04.05`	Duistermaat-Heckman theorem	Symplectic geometry
`05.04.06`	Symplectic blow-up and symplectic cut	Symplectic geometry
`05.04.07`	Souriau cocycle and non-equivariant moment maps	Symplectic geometry
`05.05.01`	Lagrangian submanifold	Symplectic geometry
`05.05.02`	Weinstein Lagrangian neighbourhood theorem	Symplectic geometry
`05.05.03`	Generating functions for symplectomorphisms	Symplectic geometry
`05.05.04`	Hamilton-Jacobi equation	Symplectic geometry
`05.05.05`	Jet bundle and total derivative	Symplectic geometry
`05.05.06`	Prolongation of vector fields and the infinitesimal symmetry criterion	Symplectic geometry
`05.05.07`	Group-invariant solutions and symmetry reduction	Symplectic geometry
`05.05.08`	Noether's second theorem and the Bianchi identity	Symplectic geometry
`05.05.09`	Generalised symmetries (Lie-Bäcklund) and recursion operators	Symplectic geometry
`05.05.10`	Lie's classification of second-order ODEs and the symmetry algorithm for ODEs	Symplectic geometry
`05.05.11`	Differential invariants and the moving-frame method	Symplectic geometry
`05.06.01`	Almost-complex structure on a symplectic manifold	Symplectic geometry
`05.06.02`	Pseudoholomorphic curve	Symplectic geometry
`05.06.03`	Newlander-Nirenberg integrability theorem	Symplectic geometry
`05.07.01`	Gromov non-squeezing theorem	Symplectic geometry
`05.07.02`	Symplectic capacity	Symplectic geometry
`05.07.04`	Eliashberg-Gromov $C^{0}$ -rigidity of $Symp$	Symplectic geometry
`05.08.01`	Arnold conjecture and Floer homology setup	Symplectic geometry
`05.08.02`	Floer homology	Symplectic geometry
`05.08.03`	Maslov index	Symplectic geometry
`05.08.04`	Conley-Zehnder index	Symplectic geometry
`05.09.01`	Kolmogorov-Arnold-Moser theorem	Symplectic geometry
`05.09.02`	Adiabatic invariants	Symplectic geometry
`05.09.03`	Birkhoff normal form	Symplectic geometry
`05.09.04`	Williamson normal form for quadratic Hamiltonians	Symplectic geometry
`05.09.05`	Euler-Arnold equations	Symplectic geometry
`05.09.06`	Nekhoroshev estimates	Symplectic geometry
`05.09.07`	Exponential accuracy of the adiabatic invariant	Symplectic geometry
`05.09.08`	Infinite-dimensional Poisson manifolds and Hamiltonian evolution equations	Symplectic geometry
`05.09.09`	Finite-gap integration and theta-function solutions	Symplectic geometry
`05.09.10`	KP hierarchy, Sato Grassmannian, and tau-functions	Symplectic geometry
`05.09.11`	Master symmetries and the Fuchssteiner construction	Symplectic geometry
`05.09.12`	Casimir functions of degenerate Poisson structures	Symplectic geometry
`05.09.E1`	Symplectic geometry and integrable systems exercise pack (Arnold Part III appendices supplement)	Symplectic geometry
`05.10.01`	Contact manifold	Symplectic geometry
`05.10.02`	Symplectisation of a contact manifold	Symplectic geometry
`05.10.03`	Gray's stability theorem	Symplectic geometry
`05.10.04`	Contact topology and Reeb dynamics	Symplectic geometry
`05.11.01`	Prequantum line bundle and the integrality condition	Symplectic geometry
`05.11.02`	Prequantisation of the spin coadjoint orbit	Symplectic geometry
`05.11.03`	Polarisation, half-densities, and metaplectic correction	Symplectic geometry
`05.11.04`	The Groenewold–van Hove no-go theorem	Symplectic geometry
`05.11.05`	Geometric quantization of coadjoint orbits and the Borel–Weil theorem	Symplectic geometry
`05.11.09`	Quantization of the relativistic particle	Symplectic geometry
`05.12.01`	Lagrangian Grassmannian and the universal Maslov class	Symplectic geometry
`05.12.03`	Legendrian singularities and wave-front evolution	Symplectic geometry
`05.12.04`	Lagrangian and Legendrian cobordism	Symplectic geometry
`05.14.01`	Helicity as a Casimir invariant of the ideal fluid	Symplectic geometry
`05.14.02`	Helicity as Asymptotic Linking Number (Arnold's Theorem)	Symplectic geometry
`05.14.03`	The Hopf invariant and the vortex-unknotting obstruction	Symplectic geometry
`05.14.04`	Ideal magnetohydrodynamics: frozen flux and magnetic helicity	Symplectic geometry
`05.14.05`	Arnold's energy-Casimir stability theorem	Symplectic geometry
`05.14.06`	KdV and Camassa-Holm as geodesics on the Bott-Virasoro group	Symplectic geometry
`05.14.07`	Beltrami fields, ABC flows, and chaotic streamlines	Symplectic geometry
`05.14.08`	Fast dynamo problem and Arnold cat-map dynamo	Symplectic geometry
`05.15.01`	Wasserstein metric and Otto's formal Riemannian calculus	Symplectic geometry
`05.15.02`	Korteweg / Madelung quantum hydrodynamics	Symplectic geometry
`06.01.01`	Holomorphic function	Riemann surfaces
`06.01.02`	Cauchy integral formula	Riemann surfaces
`06.01.03`	Residue theorem	Riemann surfaces
`06.01.04`	Analytic continuation	Riemann surfaces
`06.01.05`	Meromorphic function	Riemann surfaces
`06.01.06`	Riemann mapping theorem	Riemann surfaces
`06.01.07`	Riemann sphere	Riemann surfaces
`06.01.08`	Möbius (linear-fractional) transformations	Riemann surfaces
`06.01.10`	Cauchy-Riemann equations and harmonic conjugate	Riemann surfaces
`06.01.11`	Harmonic functions on the plane	Riemann surfaces
`06.01.12`	Maximum modulus + Schwarz lemma	Riemann surfaces
`06.01.13`	Argument principle and Rouché's theorem	Riemann surfaces
`06.01.14`	Normal families and Montel's theorem	Riemann surfaces
`06.01.15`	Gamma function Gamma(z)	Riemann surfaces
`06.01.16`	Riemann zeta function zeta(s)	Riemann surfaces
`06.01.17`	Weierstrass factorization theorem	Riemann surfaces
`06.01.18`	Mittag-Leffler theorem on C	Riemann surfaces
`06.01.19`	Schwarz-Christoffel formula	Riemann surfaces
`06.01.20`	Picard's little theorem	Riemann surfaces
`06.01.21`	Picard's great theorem	Riemann surfaces
`06.01.22`	Phragmen-Lindelof principle	Riemann surfaces
`06.01.23`	Schwarz reflection principle	Riemann surfaces
`06.01.24`	Dirichlet problem on the disc + Perron's method	Riemann surfaces
`06.01.25`	Weierstrass p-function	Riemann surfaces
`06.01.26`	Modular function and j-invariant	Riemann surfaces
`06.01.27`	Power series and Laurent series	Riemann surfaces
`06.01.28`	Index / winding number of a closed curve	Riemann surfaces
`06.01.29`	Schottky's and Bloch's theorems	Riemann surfaces
`06.01.30`	Riemann-Hurwitz for plane meromorphic / sphere maps	Riemann surfaces
`06.01.31`	Jacobi theta functions and the triple product	Riemann surfaces
`06.01.E1`	Complex analysis exercise pack I (Ahlfors Ch. 1-4 supplement)	Riemann surfaces
`06.01.E2`	Complex analysis exercise pack II (Ahlfors Ch. 5-8 supplement)	Riemann surfaces
`06.02.01`	Branch point and ramification	Riemann surfaces
`06.02.02`	Branched coverings of Riemann surfaces	Riemann surfaces
`06.02.03`	Riemann's existence theorem for algebraic curves	Riemann surfaces
`06.03.01`	Riemann surface	Riemann surfaces
`06.03.02`	Genus of a Riemann surface	Riemann surfaces
`06.03.03`	Uniformization theorem	Riemann surfaces
`06.03.04`	Uniformization via constant-curvature conformal metrics and Ricci flow on surfaces	Riemann surfaces
`06.03.05`	The prescribed-Gaussian-curvature equation on a surface (Kazdan-Warner)	Riemann surfaces
`06.04.01`	Riemann-Roch theorem for compact Riemann surfaces	Riemann surfaces
`06.04.02`	Čech cohomology of holomorphic line bundles	Riemann surfaces
`06.04.03`	Hodge decomposition on a compact Riemann surface	Riemann surfaces
`06.04.04`	Serre duality on a curve	Riemann surfaces
`06.04.05`	Hilbert-space PDE for $\overset{ˉ}{\partial}$	Riemann surfaces
`06.04.07`	Survey of sheaf cohomology on Riemann surfaces	Riemann surfaces
`06.05.01`	Divisor on a Riemann surface	Riemann surfaces
`06.05.02`	Holomorphic line bundle on a Riemann surface	Riemann surfaces
`06.05.03`	Riemann-Hurwitz formula	Riemann surfaces
`06.06.01`	Holomorphic 1-form / abelian differential	Riemann surfaces
`06.06.02`	Period matrix	Riemann surfaces
`06.06.03`	Jacobian variety	Riemann surfaces
`06.06.04`	Abel-Jacobi map	Riemann surfaces
`06.06.05`	Theta function	Riemann surfaces
`06.06.06`	Jacobi inversion theorem	Riemann surfaces
`06.06.07`	Riemann's bilinear relations	Riemann surfaces
`06.06.08`	Schottky problem	Riemann surfaces
`06.06.09`	Weierstrass points and gap sequences	Riemann surfaces
`06.07.01`	Holomorphic functions of several variables	Riemann surfaces
`06.07.02`	Hartogs phenomenon	Riemann surfaces
`06.08.01`	Gauss-Manin connection	Riemann surfaces
`06.08.02`	Variation of Hodge structure on the Jacobian	Riemann surfaces
`06.08.03`	Moduli of Riemann surfaces	Riemann surfaces
`06.09.01`	Stein Riemann surfaces	Riemann surfaces
`06.09.02`	Cartan's Theorems A and B for Stein Riemann surfaces	Riemann surfaces
`06.09.03`	Behnke-Stein theorem	Riemann surfaces
`06.09.04`	Cousin I (additive)	Riemann surfaces
`06.09.05`	Cousin II (multiplicative)	Riemann surfaces
`06.09.06`	Mittag-Leffler on RS	Riemann surfaces
`06.09.07`	Runge approximation on RS	Riemann surfaces
`06.09.08`	Survey: Cartan-Serre Stein theory in higher dim	Riemann surfaces
`06.10.01`	Domains of holomorphy and holomorphic convexity	Riemann surfaces
`06.10.02`	Plurisubharmonic functions	Riemann surfaces
`06.10.03`	Pseudoconvexity and the Levi form	Riemann surfaces
`06.10.04`	The ∂̄-equation and Hörmander's L² estimates	Riemann surfaces
`06.10.05`	Solution of the Levi problem	Riemann surfaces
`06.10.06`	Bochner-Martinelli kernel and formula	Riemann surfaces
`06.10.07`	Cauchy-Fantappiè and Henkin-Ramirez kernels	Riemann surfaces
`06.10.08`	Bergman kernel and Bergman metric	Riemann surfaces
`06.10.09`	Szegő kernel and Fefferman boundary asymptotics	Riemann surfaces
`06.10.10`	The ∂̄-Neumann problem and subelliptic estimates	Riemann surfaces
`06.10.11`	Cousin I/II and the Levi problem in $C^{n}$	Riemann surfaces
`06.10.12`	Invariant metrics: Carathéodory, Kobayashi, Bergman	Riemann surfaces
`06.10.13`	Automorphism groups and the Fefferman mapping theorem	Riemann surfaces
`06.10.14`	Weierstrass preparation and division	Riemann surfaces
`06.10.15`	Tangential CR complex, ∂̄_b, and the Lewy example	Riemann surfaces
`06.10.16`	Wong-Rosay theorem and boundary rigidity	Riemann surfaces
`06.10.17`	Local analytic Nullstellensatz and the ideal–germ correspondence	Riemann surfaces
`06.10.18`	Analytic sets: local parametrisation, dimension, and irreducible components	Riemann surfaces
`06.10.19`	The local ring of an analytic set; regular points, singular locus, Remmert–Stein	Riemann surfaces
`06.10.20`	Coherent analytic sheaves and Oka's coherence theorem	Riemann surfaces
`06.10.21`	Cartan Theorems A and B in $C^{n}$ (with proof)	Riemann surfaces
`06.10.22`	Complex spaces and coherence on them	Riemann surfaces
`06.10.E1`	Several complex variables exercise pack (Krantz supplement)	Riemann surfaces
`06.11.01`	Ideal boundary and exhaustions of an open Riemann surface	Riemann surfaces
`06.11.02`	Hilbert space of differentials; orthogonal decomposition on an open surface	Riemann surfaces
`06.11.03`	Green's function on a Riemann surface and the type problem (parabolic vs. hyperbolic)	Riemann surfaces
`06.11.04`	Null-classes O_G, O_HB, O_HD, O_AD and the classification of open surfaces	Riemann surfaces
`06.11.05`	Capacity and harmonic measure of the ideal boundary	Riemann surfaces
`06.11.06`	Extremal length and the modulus of curve families	Riemann surfaces
