All units

820 units total · 820 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
`00.02.05`	Function	Precalculus foundations
`00.03.01`	Linear equations and the line	Precalculus foundations
`00.03.02`	Quadratic equations and the quadratic formula	Precalculus foundations
`00.04.01`	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
`01.01.03`	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
`01.01.07`	Determinant: axiomatic + expansion + properties	Algebra & linear algebra
`01.01.08`	Eigenvalue, eigenvector, characteristic polynomial	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
`01.01.15`	Bilinear form / quadratic form	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
`01.02.08`	Localisation of a commutative ring	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
`01.02.16`	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
`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
`02.02.01`	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
`02.06.01`	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
`02.08.01`	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
`02.10.05`	Surface integral and parametric surfaces	Analysis
`02.11.01`	Bounded linear operators	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
`02.12.01`	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
`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
`03.03.01`	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
`03.04.01`	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.E1`	Mayer-Vietoris and degree-theory exercise pack (Bott-Tu Ch. I 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
`03.06.03`	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
`03.06.23`	Modularity of the elliptic genus	Modern geometry
`03.07.05`	Yang-Mills action	Modern geometry
`03.07.07`	BPST instanton and the Bogomolny bound	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
`03.07.14`	Penrose transform at linear level	Modern geometry
`03.07.16`	$B$ -field as a gerbe connection	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
`03.08.10`	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
`03.09.02`	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.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
`03.11.01`	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
`03.12.00`	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
`03.12.35`	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
`03.12.45`	Arithmetic square and integral fracture theorems	Modern geometry
`03.12.49`	Bialgebra, Hopf algebra, and the Milnor-Moore theorem	Differential geometry
`03.12.E1`	Rational homotopy and Sullivan minimal-model exercise pack (Bott-Tu Ch. III §19 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.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
`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
`04.04.01`	Riemann-Roch theorem for curves	Algebraic geometry
`04.04.02`	Hurwitz formula	Algebraic geometry
`04.04.03`	Elliptic curves	Algebraic geometry
`04.04.08`	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
`04.04.15`	Fulton-Lazarsfeld connectedness theorem	Algebraic geometry
`04.04.16`	Lazarsfeld's K3-vector-bundle proof of Petri	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
`04.06.01`	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
`04.08.01`	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
`04.09.05`	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	alg-geom
`04.10.08`	Kirwan stratification of the unstable locus	Algebraic geometry
`04.10.09`	Variation of GIT (VGIT)	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
`04.11.16`	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)	alg-geom
`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
`04.12.15`	Log Gromov-Witten Invariants (pointer)	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
`05.00.06`	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
`05.01.01`	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
`05.03.01`	Coadjoint orbit	Symplectic geometry
`05.03.02`	Souriau Gibbs state on a symplectic G-space	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
`05.06.01`	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
`05.08.01`	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
`05.10.01`	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
`05.11.02`	Prequantisation of the spin coadjoint orbit	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
`06.02.01`	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
`06.04.01`	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
`06.07.01`	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
`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
`07.02.01`	Maschke's 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
`07.03.01`	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
`07.04.06`	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
`07.04.12`	Spherical function on G/K	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
`07.05.07`	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
`07.06.01`	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
`07.06.14`	Engel's theorem + Lie's theorem	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`	Casimir element	Representation theory
`07.06.22`	Weyl complete-reducibility theorem	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
`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
`08.05.01`	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
`08.09.01`	Quantum-classical correspondence (Wick rotation)	Statistical field theory
`08.10.01`	Bosonic Fock space and second quantisation	Statistical field theory
`08.10.03`	φ⁴ theory and the Dyson series	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
`09.02.01`	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
`09.04.01`	Legendre transform — from Lagrangian to Hamiltonian	Classical mechanics
`09.04.02`	Hamilton's equations	Classical mechanics
`09.05.01`	Canonical transformations	Classical mechanics
`09.05.02`	Hamilton-Jacobi equation	Classical mechanics
`09.06.01`	Action-angle variables	Classical mechanics
`09.08.01`	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
`10.02.01`	Biot-Savart law and Ampere's law	Electromagnetism & special relativity
`10.03.01`	Faraday's law and electromagnetic induction	Electromagnetism & special relativity
`10.04.01`	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
`10.06.01`	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
`11.05.01`	Bose-Einstein distribution	Statistical mechanics
`11.05.02`	Fermi-Dirac distribution and electron gas	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
`12.03.01`	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
`12.06.01`	Hydrogen atom bound states	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
`12.10.01`	Path integral formulation of quantum mechanics	Quantum mechanics & QFT
`12.11.01`	Dirac equation and relativistic spin	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
`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
`13.03.01`	Riemann curvature tensor	General relativity & cosmology
`13.04.01`	Einstein field equations	General relativity & cosmology
`13.05.01`	Schwarzschild solution	General relativity & cosmology
`13.05.02`	Orbits in Schwarzschild geometry	General relativity & cosmology
`13.07.01`	Linearized GR and gravitational waves	General relativity & cosmology
`13.08.01`	FLRW cosmology and Friedmann equations	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
`14.08.01`	Chemical kinetics: rate laws and the Arrhenius equation	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
`15.10.01`	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
`18.02.01`	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
`18.07.01`	Endocrine system — hormones and regulation	Organismal biology
`18.08.01`	Renal physiology — homeostasis and the nephron	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
`19.09.01`	Population ecology — Lotka-Volterra	Ecology & evolution
`19.10.01`	Community ecology — interactions and food webs	Ecology & evolution
`19.15.01`	Origin of life — mechanistic scenarios	Ecology & evolution
`20.03.01`	The measurement problem in quantum mechanics	Philosophy
`20.05.02`	The unit of selection	Philosophy
`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
`21.04.01`	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
`21.05.01`	$ℓ$ -adic Galois Representations	Number theory
`21.06.01`	Modularity Theorem (Statement) and BSD 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
`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
`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