`07.01.01`	Group representation	Representation theory
`07.01.02`	Schur's lemma	Representation theory
`07.01.03`	Character of a representation	Representation theory
`07.01.04`	Character orthogonality	Representation theory
`07.01.05`	Regular representation	Representation theory
`07.01.06`	Tensor product of representations	Representation theory
`07.01.07`	Induced representation	Representation theory
`07.01.08`	Frobenius reciprocity	Representation theory
`07.01.09`	Non-abelian Fourier transform on a finite group	Representation theory
`07.01.10`	Artin's induction theorem	Representation theory
`07.01.11`	Brauer's induction theorem	Representation theory
`07.01.12`	Frobenius-Schur indicator	Representation theory
`07.01.13`	The irreducible representations of GL₂(𝔽_q)	Representation theory
`07.02.01`	Maschke's theorem	Representation theory
`07.02.02`	The Fong-Swan theorem	Representation theory
`07.02.03`	Grothendieck groups and the cde-triangle	Representation theory
`07.02.04`	Brauer character	Representation theory
`07.02.06`	Block theory of kG	Representation theory
`07.02.E1`	Finite-group representation exercise pack (Serre Linear Representations supplement)	Representation theory
`07.03.01`	Highest weight representation	Representation theory
`07.04.01`	Cartan-Weyl classification	Representation theory
`07.04.02`	Compact real form of a complex semisimple Lie algebra	Representation theory
`07.04.03`	Cartan involution	Representation theory
`07.04.05`	Real forms of a complex semisimple Lie algebra	Representation theory
`07.04.06`	Orthogonal symmetric Lie algebra	Representation theory
`07.04.07`	Riemannian symmetric space	Representation theory
`07.04.08`	Restricted root system	Representation theory
`07.04.09`	Iwasawa decomposition G=KAN	Representation theory
`07.04.10`	Bruhat decomposition	Representation theory
`07.04.11`	Invariant differential operators on G/K and the Harish-Chandra isomorphism	Representation theory
`07.04.12`	Spherical function on G/K	Representation theory
`07.04.13`	Classification tables of irreducible Riemannian symmetric spaces (Cartan's list)	Representation theory
`07.04.14`	Hermitian symmetric space	Representation theory
`07.05.01`	Symmetric group representation	Representation theory
`07.05.02`	Young diagram and tableau	Representation theory
`07.05.03`	Specht module	Representation theory
`07.05.04`	Schur-Weyl duality	Representation theory
`07.05.05`	Random walk on a finite group; Upper Bound Lemma	Representation theory
`07.05.06`	Association schemes, the Bose-Mesner algebra, and Krawtchouk/Hahn polynomials	Representation theory
`07.05.07`	Riffle shuffle and the 7-shuffle theorem	Representation theory
`07.05.08`	Cutoff phenomenon	Representation theory
`07.05.09`	Strong stationary time; coupling argument	Representation theory
`07.05.10`	Murnaghan-Nakayama rule	Representation theory
`07.05.11`	Spectral analysis of permutation-valued data	Representation theory
`07.05.12`	Metrics on S_n	Representation theory
`07.05.13`	Models for partially ranked data on S_n/S_{n-k}	Representation theory
`07.05.14`	De Finetti / exchangeability and the symmetric group	Representation theory
`07.05.15`	The Bernoulli-Laplace and Ehrenfest urn diffusion models	Representation theory
`07.05.16`	Wreath products and the representations of the hyperoctahedral group	Representation theory
`07.05.E1`	Lie-group and Lie-algebra representation exercise pack (Fulton-Harris supplement)	Representation theory
`07.06.01`	Lie algebra representation	Representation theory
`07.06.02`	Universal enveloping algebra	Representation theory
`07.06.03`	Root system	Representation theory
`07.06.04`	Weyl group	Representation theory
`07.06.05`	Dynkin diagram	Representation theory
`07.06.06`	Verma module	Representation theory
`07.06.07`	Weyl character formula	Representation theory
`07.06.08`	Weyl dimension formula	Representation theory
`07.06.09`	Borel-Weil theorem	Representation theory
`07.06.10`	Casimir element	Representation theory
`07.06.11`	Representations of $sl_{2} C$	Representation theory
`07.06.12`	Representations of $sl_{3} C$	Representation theory
`07.06.13`	Free Lie algebras, the Hall basis, and Magnus's theorem	Representation theory
`07.06.14`	Engel's theorem + Lie's theorem	Representation theory
`07.06.15`	The Campbell–Baker–Hausdorff formula	Representation theory
`07.06.16`	Cartan's criterion for solvability and semisimplicity	Representation theory
`07.06.17`	Cartan subalgebra	Representation theory
`07.06.18`	Root-space decomposition	Representation theory
`07.06.19`	Cartan matrix	Representation theory
`07.06.20`	Serre relations and Serre's theorem	Representation theory
`07.06.21`	The Killing form and the trace form	Representation theory
`07.06.22`	Weyl complete-reducibility theorem	Representation theory
`07.06.23`	Lie algebra cohomology and Whitehead's lemmas	Representation theory
`07.06.24`	The Hochschild-Serre spectral sequence for a Lie-algebra ideal	Representation theory
`07.06.25`	Weyl construction of the classical-group irreducibles	Representation theory
`07.06.26`	$G_{2}$ via the octonions	Representation theory
`07.06.27`	Lie superalgebras: the graded bracket, the super-Jacobi identity, and basic classification	Representation theory
`07.06.28`	Supermanifolds as ringed spaces, the functor of points, and super Lie groups	Representation theory
`07.06.29`	Composition algebras and the octonions	Representation theory
`07.06.E2`	Lie algebra structure exercise pack (Serre Lie Algebras and Lie Groups supplement)	Representation theory
`07.07.01`	Compact Lie group representation	Representation theory
`07.07.02`	Peter-Weyl theorem	Representation theory
`07.07.03`	Haar measure	Representation theory
`07.07.04`	Weyl integration formula	Representation theory
`07.07.05`	Representations of SU(2) and SO(3): the double cover, spin, and projective representations	Representation theory
`07.07.06`	Wigner's classification of the unitary irreducible representations of the Poincaré group	Representation theory
`07.07.07`	Mackey theory of induced representations and systems of imprimitivity	Representation theory
`07.07.08`	Crystallographic point groups, space groups, and the crystallographic restriction theorem	Representation theory
`07.07.09`	Representations of the Lorentz group: $SL (2, C)$ , the $(j_{1}, j_{2})$ reps, and Wigner's theorem	Representation theory
`07.07.10`	The unitary dual of $SL (2, R)$ : principal, discrete, and complementary series	Representation theory
`08.01.01`	Partition function (statistical mechanics)	Statistical field theory
`08.01.02`	Ising model	Statistical field theory
`08.01.03`	Boltzmann distribution and canonical ensemble	Statistical field theory
`08.01.04`	Free energy	Statistical field theory
`08.02.01`	Mean-field theory and Curie-Weiss model	Statistical field theory
`08.02.02`	Spontaneous symmetry breaking	Statistical field theory
`08.02.03`	Mermin-Wagner theorem	Statistical field theory
`08.03.01`	Onsager solution of the 2D Ising model (transfer matrix)	Statistical field theory
`08.03.02`	Transfer matrix	Statistical field theory
`08.04.01`	Renormalisation group (real-space block decimation)	Statistical field theory
`08.04.02`	Wilson-Fisher fixed point and universality	Statistical field theory
`08.04.03`	Beta function (renormalisation group)	Statistical field theory
`08.04.04`	Block-spin decimation	Statistical field theory
`08.04.05`	Momentum-shell (Wilson) renormalization group	Statistical field theory
`08.05.01`	Critical exponents and scaling laws	Statistical field theory
`08.05.02`	Correlation functions (statistical mechanics)	Statistical field theory
`08.06.01`	Gaussian field theory and free boson	Statistical field theory
`08.06.02`	Conformal symmetry at criticality	Statistical field theory
`08.07.01`	Path integral formulation of statistical mechanics	Statistical field theory
`08.08.01`	Wilson's lattice gauge theory	Statistical field theory
`08.08.02`	Wilson action	Statistical field theory
`08.08.03`	Effective field theory	Statistical field theory
`08.08.04`	The roughening transition and the confining string	Statistical field theory
`08.09.01`	Quantum-classical correspondence (Wick rotation)	Statistical field theory
`08.10.01`	Bosonic Fock space and second quantisation	Statistical field theory
`08.10.02`	Fokker-Planck equation and equilibrium distribution	Statistical field theory
`08.10.03`	φ⁴ theory and the Dyson series	Statistical field theory
`08.10.04`	Wick's theorem for operator products	Statistical field theory
`08.10.05`	Feynman propagator and the contour-integral representation	Statistical field theory
`08.10.06`	One-loop renormalisation in φ⁴	Statistical field theory
`08.10.07`	Wightman axioms (W1–W7)	Statistical field theory
`08.10.08`	Langevin updates and lattice numerics	Statistical field theory
`08.10.09`	Fermionic Fock space, Pauli exclusion, anticommutators	Statistical field theory
`08.10.10`	Dirac field $ψ$ and the Dirac adjoint $\overset{ˉ}{ψ}$	Statistical field theory
`08.10.11`	Supersymmetric quantum mechanics: superpotential, supercharges, and the Witten index	Statistical field theory
`08.10.12`	The Nicolai map and stochastic quantisation of supersymmetric theories	Statistical field theory
`08.10.13`	Parisi-Sourlas dimensional reduction and random-field supersymmetry	Statistical field theory
`08.10.14`	The Martin-Siggia-Rose / Janssen-De Dominicis response-field formalism	Statistical field theory
`08.10.15`	Stochastic perturbation theory and the tree expansion	Statistical field theory
`08.11.02`	Debye theory of specific heats of solids	Statistical field theory
`08.11.03`	Real gases — virial expansion and van der Waals	Statistical field theory
`08.12.01`	Fluctuation-dissipation theorem (Landau-Callen-Welton)	Statistical field theory
`08.12.02`	Equilibrium fluctuations of thermodynamic quantities	Statistical field theory
`08.13.01`	The Yang–Baxter equation and the star–triangle relation	Statistical field theory
`08.13.02`	The six-vertex (ice-type) model and the Bethe ansatz	Statistical field theory
`08.13.03`	The eight-vertex model (Baxter 1971)	Statistical field theory
`08.13.04`	The corner transfer matrix	Statistical field theory
`08.13.05`	The hard-hexagon model (Baxter 1980)	Statistical field theory
`08.13.07`	The spherical model (Berlin-Kac)	Statistical field theory
`08.13.08`	The Ising model on the Bethe lattice	Statistical field theory
`08.14.01`	Brownian motion, the Wiener measure, and the path integral	Statistical field theory
`08.14.02`	Grassmann integration and the 2D Ising model as free fermions	Statistical field theory
`08.14.03`	The large-N limit	Statistical field theory
`08.14.04`	Lattice fermions and the doubling problem	Statistical field theory
`08.14.05`	The Pfaffian and the dimer model	Statistical field theory
`08.14.06`	Pointer: matrix models and the topological expansion	Statistical field theory
`08.14.07`	The Kardar-Parisi-Zhang equation and dynamic scaling	Statistical field theory
`08.14.08`	Liouville field theory and 2D quantum gravity (KPZ-DDK scaling)	Statistical field theory
`08.15.01`	The Kosterlitz-Thouless transition (2D XY model)	Statistical field theory
`08.15.02`	The nonlinear σ-model and the O(n) renormalization group	Statistical field theory
`08.15.03`	Topological defects in ordered media	Statistical field theory
`09.01.01`	Kinematics — position, velocity, acceleration	Classical mechanics
`09.01.02`	Newton's laws of motion	Classical mechanics
`09.01.03`	Conservation laws — energy, momentum, angular momentum	Classical mechanics
`09.01.04`	Two-body central-force problem, Kepler orbits, and Rutherford scattering	Classical mechanics
`09.02.01`	The action principle and variational calculus	Classical mechanics
`09.02.02`	Euler-Lagrange equations	Classical mechanics
`09.03.01`	Noether's theorem — symmetries and conservation laws	Classical mechanics
`09.03.03`	Quantum free particle as a representation of $E (3)$	Quantum mechanics
`09.04.01`	Legendre transform — from Lagrangian to Hamiltonian	Classical mechanics
`09.04.02`	Hamilton's equations	Classical mechanics
`09.04.07`	Complex structures and quantization; squeezed states	Quantum mechanics
`09.05.01`	Canonical transformations	Classical mechanics
`09.05.02`	Hamilton-Jacobi equation	Classical mechanics
`09.06.01`	Action-angle variables	Classical mechanics
`09.07.01`	Continuum Mechanics and Field Theory	Classical mechanics
`09.08.01`	KAM theorem and chaos	Classical mechanics
`10.01.01`	Coulomb's law and Gauss's law	Electromagnetism & special relativity
`10.01.02`	Laplace equation and boundary value problems	Electromagnetism & special relativity
`10.01.03`	Conductors, capacitance, and electrostatic energy	Electromagnetism & special relativity
`10.01.04`	Dielectrics, polarization P, and the electric displacement D	Electromagnetism & special relativity
`10.01.05`	Multipole expansion of the electrostatic potential	Electromagnetism & special relativity
`10.02.01`	Biot-Savart law and Ampere's law	Electromagnetism & special relativity
`10.03.01`	Faraday's law and electromagnetic induction	Electromagnetism & special relativity
`10.03.03`	Energy and momentum in the electromagnetic field: Poynting vector, Maxwell stress tensor, conservation laws	Electromagnetism & special relativity
`10.04.01`	Maxwell's equations in differential form	Electromagnetism & special relativity
`10.04.02`	EM waves and the wave equation	Electromagnetism & special relativity
`10.05.01`	Special relativity — postulates and Lorentz transformations	Electromagnetism & special relativity
`10.05.02`	Relativistic kinematics and dynamics	Electromagnetism & special relativity
`10.05.03`	Four-velocity, four-momentum, and the relativistic energy-momentum identity	Electromagnetism & special relativity
`10.06.01`	Covariant electrodynamics — Faraday tensor	Electromagnetism & special relativity
`10.07.01`	Radiation from accelerating charges — Larmor formula	Electromagnetism & special relativity
`11.01.01`	First and second laws of thermodynamics	Statistical mechanics
`11.01.02`	Thermodynamic potentials and Legendre transforms	Statistical mechanics
`11.02.01`	Maxwell-Boltzmann distribution from kinetic theory	Statistical mechanics
`11.03.01`	Microcanonical ensemble	Statistical mechanics
`11.04.01`	Canonical ensemble and partition function	Statistical mechanics
`11.04.02`	Souriau Gibbs state on a symplectic $G$ -space	Statistical mechanics
`11.05.01`	Bose-Einstein distribution	Statistical mechanics
`11.05.02`	Fermi-Dirac distribution and electron gas	Statistical mechanics
`11.05.03`	Blackbody radiation: Planck distribution, Stefan-Boltzmann law, Wien displacement law	Statistical mechanics
`11.05.04`	Bose-Einstein condensation and the critical temperature	Statistical mechanics
`11.05.05`	Fermi gas: heat capacity, electron specific heat, and Pauli paramagnetism	Statistical mechanics
`11.05.06`	Photon gas, phonon gas, and the Debye model of solids	Statistical mechanics
`11.06.01`	Ising model and phase transitions	Statistical mechanics
`11.07.01`	Critical phenomena and renormalization group	Statistical mechanics
`12.01.01`	Wave-particle duality and the double-slit	Quantum mechanics & QFT
`12.01.02`	Stern-Gerlach and spin-1/2	Quantum mechanics & QFT
`12.02.01`	Hilbert-space formalism of quantum mechanics	Quantum mechanics & QFT
`12.02.02`	Operators, observables, and Hermiticity	Quantum mechanics & QFT
`12.02.03`	Density matrix and pure / mixed states	Quantum mechanics & QFT
`12.03.01`	Schrödinger and Heisenberg pictures	Quantum mechanics & QFT
`12.04.01`	Particle in a box	Quantum mechanics & QFT
`12.04.02`	Quantum harmonic oscillator	Quantum mechanics & QFT
`12.05.01`	Angular momentum operators and SU(2) representations	Quantum mechanics & QFT
`12.05.02`	Spherical harmonics and Legendre polynomials	Quantum mechanics & QFT
`12.05.03`	Addition of angular momenta and Clebsch-Gordan coefficients	Quantum mechanics & QFT
`12.05.04`	Free Klein-Gordon scalar quantum field	Quantum mechanics & QFT
`12.05.05`	Free Dirac spin-1/2 quantum field	Quantum mechanics & QFT
`12.05.06`	Free Maxwell / massive vector fields; photon and Proca	Quantum mechanics & QFT
`12.05.07`	Molecular vibrations and spectroscopic selection rules via symmetry	Quantum mechanics & QFT
`12.06.01`	Hydrogen atom bound states	Quantum mechanics & QFT
`12.06.04`	Crossing symmetry; CPT theorem at the $S$ -matrix level	Quantum mechanics & QFT
`12.07.01`	Time-independent perturbation theory	Quantum mechanics & QFT
`12.07.02`	Time-dependent perturbation theory and Fermi's golden rule	Quantum mechanics & QFT
`12.07.03`	Variational method (Rayleigh-Ritz) in quantum mechanics	Quantum mechanics & QFT
`12.07.04`	WKB approximation and Bohr-Sommerfeld quantisation	Quantum mechanics & QFT
`12.07.05`	Stark and Zeeman effects in LL3 framing	Quantum mechanics & QFT
`12.07.07`	Adiabatic theorem and Berry phase preview	Quantum mechanics & QFT
`12.07.08`	Berry phase and the geometric phase	Quantum mechanics & QFT
`12.08.01`	Scattering Theory	Quantum mechanics & QFT
`12.08.02`	Born approximation and the Lippmann-Schwinger equation	Quantum mechanics & QFT
`12.08.03`	Partial-wave expansion and phase shifts	Quantum mechanics & QFT
`12.08.04`	Inelastic collisions and the distorted-wave Born approximation	Quantum mechanics & QFT
`12.09.01`	Identical Particles and Many-Body Quantum Mechanics	Quantum mechanics & QFT
`12.09.02`	Exchange interaction and the helium atom	Quantum mechanics & QFT
`12.09.03`	Hartree-Fock self-consistent field method	Quantum mechanics & QFT
`12.09.04`	Multi-electron atomic structure and LS coupling	Quantum mechanics & QFT
`12.09.05`	Diatomic molecule and the Born-Oppenheimer approximation	Quantum mechanics & QFT
`12.10.01`	Path integral formulation of quantum mechanics	Quantum mechanics & QFT
`12.11.01`	Dirac equation and relativistic spin	Quantum mechanics & QFT
`12.11.02`	Klein-Gordon equation in external EM field: Coulomb and uniform-magnetic cases	Quantum mechanics & QFT
`12.11.03`	Dirac equation in a Coulomb field	Quantum mechanics & QFT
`12.11.04`	Klein paradox	Quantum mechanics & QFT
`12.11.05`	Furry's theorem and charge-conjugation symmetry of QED	Quantum mechanics & QFT
`12.12.01`	Canonical Quantum Field Theory	Quantum mechanics & QFT
`12.12.02`	Coulomb gauge vs Lorenz gauge in QED	Quantum mechanics & QFT
`12.12.03`	Compton scattering and the Klein-Nishina formula	Quantum mechanics & QFT
`12.12.04`	Møller scattering (electron-electron)	Quantum mechanics & QFT
`12.12.05`	Bhabha scattering (electron-positron)	Quantum mechanics & QFT
`12.12.06`	Bethe-Heitler bremsstrahlung and pair production	Quantum mechanics & QFT
`12.13.01`	Bosonic Fock space and second quantisation	Quantum mechanics & QFT
`12.13.02`	Fermionic Fock space, Pauli exclusion, and anticommutators	Quantum mechanics & QFT
`12.13.03`	Cluster decomposition and the connected S-matrix	Quantum mechanics & QFT
`12.14.01`	CCR algebra, Weyl algebra, and quasi-free states	Quantum mechanics & QFT
`12.14.02`	The Heisenberg group, the Schrödinger representation, and Stone–von Neumann as quantization	Quantum mechanics & QFT
`12.15.01`	Time-reversal symmetry and Kramers' degeneracy	Quantum mechanics & QFT
`12.15.02`	Parity, discrete-symmetry groups, and the Wigner-Eckart theorem	Quantum mechanics & QFT
`12.16.01`	Electron self-energy and mass renormalization at one loop	Quantum mechanics & QFT
`12.16.02`	One-loop QED vertex function and the anomalous magnetic moment	Quantum mechanics & QFT
`12.16.03`	Vacuum polarization at one loop and the Uehling potential	Quantum mechanics & QFT
`12.16.04`	Lamb shift from one-loop QED	Quantum mechanics & QFT
`12.16.05`	Infrared divergences and the Bloch-Nordsieck cancellation in QED	Quantum mechanics & QFT
`12.16.06`	Power counting, the superficial degree of divergence, and renormalizability classification	Quantum mechanics & QFT
`12.17.01`	Density matrix, pure states, and mixed states	Quantum mechanics & QFT
`12.17.02`	Entanglement, Schmidt decomposition, and entanglement entropy	Quantum mechanics & QFT
`12.17.03`	Bell inequalities, CHSH inequality, and the Tsirelson bound	Quantum mechanics & QFT
`12.17.07`	Quantum teleportation and superdense coding	Quantum mechanics & QFT
`12.18.01`	The Higgs mechanism: spontaneously broken gauge symmetry	Quantum mechanics & QFT
`12.18.02`	The Goldstone theorem and effective Goldstone Lagrangians	Quantum mechanics & QFT
`12.18.03`	Asymptotic freedom and the running gauge coupling	Quantum mechanics & QFT
`12.18.04`	Theta-vacua, the vacuum angle, and the strong-CP problem	Quantum mechanics & QFT
`12.18.05`	The chiral (Adler-Bell-Jackiw) anomaly from the triangle diagram	Quantum mechanics & QFT
`12.18.06`	Operator product expansion and short-distance behaviour	Quantum mechanics & QFT
`12.18.13`	Vortices (Nielsen-Olesen / Abrikosov flux tubes)	Quantum mechanics & QFT
`12.18.16`	Lattice gauge theory and confinement (QFT pointer)	Quantum mechanics & QFT
`12.19.01`	The supersymmetry algebra: Coleman-Mandula, Haag-Lopuszanski-Sohnius, and the graded extension of Poincare	Quantum mechanics & QFT
`12.19.02`	Superspace, superfields, and Berezin integration	Quantum mechanics & QFT
`12.19.03`	Supermultiplets and the Wess-Zumino model: Bose-Fermi degeneracy, the superpotential, and the auxiliary F-field	Quantum mechanics & QFT
`12.19.04`	Super-Yang-Mills and the non-renormalization theorem: the vector superfield, Wess-Zumino gauge, $W^{a} W_{a}$ , and supergraphs	Quantum mechanics & QFT
`12.19.05`	Spontaneous SUSY breaking: the Goldstino theorem, O'Raifeartaigh and Fayet-Iliopoulos, the supertrace sum rule, and the field-theory Witten index	Quantum mechanics & QFT
`12.19.06`	Supersymmetric QCD and Seiberg duality: the moduli space of vacua, holomorphy, and N=1 electric-magnetic duality	Quantum mechanics & QFT
`13.01.01`	The equivalence principle	General relativity & cosmology
`13.02.01`	Tensors on smooth manifolds	General relativity & cosmology
`13.02.02`	Geodesics and parallel transport	General relativity & cosmology
`13.02.03`	Cartan tetrad and spin-connection formulation of general relativity	General relativity & cosmology
`13.03.01`	Riemann curvature tensor	General relativity & cosmology
`13.04.01`	Einstein field equations	General relativity & cosmology
`13.04.02`	Einstein-Hilbert action and variational derivation of the Einstein equations	General relativity & cosmology
`13.04.03`	Palatini first-order variational formulation of general relativity	General relativity & cosmology
`13.04.04`	Stress-energy tensor as functional derivative of the matter action	General relativity & cosmology
`13.05.01`	Schwarzschild solution	General relativity & cosmology
`13.05.02`	Orbits in Schwarzschild geometry	General relativity & cosmology
`13.05.03`	Solar-system tests of general relativity: perihelion precession, light bending, Shapiro time delay, gravitational redshift, frame-dragging	General relativity & cosmology
`13.05.04`	Kerr black hole, ergosphere, and the Penrose process	General relativity & cosmology
`13.06.01`	Black Holes	General relativity & cosmology
`13.06.03`	Black hole thermodynamics: the four laws, Bekenstein-Hawking entropy, and the area theorem	General relativity & cosmology
`13.06.04`	Hawking radiation: Bogoliubov derivation, thermal spectrum, and black-hole evaporation	General relativity & cosmology
`13.07.01`	Linearized GR and gravitational waves	General relativity & cosmology
`13.07.02`	Null infinity, the BMS group, and the Bondi-Sachs mass-loss formula	General relativity & cosmology
`13.08.01`	FLRW cosmology and Friedmann equations	General relativity & cosmology
`13.09.01`	Globally hyperbolic Lorentzian manifolds	General relativity & cosmology
`13.09.02`	Klein-Gordon equation on a globally hyperbolic spacetime	General relativity & cosmology
`13.09.03`	Hadamard states via the wave-front-set criterion	General relativity & cosmology
`13.09.04`	Existence of Hadamard states via the FNW deformation argument	General relativity & cosmology
`13.09.05`	Hadamard states by pseudo-differential calculus (Gérard-Wrochna)	General relativity & cosmology
`13.09.06`	Wick polynomials in curved spacetime via Hadamard parametrix subtraction	General relativity & cosmology
`13.09.07`	Time-ordered products and Hollands-Wald renormalisation on curved spacetimes	General relativity & cosmology
`13.09.08`	Bunch-Davies state on de Sitter spacetime	General relativity & cosmology
`13.09.09`	Unruh effect via the Bisognano-Wichmann theorem	General relativity & cosmology
`13.09.10`	Hartle-Hawking and Unruh states on Schwarzschild	General relativity & cosmology
`13.09.11`	Quantum energy inequalities (Fewster)	General relativity & cosmology
`13.09.12`	The Peierls bracket and the covariant phase space of an interacting field theory	General relativity & cosmology
`14.01.01`	Atomic structure and electron configurations	General & physical chemistry
`14.02.01`	Lewis structures and VSEPR	General & physical chemistry
`14.02.02`	Hybridization and valence bond theory	General & physical chemistry
`14.03.01`	Stoichiometry and gas laws	General & physical chemistry
`14.04.01`	Hydrogen atom quantum chemistry	General & physical chemistry
`14.05.02`	Molecular orbital theory for homonuclear diatomics	General & physical chemistry
`14.06.01`	Chemical thermodynamics: free energies and equilibrium	General & physical chemistry
`14.07.01`	Statistical Mechanics for Chemistry	General & physical chemistry
`14.08.01`	Chemical kinetics: rate laws and the Arrhenius equation	General & physical chemistry
`14.09.01`	Solutions and Phase Equilibria	General & physical chemistry
`14.10.01`	Acid-base chemistry: Bronsted-Lowry, Lewis, and pKa	General & physical chemistry
`14.11.01`	Electrochemistry: the Nernst equation and electrochemical cells	General & physical chemistry
`14.12.01`	UV-Vis, IR, and NMR — fundamentals of molecular spectroscopy	General & physical chemistry
`15.01.01`	Structure of organic molecules — stereochemistry	Organic chemistry
`15.02.01`	Functional groups and nomenclature	Organic chemistry
`15.03.01`	Acids and bases in organic chemistry	Organic chemistry
`15.04.02`	SN1 vs SN2 substitution mechanisms	Organic chemistry
`15.05.01`	Electrophilic addition to alkenes	Organic chemistry
`15.06.01`	Aromatic chemistry — EAS, Huckel	Organic chemistry
`15.07.01`	Carbonyl chemistry — nucleophilic addition	Organic chemistry
`15.08.01`	Radical and Pericyclic Reactions	Organic chemistry
`15.09.01`	Organometallic Methods in Synthesis	Organic chemistry
`15.10.01`	Retrosynthetic analysis	Organic chemistry
`15.11.01`	NMR spectroscopy of organic molecules	Organic chemistry
`15.12.01`	Amino acids and protein chemistry	Organic chemistry
`15.13.01`	Nucleic acid chemistry	Organic chemistry
`15.14.01`	Enzyme mechanism	Organic chemistry
`16.01.01`	Periodic trends quantified	Inorganic chemistry
`16.02.01`	Symmetry and group theory in chemistry	Inorganic chemistry
`16.03.01`	Crystal field theory fundamentals	Inorganic chemistry
`16.03.02`	Crystal field splitting in octahedral complexes	Inorganic chemistry
`16.04.01`	Coordination chemistry	Inorganic chemistry
`16.04.02`	Crystal field stabilization energy and the spectrochemical series	Inorganic chemistry
`16.05.01`	Organometallic chemistry	Inorganic chemistry
`16.06.01`	Bioinorganic chemistry	Inorganic chemistry
`16.07.01`	Solid-state chemistry	Inorganic chemistry
`17.01.01`	Biomolecules in cells — overview	Molecular & cellular biology
`17.02.01`	Cell membranes: structure	Molecular & cellular biology
`17.02.02`	Membrane transport — passive and active	Molecular & cellular biology
`17.03.01`	Cellular organization: organelles	Molecular & cellular biology
`17.03.02`	Cytoskeleton and contractile proteins	Molecular & cellular biology
`17.04.01`	Cellular respiration: glycolysis and CAC	Molecular & cellular biology
`17.04.02`	Oxidative phosphorylation and ATP synthesis	Molecular & cellular biology
`17.04.03`	Photosynthesis: light and dark reactions	Molecular & cellular biology
`17.05.01`	DNA replication	Molecular & cellular biology
`17.05.02`	Transcription	Molecular & cellular biology
`17.05.03`	Translation	Molecular & cellular biology
`17.06.01`	Mutation and repair	Molecular & cellular biology
`17.07.01`	Cell signaling: receptors and GPCRs	Molecular & cellular biology
`17.07.02`	Receptor tyrosine kinases and the MAPK signaling cascade	Molecular & cellular biology
`17.08.01`	Cell cycle and mitosis	Molecular & cellular biology
`17.09.01`	Resting membrane potential and ion channels	Molecular & cellular biology
`17.09.02`	The action potential — ionic basis	Molecular & cellular biology
`17.10.01`	Innate immunity at the molecular level	Molecular & cellular biology
`18.01.01`	Body plans and organization	Organismal biology
`18.02.01`	Cardiovascular physiology — the heart	Organismal biology
`18.02.02`	Cardiac action potentials, pacemaker physiology, and the ECG	Organismal biology
`18.03.01`	Respiratory physiology — gas exchange and transport	Organismal biology
`18.04.01`	Skeletal muscle physiology	Organismal biology
`18.04.02`	Muscle contraction — the actin-myosin cycle	Organismal biology
`18.05.01`	Nervous system — gross anatomy and systems	Organismal biology
`18.06.01`	Digestive physiology and nutrition	Organismal biology
`18.07.01`	Endocrine system — hormones and regulation	Organismal biology
`18.08.01`	Renal physiology — homeostasis and the nephron	Organismal biology
`18.09.01`	Reproductive biology	Organismal biology
`18.10.01`	Immunology	Organismal biology
`18.11.01`	Embryology and morphogenesis	Organismal biology
`19.01.01`	Mendelian genetics — segregation and dominance	Ecology & evolution
`19.02.01`	Hardy-Weinberg equilibrium	Ecology & evolution
`19.02.05`	Wright-Fisher model and the diffusion approximation	Ecology & evolution
`19.03.01`	Natural selection — directional, stabilizing, and disruptive	Ecology & evolution
`19.03.02`	Sexual selection	Ecology & evolution
`19.03.03`	Kin selection and Hamilton's rule	Ecology & evolution
`19.04.01`	Genetic drift	Ecology & evolution
`19.05.01`	Quantitative genetics — heritability and the breeder's equation	Ecology & evolution
`19.06.01`	Speciation — allopatric and sympatric	Ecology & evolution
`19.07.01`	Phylogenetics — tree reconstruction	Ecology & evolution
`19.08.01`	Macroevolution	Ecology & evolution
`19.09.01`	Population ecology — Lotka-Volterra	Ecology & evolution
`19.10.01`	Community ecology — interactions and food webs	Ecology & evolution
`19.11.01`	Ecosystem ecology	Ecology & evolution
`19.12.01`	Biogeography	Ecology & evolution
`19.13.01`	Coevolution	Ecology & evolution
`19.14.01`	Conservation biology	Ecology & evolution
`19.15.01`	Origin of life — mechanistic scenarios	Ecology & evolution
`20.01.01`	Epistemology: knowledge, justification, and truth	Philosophy
`20.02.01`	Theories of justice: Rawls, Nozick, and fairness	Philosophy
`20.02.02`	Rights: natural, human, and legal	Philosophy
`20.02.03`	Freedom and liberty: negative, positive, and free will	Philosophy
`20.02.04`	The trolley problem and moral dilemmas	Philosophy
`20.02.05`	The good life: eudaimonia, flourishing, and meaning	Philosophy
`20.02.06`	Ethics of artificial intelligence	Philosophy
`20.03.01`	The measurement problem in quantum mechanics	Philosophy
`20.04.01`	Aesthetics: beauty, art, and judgment	Philosophy
`20.05.02`	The unit of selection	Philosophy
`20.06.01`	Consciousness: the hard problem, qualia, and the mind-body debate	Philosophy
`20.07.01`	Democratic theory: participation, deliberation, and representation	Philosophy
`20.08.01`	Philosophy of science: demarcation, falsification, and paradigms	Philosophy
`20.09.01`	Philosophy of mathematics: Platonism, constructivism, and the nature of numbers	Philosophy
`20.10.01`	Confucianism: ethics, society, and the exemplary person	Philosophy
`20.11.01`	Buddhism: the Four Noble Truths, the Eightfold Path, and the question of suffering	Philosophy
`20.12.01`	Advaita Vedanta and Hindu philosophy: Brahman, Atman, and the question of reality	Philosophy
`20.13.01`	Daoism: wu wei, the Dao, and natural harmony	Philosophy
`21.01.01`	Divisibility, GCD, Bézout's identity, and the Euclidean algorithm	Number theory
`21.01.02`	Primes, the fundamental theorem of arithmetic, and the infinitude of primes	Number theory
`21.01.03`	Congruences, the Chinese remainder theorem, and the ring structure of ℤ/nℤ	Number theory
`21.01.04`	Fermat's little theorem, Euler's theorem, and Wilson's theorem	Number theory
`21.01.05`	Primitive roots and the structure of $(Z / n Z)^{\times}$	Number theory
`21.01.06`	Quadratic residues, the Legendre symbol, and Euler's criterion	Number theory
`21.01.07`	Quadratic reciprocity (Gauss's theorema aureum)	Number theory
`21.01.08`	Pell equation and continued fractions	Number theory
`21.02.01`	Finite fields $F_{q}$ — structure and squares	Number theory
`21.02.02`	Quadratic reciprocity via Gauss sums	Number theory
`21.02.03`	$p$ -adic numbers $Q_{p}$ and $Z_{p}$	Number theory
`21.02.04`	Hensel's lemma	Number theory
`21.02.05`	Hilbert symbol $(a, b)_{v}$ and the product formula	Number theory
`21.02.06`	Witt's theorem: cancellation and the Witt decomposition	Number theory
`21.02.07`	Number fields: ring of integers, ideal class group, and the Dirichlet unit theorem	Number theory
`21.02.08`	Hasse-Minkowski theorem	Number theory
`21.02.09`	The Brauer-Manin obstruction	Number theory
`21.03.01`	Riemann Zeta Function $ζ (s)$	Number theory
`21.03.02`	Dirichlet $L$ -functions $L (s, χ)$	Number theory
`21.03.03`	Dedekind Zeta Function, Hecke $L$ -Functions, Artin $L$ -Functions	Number theory
`21.03.04`	Dirichlet density	Number theory
`21.04.01`	Modular Forms on $SL_{2} (Z)$	Number theory
`21.04.02`	Hecke Operators and Hecke Algebra	Number theory
`21.04.03`	Eichler-Shimura Correspondence	Number theory
`21.04.04`	Theta Series of Quadratic Forms and Sums of Squares	Number theory
`21.04.05`	Ramanujan $τ$ -function and Ramanujan conjectures	Number theory
`21.05.01`	$ℓ$ -adic Galois Representations	Number theory
`21.06.01`	Modularity Theorem (Statement) and BSD Conjecture	Number theory
`21.06.02`	Sato-Tate conjecture	Number theory
`21.07.01`	$Z_{p}$ -extensions and Iwasawa Theory	Number theory
`21.07.02`	$p$ -adic $L$ -functions and the Iwasawa Main Conjecture	Number theory
`21.09.01`	Arakelov geometry and arithmetic surfaces (survey)	Number theory
`21.09.02`	Faltings / Mordell theorem	Number theory
`21.09.03`	Heights and the Néron–Tate canonical height	Number theory
`21.10.01`	Langlands Philosophy Survey	Number theory
`21.11.01`	Arithmetic Functions, Dirichlet Convolution, and Möbius Inversion	Number theory
`21.11.02`	Average Orders of Arithmetic Functions and the Summation Toolkit	Number theory
`21.11.03`	Chebyshev's Bounds, Bertrand's Postulate, and Mertens' Theorems	Number theory
`21.11.04`	Perron's Formula and Mellin Inversion	Number theory
`21.11.05`	The Selberg-Delange Method	Number theory
`21.12.01`	The von Mangoldt Function, the Chebyshev Psi Function, and the Logarithmic Derivative of Zeta	Number theory
`21.12.02`	The Prime Number Theorem via Contour Integration	Number theory
`21.12.03`	Effective Zero-Free Regions for Zeta and the Prime Number Theorem Error Term	Number theory
`21.12.04`	The Riemann-von Mangoldt Explicit Formula	Number theory
`21.13.01`	Zero-Free Regions for Dirichlet L-Functions and Exceptional (Siegel) Zeros	Number theory
`21.13.02`	Siegel's Theorem on the Exceptional Zero	Number theory
`21.13.03`	The Prime Number Theorem in Arithmetic Progressions and Siegel-Walfisz	Number theory
`21.13.04`	The Polya-Vinogradov Inequality	Number theory
`21.13.05`	The Approximate Functional Equation, Analytic Conductor, and Convexity Bound	Number theory
`21.14.01`	The Large Sieve Inequality and Brun-Titchmarsh	Number theory
`21.14.02`	The Bombieri-Vinogradov Theorem	Number theory
`21.14.03`	Mean Values of Multiplicative Functions: Halász's Theorem	Number theory
`21.14.04`	Combinatorial Sieve Methods: Brun and Selberg	Number theory
`21.14.05`	Kloosterman Sums and the Kuznetsov Spectral Formula	Number theory
`21.15.01`	Poisson and Voronoi Summation	Number theory
`21.15.02`	Weyl Sums, Weyl Differencing, and Equidistribution	Number theory
`21.15.03`	van der Corput's Method for Exponential Sums	Number theory
`21.15.04`	Gauss, Jacobi, Kloosterman, and Salié Sums; the Weil Bound	Number theory
`21.15.05`	The Vinogradov Mean Value Theorem	Number theory
`21.16.01`	The Partition Function, Generating Functions, and the Pentagonal Number Theorem	Number theory
`21.16.02`	The Hardy-Ramanujan-Rademacher Asymptotics via the Circle Method	Number theory
`22.01.01`	Nouns	Grammar
`22.01.02`	Verbs	Grammar
`22.01.03`	Sentences: subject and predicate	Grammar
`22.01.04`	Pronouns	Grammar
`22.01.05`	Adjectives	Grammar
`22.01.06`	Adverbs	Grammar
`22.01.07`	Prepositions	Grammar
`22.01.08`	Conjunctions	Grammar
`22.01.09`	Interjections	Grammar
`22.01.10`	Noun phrases and verb phrases	Grammar
`22.01.11`	Subject-verb agreement	Grammar
`22.01.12`	Verb tense: present, past, future	Grammar
`22.01.13`	Perfect and progressive aspects	Grammar
`22.01.14`	Active and passive voice	Grammar
`22.01.15`	Clauses: independent and dependent	Grammar
`22.01.16`	Compound and complex sentences	Grammar
`22.01.17`	Relative clauses	Grammar
`22.01.18`	Punctuation: end marks and commas	Grammar
`22.01.19`	Punctuation: semicolons, colons, dashes	Grammar
`22.01.20`	Apostrophes and quotation marks	Grammar
`22.01.21`	Common errors: fragments, run-ons, dangling modifiers	Grammar
`22.01.22`	Parallel structure	Grammar
`22.01.23`	Pronoun case and reference	Grammar
`22.01.24`	Capitalization conventions	Grammar
`22.02.01`	Writing a clear sentence	Writing
`22.02.02`	Paragraph structure	Writing
`22.02.03`	Transitions and flow	Writing
`22.02.04`	Thesis statement	Writing
`22.02.05`	Structuring an argument	Writing
`22.02.06`	Using evidence	Writing
`22.02.07`	Counterargument and rebuttal	Writing
`22.02.08`	Introduction and conclusion	Writing
`22.02.09`	Citation and attribution	Writing
`22.02.10`	Revision and editing	Writing
`22.02.11`	Style and voice	Writing
`22.03.01`	Literal vs Figurative Language	Literature techniques
`22.03.02`	Metaphor and Simile	Literature techniques
`22.03.03`	Symbolism and Allegory	Literature techniques
`22.03.04`	Irony	Literature techniques
`22.03.05`	Foreshadowing and Suspense	Literature techniques
`22.03.06`	Point of View	Literature techniques
`22.03.07`	Tone and Mood	Literature techniques
`22.03.08`	Theme	Literature techniques
`22.03.09`	Motif and Repetition	Literature techniques
`22.03.10`	Unreliable Narration	Literature techniques
`22.03.11`	Satire and Parody	Literature techniques
`22.03.12`	Imagery and Sensory Detail	Literature techniques
`22.03.13`	Allusion	Literature techniques
`22.03.14`	Personification	Literature techniques
`22.03.15`	Hyperbole and Understatement	Literature techniques
`22.04.01`	Reading guide: The Catcher in the Rye (Salinger)	Literature techniques
`22.04.02`	Reading guide: The Great Gatsby (Fitzgerald)	Literature techniques
`22.04.03`	Reading guide: The Old Man and the Sea (Hemingway)	Literature techniques
`22.04.04`	Reading guide: Nineteen Eighty-Four (Orwell)	Literature techniques
`23.01.01`	Scarcity and choice	Economics
`23.01.02`	Opportunity Cost	Economics
`23.01.03`	Supply and Demand	Economics
`23.01.04`	Market Equilibrium	Economics
`23.01.05`	Elasticity	Economics
`23.01.06`	Price Controls	Economics
`23.01.07`	Consumer and Producer Surplus	Economics
`23.01.08`	Costs of Production	Economics
`23.01.09`	Perfect Competition	Economics
`23.01.10`	Monopoly	Economics
`23.01.11`	Oligopoly and Monopolistic Competition	Economics
`23.01.12`	Profit Maximization	Economics
`23.01.13`	Labor Markets and Wages	Economics
`23.01.14`	Money and Banking	Economics
`23.01.15`	Inflation and Deflation	Economics
`23.01.16`	GDP and economic measurement	Economics
`23.01.17`	Unemployment	Economics
`23.01.18`	Fiscal policy	Economics
`23.01.19`	Monetary policy	Economics
`23.01.20`	International trade and comparative advantage	Economics
`23.01.21`	Exchange rates	Economics
`23.01.22`	Game theory basics	Economics
`23.01.23`	Externalities and public goods	Economics
`23.01.24`	Income inequality and redistribution	Economics
`23.01.25`	Behavioral economics	Economics
`23.01.26`	Market failures	Economics
`23.01.27`	Economic systems	Economics
`23.01.28`	Development economics	Economics
`23.01.29`	Personal finance: budgeting, saving, compound interest	Economics
`23.01.30`	Personal finance: credit, debt, investing	Economics
`23.02.01`	What is government	Civics
`23.02.02`	Types of government	Civics
`23.02.03`	What is a constitution	Civics
`23.02.04`	Separation of powers	Civics
`23.02.05`	The legislature	Civics
`23.02.06`	The executive	Civics
`23.02.07`	The judiciary	Civics
`23.02.08`	How a law is made	Civics
`23.02.09`	Electoral systems	Civics
`23.02.10`	Political parties and interest groups	Civics
`23.02.11`	Rights and civil liberties	Civics
`23.02.12`	Federalism and local government	Civics
`23.02.13`	International organizations	Civics
`23.02.14`	Treaties and international law	Civics
`23.02.15`	Citizenship and civic participation	Civics
`23.03.01`	Maps and Map Projections	Geography
`23.03.02`	Latitude, Longitude, and Coordinate Systems	Geography
`23.03.03`	Continents and Oceans	Geography
`23.03.04`	Landforms	Geography
`23.03.05`	Climate Zones and Biomes	Geography
`23.03.06`	Population Distribution and Density	Geography
`23.03.07`	Urbanization and Settlement	Geography
`23.03.08`	Natural Resources and Distribution	Geography
`23.03.09`	Cultural Geography	Geography
`23.03.10`	Political Geography	Geography
`23.03.11`	Human Migration	Geography
`23.03.12`	Environmental Geography	Geography
`24.01.00`	Numerical-PDE chapter README and notation crosswalk	Numerical analysis & PDE
`24.01.01`	Propositional logic and truth tables	Logic
`24.01.01`	Sobolev spaces $H^{s}$ and $W^{k, p}$	Numerical analysis & PDE
`24.01.02`	Sobolev spaces of differential forms $H Λ^{k}$	Numerical analysis & PDE
`24.01.03`	Weak / variational formulation of elliptic PDE	Numerical analysis & PDE
`24.01.04`	Babuška-Brezzi (inf-sup) condition for saddle-point problems	Numerical analysis & PDE
`24.02.01`	Predicate logic and quantifiers	Logic
`24.02.01`	Classical conforming FEM — Galerkin, Céa, Bramble-Hilbert	Numerical analysis & PDE
`24.02.02`	Mixed FEM for the Poisson equation (Raviart-Thomas)	Numerical analysis & PDE
`24.03.01`	Informal fallacies and argument analysis	Logic
`24.03.01`	Whitney forms $W^{k}$	Numerical analysis & PDE
`24.03.02`	Nédélec first-kind edge elements and $P_{r}^{-} Λ^{1}$	Numerical analysis & PDE
`24.03.03`	Polynomial differential form spaces $P_{r} Λ^{k}$ and $P_{r}^{-} Λ^{k}$	Numerical analysis & PDE
`24.03.04`	Discrete de Rham complex and the FEEC subcomplex axiom	Numerical analysis & PDE
`24.03.05`	Bounded cochain projection and the commuting diagram	Numerical analysis & PDE
`24.03.06`	FEEC convergence theorem (Arnold-Falk-Winther)	Numerical analysis & PDE
`24.03.07`	Abstract Hilbert complexes, the abstract Hodge decomposition, and abstract Galerkin stability	Numerical analysis & PDE
`24.04.01`	Deductive reasoning and syllogisms	Logic
`24.04.01`	Mixed FEM for the Hodge Laplacian	Numerical analysis & PDE
`24.04.02`	Maxwell equations and FEEC edge elements	Numerical analysis & PDE
`24.04.03`	Linearised elasticity via AFW symmetric-tensor mixed elements	Numerical analysis & PDE
`24.04.04`	Smooth FEEC pointer (Falk-Neilan)	Numerical analysis & PDE
`24.04.05`	Isogeometric exterior calculus pointer	Numerical analysis & PDE
`24.04.06`	Virtual element exterior calculus pointer	Numerical analysis & PDE
`24.04.07`	Eigenvalue approximation and discrete compactness in FEEC	Numerical analysis & PDE
`24.04.E1`	Finite element exterior calculus exercise pack (Arnold-Falk-Winther supplement)	Numerical analysis & PDE
`24.05.01`	Inductive reasoning, analogy, and causation	Logic
`24.06.01`	Decision theory and Bayesian reasoning	Logic
`24.07.01`	Cognitive biases and rationality	Logic
`24.08.01`	Critical thinking in media, science, and everyday life	Logic
`25.01.01`	Computational thinking and algorithms	Computer science
`25.01.01`	Propositional logic and truth tables	Logic
`25.02.01`	Data structures: arrays, trees, and graphs	Computer science
`25.02.01`	Predicate logic and quantifiers	Logic
`25.03.01`	Algorithmic complexity and Big-O notation	Computer science
`25.03.01`	Informal fallacies and argument analysis	Logic
`25.04.01`	Programming paradigms: functional, OOP, and beyond	Computer science
`25.04.01`	Deductive reasoning and syllogisms	Logic
`25.05.01`	Operating systems: processes and memory	Computer science
`25.06.01`	Computer networks and internet architecture	Computer science
`25.07.01`	Databases: relational, NoSQL, and data modeling	Computer science
`25.08.01`	Cybersecurity, encryption, and privacy	Computer science
`25.09.01`	Artificial intelligence and machine learning	Computer science
`25.10.01`	Software engineering and design patterns	Computer science
`25.11.01`	Distributed systems and consensus	Computer science
`25.12.01`	Computing ethics and societal impact	Computer science
`26.01.01`	Descriptive statistics: central tendency and variability	Statistics
`26.02.01`	Probability theory: rules and distributions	Statistics
`26.03.01`	Random variables and expected value	Statistics
`26.04.01`	Sampling distributions and the Central Limit Theorem	Statistics
`26.05.01`	Hypothesis testing, p-values, and confidence intervals	Statistics
`26.06.01`	Correlation and regression analysis	Statistics
`26.07.01`	Bayesian statistics: prior and posterior	Statistics
`26.08.01`	Nonparametric methods and resampling	Statistics
`26.09.01`	Experimental design and ANOVA	Statistics
`26.10.01`	Statistical literacy, misuse, and data ethics	Statistics
`27.01.01`	Plate tectonics and continental drift	Earth science
`27.02.01`	Minerals, rocks, and the rock cycle	Earth science
`27.03.01`	Earthquakes, volcanoes, and geologic hazards	Earth science
`27.04.01`	Atmosphere, weather, and climate basics	Earth science
`27.05.01`	Oceanography: currents, tides, and marine ecosystems	Earth science
`27.06.01`	Hydrology: the water cycle and groundwater	Earth science
`27.07.01`	Climate change: evidence, impacts, and mitigation	Earth science
`27.08.01`	Earth history and the geologic time scale	Earth science
`28.01.01`	The solar system: planets, moons, and small bodies	Astronomy
`28.02.01`	Stars and stellar evolution	Astronomy
`28.03.01`	Galaxies and the Milky Way	Astronomy
`28.04.01`	Cosmology: the Big Bang, expansion, and fate of the universe	Astronomy
`28.05.01`	Exoplanets: detection methods and habitability	Astronomy
`28.06.01`	Space exploration: history and future	Astronomy
`29.01.01`	Introduction to psychology and research methods	Psychology
`29.02.01`	Neuroscience: brain and behaviour	Psychology
`29.03.01`	Sensation and perception: how the brain constructs reality from sensory data	Psychology
`29.04.01`	Learning and memory: conditioning, cognitive maps, encoding, storage, retrieval, and forgetting	Psychology
`29.05.01`	Cognition and intelligence: thinking, reasoning, and the measurement of mind	Psychology
`29.06.01`	Developmental Psychology Across the Lifespan	Psychology
`29.07.01`	Social psychology: social influence, group dynamics, prejudice, and relationships	Psychology
`29.08.01`	Personality theories and assessment	Psychology
`29.09.01`	Psychological disorders: diagnosis, controversy, and the limits of classification	Psychology
`29.10.01`	Therapy and treatment approaches	Psychology
`29.11.01`	Motivation and emotion: drives, needs, feelings, and the forces that shape behaviour	Psychology
`29.12.01`	Cross-Cultural and Indigenous Psychology	Psychology
`30.01.01`	The sociological imagination and research methods	Sociology
`30.02.01`	Culture and society: a global perspective	Sociology
`30.03.01`	Socialization and Identity Formation	Sociology
`30.04.01`	Social stratification: class, race, gender, and caste	Sociology
`30.05.01`	Social institutions: family, education, religion, and media	Sociology
`30.06.01`	Deviance and social control: who defines normal, who punishes difference, and why it matters	Sociology
`30.07.01`	Globalization and social movements: how people organize across borders to challenge power	Sociology
`30.08.01`	Urbanization and demography: the global movement of people and the cities that shape their lives	Sociology
`31.01.01`	Anthropology: the four fields and holism	Anthropology
`31.02.01`	Cultural anthropology: ethnography and fieldwork	Anthropology
`31.03.01`	Archaeology: material culture and excavation	Anthropology
`31.04.01`	Biological anthropology: evolution and hominins	Anthropology
`31.05.01`	Linguistic anthropology: language, culture, and society	Anthropology
`31.06.01`	Applied anthropology: globalization, ethics, and decolonization	Anthropology
`32.01.01`	Prehistory and human migration out of Africa	World history
`32.02.01`	Mesopotamia and the Fertile Crescent	World history
`32.03.01`	Ancient Egypt and Nubia	World history
`32.04.01`	Indus Valley Civilization and Vedic India	World history
`32.05.01`	Ancient China: Shang through Han dynasties	World history
`32.06.01`	Classical Greece and the Hellenistic world	World history
`32.07.01`	Roman Republic and Empire: from founding myths to the fall and beyond	World history
`32.08.01`	Classical India: Mauryan Empire, Gupta Golden Age, and South Indian Kingdoms	World history
`32.09.01`	Pre-Columbian Americas: Olmec, Maya, Aztec, Inca, and North American civilizations	World history
`32.10.01`	Islamic Golden Age and the Caliphates	World history
`32.11.01`	Medieval Europe and the Crusades	World history
`32.12.01`	Sub-Saharan African kingdoms	World history
`32.13.01`	The Mongol Empire and its legacy	World history
`32.14.01`	Age of Exploration: Multiple Perspectives	World history
`32.15.01`	Colonialism and imperialism: colonizer and colonized	World history
`32.16.01`	The Atlantic Slave Trade: African, European, and American Perspectives	World history
`32.17.01`	Enlightenment and Revolutions: American, French, Haitian, and Latin American	World history
`32.18.01`	Industrial Revolution and its global consequences	World history
`32.19.01`	Meiji Japan, Qing collapse, and the Scramble for Africa	World history
`32.20.01`	World War I: Global Perspectives	World history
`32.21.01`	Interwar Period and the Rise of Totalitarianism: Fascism, Communism, and Liberal Democracy in Crisis	World history
`32.22.01`	World War II: Global Theaters and Multi-Perspective Histories	World history
`32.23.01`	Decolonization: India, Algeria, Vietnam, Congo, and the end of empire	World history
`32.24.01`	The Cold War: US, Soviet, Chinese, and Non-Aligned Perspectives	World history
`32.25.01`	Globalization, Neoliberalism, and the Post-Colonial World	World history
`32.26.01`	The 21st Century: Digital Revolution, Climate Crisis, and Shifting Power	World history
`33.01.01`	Ancient science: Mesopotamia, Greece, China, and India	History of science
`33.02.01`	Islamic Golden Age and medieval European science	History of science
`33.03.01`	The Scientific Revolution: Copernicus to Newton	History of science
`33.04.01`	Industrial Revolution, chemistry, and electromagnetism	History of science
`33.05.01`	The relativity and quantum revolutions	History of science
`33.06.01`	Genetics, DNA, and the molecular biology revolution	History of science
`33.07.01`	The digital revolution: computing and the internet	History of science
`33.08.01`	Contemporary science: challenges, open science, and the future	History of science
`34.01.01`	Music fundamentals: rhythm, melody, and harmony	Music & art
`34.02.01`	Music history: Western and world traditions	Music & art
`34.03.01`	Visual art: elements, principles, and composition	Music & art
`34.04.01`	Art history: cave paintings to contemporary	Music & art
`34.05.01`	Film and photography as visual storytelling	Music & art
`34.06.01`	Architecture and design of the built environment	Music & art
`34.07.01`	Aesthetics theory: taste, judgment, and culture	Music & art
`34.08.01`	Digital media, art, and technology	Music & art
`35.01.01`	The human body: organ systems and homeostasis	Health & medicine
`35.02.01`	Infectious disease, immunity, and vaccines	Health & medicine
`35.03.01`	Chronic disease: cardiovascular disease, diabetes, and cancer	Health & medicine
`35.04.01`	Nutrition: macronutrients, micronutrients, and diet	Health & medicine
`35.05.01`	Mental health: disorders, stigma, and treatment	Health & medicine
`35.06.01`	Public health, epidemiology, and health systems	Health & medicine
`35.07.01`	Pharmacology: how drugs work and ethics	Health & medicine
`35.08.01`	Future of medicine: genomics, AI, and global health	Health & medicine
`36.01.01`	Media foundations: history and theory	Media literacy
`36.02.01`	News and journalism: verification and source evaluation	Media literacy
`36.03.01`	Propaganda and persuasion: rhetorical analysis	Media literacy
`36.04.01`	Digital literacy: algorithms, filter bubbles, and echo chambers	Media literacy
`36.05.01`	Visual literacy: images, data visualization, and manipulation	Media literacy
`36.06.01`	Media ethics: responsible consumption and production	Media literacy
`37.01.01`	Probability Spaces and the Kolmogorov Extension Theorem	Probability & stochastics
`37.02.01`	The Borel-Cantelli Lemmas and the Kolmogorov 0-1 Law	Probability & stochastics
`37.02.02`	The Strong Law of Large Numbers	Probability & stochastics
`37.02.03`	The Ergodic Theorems: Birkhoff, von Neumann, and Kingman	Probability & stochastics
`37.03.01`	Characteristic Functions, Inversion, and the Lévy Continuity Theorem	Probability & stochastics
`37.03.02`	The Lindeberg–Feller Central Limit Theorem	Probability & stochastics
`37.03.03`	Donsker's Invariance Principle and the Functional Central Limit Theorem	Probability & stochastics
`37.04.01`	Discrete-Time Martingales, Stopping Times, and Optional Stopping	Probability & stochastics
`37.04.02`	Doob's Upcrossing Inequality and the Almost-Sure Martingale Convergence Theorem	Probability & stochastics
`37.04.03`	Doob's Maximal and L^p Inequalities, Uniform Integrability, and L^p-Bounded Martingales	Probability & stochastics
`37.04.04`	Kakutani's Theorem on Product Martingales and Absolute Continuity of Product Measures	Probability & stochastics
`37.05.01`	The Markov Property, Transition Matrices, and the Chapman–Kolmogorov Equations	Probability & stochastics
`37.05.02`	Class Structure, Irreducibility, and Periodicity	Probability & stochastics
`37.05.03`	Hitting Probabilities and Expected Hitting Times	Probability & stochastics
`37.05.04`	The Strong Markov Property and the Recurrence/Transience Dichotomy	Probability & stochastics
`37.05.05`	Invariant Measures and Distributions; Positive and Null Recurrence	Probability & stochastics
`37.05.06`	Convergence to Equilibrium via Coupling	Probability & stochastics
`37.05.07`	The Ergodic Theorem for Markov Chains and Detailed Balance	Probability & stochastics
`37.05.08`	Continuous-Time Markov Chains I: Q-Matrices, Jump Chains, and Holding Times	Probability & stochastics
`37.05.09`	Continuous-Time Markov Chains II: The Kolmogorov Backward and Forward Equations	Probability & stochastics
`37.05.10`	Recurrence, Invariant Distributions, and Convergence for Continuous-Time Chains	Probability & stochastics
`37.05.11`	The Poisson Process: Equivalent Characterizations	Probability & stochastics
`37.05.12`	Birth–Death Processes and Queueing Chains	Probability & stochastics
`37.06.01`	Continuous Local Martingales, Quadratic Variation, and the Doob–Meyer Decomposition	Probability & stochastics
`37.06.02`	The Brownian Martingale Representation Theorem	Probability & stochastics
`37.06.03`	Brownian Local Time and Tanaka's Formula	Probability & stochastics
`37.07.01`	The Large Deviation Principle: Rate Functions, Bounds, and Goodness	Probability & stochastics
`37.07.02`	Cramér's Theorem and the Legendre-Fenchel Rate Function	Probability & stochastics
`37.07.03`	The Legendre-Fenchel Transform and Convex Duality of Rate Functions	Probability & stochastics
`37.07.04`	The Gärtner-Ellis Theorem	Probability & stochastics
`37.07.05`	Sanov's Theorem and the Large Deviation Principle for Empirical Measures	Probability & stochastics
`37.07.06`	Relative Entropy as a Rate Function and the Donsker-Varadhan Variational Formula	Probability & stochastics
`37.07.07`	Varadhan's Integral Lemma and the Laplace Principle	Probability & stochastics
`37.07.08`	The Contraction Principle and the Inverse Contraction Principle	Probability & stochastics
`37.07.09`	Exponential Tightness, Exponential Approximation, and the Dawson–Gärtner Projective Limit	Probability & stochastics
`37.07.10`	Schilder's Theorem: Small-Noise Large Deviations for Brownian Motion	Probability & stochastics
`37.07.11`	Freidlin–Wentzell Theory: Large Deviations for Small-Noise Diffusions	Probability & stochastics
`37.08.01`	The Wigner Semicircle Law and the Moment Method	Probability & stochastics
`37.08.02`	The Stieltjes Transform and the Semicircle Law via the Resolvent	Probability & stochastics
`37.08.03`	Gaussian Ensembles GOE/GUE/GSE and the Joint Eigenvalue Density	Probability & stochastics
`37.08.04`	Determinantal Point Processes and Sine-Kernel Bulk Universality	Probability & stochastics
`37.08.05`	The Airy Kernel and the Tracy-Widom Edge Law	Probability & stochastics
`37.08.06`	The Largest Eigenvalue and the Operator-Norm Bound	Probability & stochastics
`37.08.07`	Spectral Concentration: Log-Sobolev and the Herbst Argument	Probability & stochastics
`37.08.08`	Free Probability: Freeness, Free Convolution, and the R-Transform	Probability & stochastics
`37.08.09`	The Ben Arous–Guionnet Large Deviation Principle for the Empirical Spectral Measure	Probability & stochastics
`38.01.01`	Dynamical Systems, Orbits, and Limit Sets	Dynamical systems & ergodic theory
`38.01.02`	Minimality and Recurrence	Dynamical systems & ergodic theory
`38.01.03`	Topological Transitivity, Topological Mixing, and Devaney Chaos	Dynamical systems & ergodic theory
`38.01.04`	Circle Rotations and Unique Ergodicity	Dynamical systems & ergodic theory
`38.01.05`	The Poincaré Rotation Number and Denjoy's Theorem	Dynamical systems & ergodic theory
`38.02.01`	Shift Spaces and Subshifts	Dynamical systems & ergodic theory
`38.02.02`	Shifts of Finite Type, Transition Matrices, and Coding	Dynamical systems & ergodic theory
`38.02.03`	Perron-Frobenius Theory, SFT Growth Rate, and Subshift Entropy	Dynamical systems & ergodic theory
`38.03.01`	Hyperbolic Sets, Anosov and Axiom-A Systems, and the Smale Spectral Decomposition	Dynamical systems & ergodic theory
`38.03.02`	The Smale Horseshoe and the Smale-Birkhoff Homoclinic Theorem	Dynamical systems & ergodic theory
`38.03.03`	The Hadamard-Perron Stable and Unstable Manifold Theorem	Dynamical systems & ergodic theory
`38.03.04`	Shadowing and Structural Stability	Dynamical systems & ergodic theory
`38.04.01`	Measure-Preserving Systems, Poincaré Recurrence, and the Kac Formula	Dynamical systems & ergodic theory
`38.04.02`	Ergodicity, Unique Ergodicity, and Equidistribution	Dynamical systems & ergodic theory
`38.05.01`	The Mixing Hierarchy: Mixing, Weak Mixing, and Ergodicity	Dynamical systems & ergodic theory
`38.05.02`	Spectral Theory of Dynamical Systems and the Halmos-von Neumann Theorem	Dynamical systems & ergodic theory
`38.06.01`	Topological Entropy	Dynamical systems & ergodic theory
`38.06.02`	Kolmogorov-Sinai Entropy and the Generator Theorem	Dynamical systems & ergodic theory
`38.06.03`	The Shannon-McMillan-Breiman Theorem	Dynamical systems & ergodic theory
`38.06.04`	Topological Pressure, the Variational Principle, and Equilibrium States	Dynamical systems & ergodic theory
`38.07.01`	The Oseledets Multiplicative Ergodic Theorem and Lyapunov Exponents	Dynamical systems & ergodic theory
`38.07.02`	Pesin Theory and the Entropy Formula	Dynamical systems & ergodic theory
`38.07.03`	The Hopf Argument for Ergodicity of Geodesic and Anosov Flows	Dynamical systems & ergodic theory
`38.07.04`	The Livšic Cohomological Rigidity Theorem	Dynamical systems & ergodic theory
`39.01.01`	C*-Algebras: Axioms, Spectrum, and the Continuous Functional Calculus	Operator algebras & NCG
`39.01.02`	Commutative C*-Algebras and Gelfand Duality	Operator algebras & NCG
`39.01.03`	States, the GNS Construction, and the Gelfand-Naimark Representation Theorem	Operator algebras & NCG
`39.01.04`	The Toeplitz Algebra, Cuntz Algebras, and Extensions	Operator algebras & NCG
`39.02.01`	AF Algebras, Bratteli Diagrams, and the Irrational Rotation Algebra	Operator algebras & NCG
`39.02.02`	Operator K-Theory: K_0 and K_1 of C*-Algebras	Operator algebras & NCG
`39.02.03`	The Six-Term Exact Sequence, Bott Periodicity, and AF Classification	Operator algebras & NCG
`39.03.01`	Von Neumann Algebras and the Bicommutant Theorem	Operator algebras & NCG
`39.03.02`	The Predual, Normal States, and the σ-Weak Topology	Operator algebras & NCG
`39.03.03`	The Kaplansky Density Theorem	Operator algebras & NCG
`39.03.04`	Comparison of Projections and the Murray-von Neumann Type Classification	Operator algebras & NCG
`39.03.05`	Traces, Continuous Dimension, and the II_1 Factor	Operator algebras & NCG
`39.04.01`	Cyclic and Separating Vectors and the Standard Form	Operator algebras & NCG
`39.04.02`	Tomita's Theorem: the Modular Operator and Modular Conjugation	Operator algebras & NCG
`39.04.03`	The Modular Automorphism Group and the KMS Condition	Operator algebras & NCG
`39.04.04`	The Connes Classification of Type III Factors	Operator algebras & NCG
`39.05.01`	Completely Positive Maps and the Stinespring Dilation Theorem	Operator algebras & NCG
`39.05.02`	Operator Systems, Arveson's Extension Theorem, and the Choi-Effros Theorem	Operator algebras & NCG
`39.05.03`	Tensor Products of C*-Algebras: the Minimal and Maximal Norms	Operator algebras & NCG
`39.05.04`	Nuclear C*-Algebras and the Completely Positive Approximation Property	Operator algebras & NCG
`39.05.05`	Exact C*-Algebras and Nuclear Embeddability	Operator algebras & NCG
`39.05.06`	Amenable Groups: Invariant Means, the Følner Condition, and Paradoxical Decompositions	Operator algebras & NCG
`39.05.07`	Group C*-Algebras: Amenability and Nuclearity	Operator algebras & NCG
`39.05.08`	Quasidiagonality	Operator algebras & NCG
`39.05.09`	Group Approximation Properties and the Connes Embedding Problem	Operator algebras & NCG
`39.05.10`	Exact Groups, Amenable Actions, and Property A	Operator algebras & NCG
`39.06.01`	Spectral Triples and the Reconstruction Theorem	Operator algebras & NCG
`39.06.02`	The Connes Distance Formula	Operator algebras & NCG
`39.06.03`	Fredholm Modules and the K-Theory/K-Homology Index Pairing	Operator algebras & NCG
`39.06.04`	The Noncommutative Torus and Its Geometry	Operator algebras & NCG
`39.06.05`	The Dixmier Trace and the Noncommutative Integral	Operator algebras & NCG
`39.06.06`	The Connes-Moscovici Local Index Formula	Operator algebras & NCG
`39.07.01`	Cyclic Cohomology and the Pairing with K-Theory	Operator algebras & NCG
`39.07.02`	The Chern Character in K-Homology	Operator algebras & NCG
`40.01.01`	Basic Counting and the Twelvefold Way	Combinatorics & graph theory
`40.01.02`	Inclusion-Exclusion and the Sieve	Combinatorics & graph theory
`40.01.03`	Generating Functions: Ordinary, Exponential, and the Exponential Formula	Combinatorics & graph theory
`40.01.04`	Rational Generating Functions, the Transfer-Matrix Method, and P-Partitions	Combinatorics & graph theory
`40.01.05`	Permutation Statistics: Descents, the Major Index, and Eulerian Polynomials	Combinatorics & graph theory
`40.01.06`	q-Analogues, Gaussian Binomial Coefficients, and the Combinatorics of Partitions	Combinatorics & graph theory
`40.01.07`	Trees, Cayley's Formula, the Matrix-Tree Theorem, and Lagrange Inversion	Combinatorics & graph theory
`40.02.01`	Posets, Lattices, and Birkhoff's Representation Theorem	Combinatorics & graph theory
`40.02.02`	The Incidence Algebra, the Möbius Function of a Poset, and Möbius Inversion	Combinatorics & graph theory
`40.02.03`	Eulerian Posets, Face Lattices, and the Characteristic Polynomial	Combinatorics & graph theory
`40.03.01`	The Ring of Symmetric Functions and Its Bases	Combinatorics & graph theory
`40.03.02`	Schur Functions: the Combinatorial Definition and the Jacobi-Trudi Determinant	Combinatorics & graph theory
`40.03.03`	The Cauchy Identity, Dual Bases, and the Hall Inner Product	Combinatorics & graph theory
`40.03.04`	The Robinson-Schensted-Knuth Correspondence	Combinatorics & graph theory
`40.03.05`	The Littlewood-Richardson Rule, Skew Schur Functions, and Jeu de Taquin	Combinatorics & graph theory
`40.03.06`	The Frobenius Characteristic Map and the Symmetric-Function Dictionary	Combinatorics & graph theory
`40.03.07`	Plane Partitions and the MacMahon Box Formula	Combinatorics & graph theory
`40.03.08`	Quasisymmetric Functions and Gessel's Fundamental Basis	Combinatorics & graph theory
`40.04.01`	Graphs, Basic Invariants, and the Foundational Lemmas	Combinatorics & graph theory
`40.04.02`	Matchings I: König's Theorem and Hall's Marriage Theorem	Combinatorics & graph theory
`40.04.03`	Matchings II: Tutte's 1-Factor Theorem and the Tutte-Berge Formula	Combinatorics & graph theory
`40.04.04`	Connectivity and Menger's Theorem	Combinatorics & graph theory
`40.04.05`	Planar Graphs: Euler's Formula, Kuratowski's and Wagner's Theorems	Combinatorics & graph theory
`40.04.06`	Vertex Colouring: Brooks' Theorem and the Chromatic Polynomial	Combinatorics & graph theory
`40.04.07`	Edge Colouring and List Colouring: Vizing's Theorem and Choosability	Combinatorics & graph theory
`40.04.08`	Map Colouring: the Five-Colour Theorem and the Four-Colour Theorem	Combinatorics & graph theory
`40.04.09`	Network Flows: Max-Flow Min-Cut and Nowhere-Zero Flows	Combinatorics & graph theory
`40.04.10`	Graph Minors and the Robertson-Seymour Theorem	Combinatorics & graph theory
`40.04.11`	Hamilton Cycles: Dirac, Ore, and Chvátal-Erdős	Combinatorics & graph theory
`40.05.01`	Extremal Graph Theory: Turán's Theorem and Erdős-Stone-Simonovits	Combinatorics & graph theory
`40.05.02`	Bipartite Extremal Problems: the Kővári-Sós-Turán Theorem	Combinatorics & graph theory
`40.05.03`	The Szemerédi Regularity Lemma and the Triangle Removal Lemma	Combinatorics & graph theory
`40.05.04`	Ramsey's Theorem and Ramsey Numbers	Combinatorics & graph theory
`40.06.01`	Balanced Incomplete Block Designs and Fisher's Inequality	Combinatorics & graph theory
`40.06.02`	Symmetric Designs and the Bruck-Ryser-Chowla Theorem	Combinatorics & graph theory
`40.06.03`	Finite Projective and Affine Planes, MOLS, and the 36-Officers Problem	Combinatorics & graph theory
`40.06.04`	Steiner Systems and the Existence of Steiner Triple Systems	Combinatorics & graph theory
`40.06.05`	Hadamard Matrices and the Paley Construction	Combinatorics & graph theory
`40.06.06`	Linear Codes and the Hamming, Singleton, and Gilbert-Varshamov Bounds	Combinatorics & graph theory
`40.06.07`	Perfect Codes: the Hamming and Golay Codes	Combinatorics & graph theory
`40.06.08`	Cyclic Codes: BCH, Reed-Solomon, Reed-Muller, and Assmus-Mattson	Combinatorics & graph theory
`40.06.09`	Strongly Regular Graphs and the Design-Graph Dictionary	Combinatorics & graph theory
`40.06.10`	Pólya-Redfield Enumeration and the Cycle Index	Combinatorics & graph theory
`40.07.01`	The Probabilistic Method: First-Moment and Counting Arguments	Combinatorics & graph theory
`40.07.02`	Linearity of Expectation and the Deletion Method	Combinatorics & graph theory
`40.07.03`	The Second-Moment Method and Thresholds for Random Graphs	Combinatorics & graph theory
`40.07.04`	The Lovász Local Lemma and the Moser-Tardos Algorithm	Combinatorics & graph theory
`40.07.05`	Concentration for Combinatorial Functionals: Azuma and the Bounded-Differences Method	Combinatorics & graph theory
`40.07.06`	The Entropy Method and Shearer's Lemma	Combinatorics & graph theory
`40.07.07`	Correlation Inequalities: FKG, Harris, and the Janson Inequalities	Combinatorics & graph theory
`40.07.08`	Combinatorial Discrepancy: Spencer's Theorem and the Beck-Fiala Bound	Combinatorics & graph theory
`40.07.09`	The Rödl Nibble and the Semi-Random Method	Combinatorics & graph theory
`40.08.01`	The Symbolic Method for Unlabelled Structures	Combinatorics & graph theory
`40.08.02`	The Symbolic Method for Labelled Structures	Combinatorics & graph theory
`40.08.03`	Meromorphic Coefficient Asymptotics	Combinatorics & graph theory
`40.08.04`	Singularity Analysis and the Transfer Theorems	Combinatorics & graph theory
`40.08.05`	Asymptotics of Tree Families and Simple Varieties of Trees	Combinatorics & graph theory
`40.08.06`	The Saddle-Point Method for Asymptotic Enumeration	Combinatorics & graph theory
`40.08.07`	Limit Laws and the Quasi-Powers Theorem	Combinatorics & graph theory
`41.01.01`	Categories, Functors, and the Duality Principle	Category theory
`41.01.02`	Natural Transformations, Functor Categories, and Equivalence of Categories	Category theory
`41.02.01`	Limits and Colimits as Universal Cones	Category theory
`41.02.02`	Constructing Limits: Products, Equalizers, Preservation, and Filtered Colimits	Category theory
`41.03.01`	Adjunctions: Hom-Set and Unit-Counit Definitions	Category theory
`41.03.02`	RAPL, Reflective Subcategories, and the Adjoint Functor Theorems	Category theory
`41.04.01`	Representable Functors and Universal Elements	Category theory
`41.04.02`	The Yoneda Lemma, the Yoneda Embedding, and Density	Category theory
`41.05.01`	Monads, Eilenberg-Moore Algebras, and the Kleisli Category	Category theory
`41.05.02`	Beck's Monadicity Theorem and Lawvere Theories	Category theory
`41.06.01`	Ends, Coends, and the Calculus of (Co)ends	Category theory
`41.06.02`	Kan Extensions: All Concepts Are Kan Extensions	Category theory
`41.07.01`	Monoidal Categories and Mac Lane's Coherence Theorem	Category theory
`42.01.01`	Propositional Logic as a Formal System	Mathematical logic
`42.01.02`	The Compactness Theorem for Propositional Logic	Mathematical logic
`42.01.03`	First-Order Languages: Syntax and Unique Readability	Mathematical logic
`42.01.04`	Structures and Tarski's Definition of Truth	Mathematical logic
`42.01.05`	A Deductive Calculus for First-Order Logic and Soundness	Mathematical logic
`42.01.06`	Gödel's Completeness Theorem and the Henkin Construction	Mathematical logic
`42.01.07`	Compactness and the Löwenheim-Skolem Theorems for First-Order Logic	Mathematical logic
`42.01.08`	Representability of Recursive Functions in Arithmetic	Mathematical logic
`42.01.09`	Gödel Numbering, the Fixed-Point Lemma, and the Incompleteness Theorems	Mathematical logic
`42.01.10`	The Entscheidungsproblem, Church's Theorem, and Decidable Theories	Mathematical logic
`42.02.01`	Structures, Embeddings, and Elementary Equivalence	Mathematical logic
`42.02.02`	The Compactness Theorem and the Method of Diagrams	Mathematical logic
`42.02.03`	Types and the Omitting Types Theorem	Mathematical logic
`42.02.04`	Saturation, Homogeneity, and Monster Models	Mathematical logic
`42.02.05`	Quantifier Elimination and Model-Completeness	Mathematical logic
`42.02.06`	Categoricity: Ryll-Nardzewski, Morley, and Baldwin-Lachlan	Mathematical logic
`42.02.07`	Strongly Minimal Sets, Morley Rank, and Stability	Mathematical logic
`42.02.08`	O-Minimality and the Cell Decomposition Theorem	Mathematical logic
`42.02.09`	Indiscernibles and Ehrenfeucht-Mostowski Models	Mathematical logic
`42.03.01`	The ZFC Axioms and the Cumulative Hierarchy	Mathematical logic
`42.03.02`	Ordinals, Transfinite Induction, and Recursion	Mathematical logic
`42.03.03`	Cardinals and the Arithmetic of the Infinite	Mathematical logic
`42.03.04`	Cofinality, Cardinal Exponentiation, and the Singular Cardinals Hypothesis	Mathematical logic
`42.03.05`	The Axiom of Choice and Its Equivalents	Mathematical logic
`42.03.06`	The Constructible Universe L and the Consistency of GCH	Mathematical logic
`42.03.07`	Forcing I: Posets, Generic Filters, Names, and the Fundamental Theorem	Mathematical logic
`42.03.08`	Forcing II: Cohen and the Independence of the Continuum Hypothesis	Mathematical logic
`42.03.09`	Martin's Axiom, Iterated Forcing, and Large Cardinals	Mathematical logic
`42.03.10`	Club Sets, Stationary Sets, and Fodor's Lemma	Mathematical logic
`42.04.01`	Models of Computation and the Church-Turing Thesis	Mathematical logic
`42.04.02`	The Halting Problem, Undecidability, and the Recursion Theorem	Mathematical logic
`42.04.03`	Computably Enumerable Sets: Creative and Simple Sets	Mathematical logic
`42.04.04`	Turing Reducibility, Oracles, and the Jump	Mathematical logic
`42.04.05`	The Arithmetical Hierarchy and Post's Theorem	Mathematical logic
`42.04.06`	The Turing Degrees and the Priority Method	Mathematical logic
`42.04.07`	Unsolvable Problems: the Word Problem and Hilbert's Tenth	Mathematical logic
`42.04.08`	Kolmogorov Complexity and Algorithmic Randomness	Mathematical logic
`42.05.01`	Sequent Calculus, Cut-Elimination, and the Consistency of Arithmetic	Mathematical logic
`43.01.01`	Floating-point arithmetic and the IEEE model	Numerical analysis & scientific computing
`43.01.02`	Conditioning and condition numbers of problems	Numerical analysis & scientific computing
`43.01.03`	Backward stability and backward-error analysis of algorithms	Numerical analysis & scientific computing
`43.02.01`	The Bisection Method and the Scalar Root-Finding Problem	Numerical analysis & scientific computing
`43.02.02`	Fixed-Point Iteration, Contraction Convergence, and Order of Convergence	Numerical analysis & scientific computing
`43.02.03`	Newton's Method and the Secant Method: Superlinear and Quadratic Convergence	Numerical analysis & scientific computing
`43.03.01`	Gaussian elimination, LU factorization, and its stability	Numerical analysis & scientific computing
`43.03.02`	Cholesky factorization and the symmetric positive-definite solve	Numerical analysis & scientific computing
`43.03.03`	Perturbation theory and a posteriori error for linear systems	Numerical analysis & scientific computing
`43.03.08`	Symmetric-indefinite factorisation: the Bunch-Kaufman LDLᵀ algorithm	Numerical analysis & scientific computing
`43.04.01`	Least squares: normal equations vs QR vs SVD conditioning	Numerical analysis & scientific computing
`43.04.08`	Updating and downdating matrix factorisations	Numerical analysis & scientific computing
`43.05.08`	Total least squares and the generalised SVD	Numerical analysis & scientific computing
`43.06.01`	Power iteration, inverse iteration, and Rayleigh quotient iteration	Numerical analysis & scientific computing
`43.06.02`	Reduction to Hessenberg/tridiagonal form	Numerical analysis & scientific computing
`43.06.03`	The QR algorithm for eigenvalues, with shifts	Numerical analysis & scientific computing
`43.06.04`	Bauer-Fike and the conditioning of eigenvalues	Numerical analysis & scientific computing
`43.06.10`	The generalised eigenvalue problem Ax=λBx and the QZ algorithm	Numerical analysis & scientific computing
`43.06.11`	Computing matrix functions: the matrix exponential	Numerical analysis & scientific computing
`43.06.12`	Sylvester and Lyapunov matrix equations: the Bartels-Stewart algorithm	Numerical analysis & scientific computing
`43.07.01`	Stationary iterative methods: Jacobi, Gauss-Seidel, SOR	Numerical analysis & scientific computing
`43.07.02`	Arnoldi and Lanczos iterations	Numerical analysis & scientific computing
`43.07.03`	GMRES	Numerical analysis & scientific computing
`43.07.04`	The conjugate gradient method	Numerical analysis & scientific computing
`43.07.05`	Preconditioning	Numerical analysis & scientific computing
`43.08.01`	Polynomial interpolation: existence, uniqueness, and the Lagrange form	Numerical analysis & scientific computing
`43.08.02`	Interpolation error and the Runge phenomenon	Numerical analysis & scientific computing
`43.08.03`	Hermite interpolation and piecewise / cubic spline interpolation	Numerical analysis & scientific computing
`43.08.04`	Best uniform approximation, minimax, and Chebyshev polynomials	Numerical analysis & scientific computing
`43.08.05`	Least-squares approximation and orthogonal polynomials	Numerical analysis & scientific computing
`43.09.01`	Newton-Cotes rules and their error via the Peano kernel	Numerical analysis & scientific computing
`43.09.02`	Composite rules, Euler-Maclaurin, and Romberg / adaptive quadrature	Numerical analysis & scientific computing
`43.09.03`	Gauss quadrature via orthogonal polynomials	Numerical analysis & scientific computing
`43.10.01`	One-step methods: Euler, trapezoidal, Runge-Kutta; consistency and order	Numerical analysis & scientific computing
`43.10.02`	Linear multistep methods: Adams and BDF families, order via the characteristic polynomials	Numerical analysis & scientific computing
`43.10.03`	Zero-stability, the root condition, and the Dahlquist equivalence theorem	Numerical analysis & scientific computing
`43.10.04`	Absolute stability, stability regions, and the linear test equation	Numerical analysis & scientific computing
`43.10.05`	Stiff equations, A-stability, and the Dahlquist second barrier	Numerical analysis & scientific computing
`43.10.06`	Finite-difference methods for two-point boundary-value problems	Numerical analysis & scientific computing
`43.11.01`	Finite differences for the elliptic BVP: the 5-point Laplacian and its convergence	Numerical analysis & scientific computing
`43.11.02`	The method of lines and stability for parabolic problems	Numerical analysis & scientific computing
`43.11.03`	Von Neumann stability analysis and the CFL condition	Numerical analysis & scientific computing
`43.11.04`	Hyperbolic finite differences: upwind, Lax-Friedrichs, Lax-Wendroff; numerical diffusion and dispersion	Numerical analysis & scientific computing
`43.11.05`	The Lax-Richtmyer equivalence theorem for finite-difference schemes	Numerical analysis & scientific computing