Lean formalization
Each Babel Bible unit declares a Lean status. full means the proof is checked end-to-end. partial means statements compile but Mathlib gaps gate the proofs. none means Mathlib coverage is too thin to state the theorem formally; the unit ships with a named human reviewer.
8
full
193
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Mathlib gaps
Precalculus foundations
25 units · full: 1 · partial: 6 · none: 18 · stub: 0
| id | title | status | module |
|---|---|---|---|
00.01.01 | Real numbers, integers, rationals | partial | Codex.Precalc.Numbers.Reals |
00.01.02 | Absolute value and the triangle inequality | partial | Codex.Precalc.Numbers.AbsoluteValue |
00.01.03 | Polynomials and rational expressions | partial | Codex.Precalc.Numbers.Polynomials |
00.02.05 | Function | full | Codex.Precalc.SetAndFunction.Function |
00.03.01 | Linear equations and the line | partial | Codex.Precalc.Equations.LinesPlane |
00.03.02 | Quadratic equations and the quadratic formula | partial | Codex.Precalc.Equations.QuadraticFormula |
00.04.01 | Inequalities (linear and quadratic) | partial | Codex.Precalc.Inequalities.Basic |
00.05.01 | Real exponents and exponential function | none | — |
00.05.02 | Logarithms as inverses of exponentials | none | — |
00.05.03 | Complex numbers (introductory) | none | — |
00.06.01 | Right-triangle trigonometry | none | — |
00.06.02 | Inverse trigonometric functions | none | — |
00.06.03 | Law of sines and law of cosines | none | — |
00.07.01 | Unit-circle trigonometry | none | — |
00.07.02 | Trigonometric identities: sum, difference, and double-angle | none | — |
00.08.01 | Trigonometric identities (addition formulas) | none | — |
00.08.02 | Law of sines and law of cosines | none | — |
00.09.01 | Cartesian coordinates and distance in the plane | none | — |
00.10.01 | Conic sections (parabola, ellipse, hyperbola) | none | — |
00.11.01 | Polar coordinates and parametric curves | none | — |
00.11.02 | Conic-section parametrisations and intersections | none | — |
00.12.01 | Mathematical induction | none | — |
00.12.02 | Binomial theorem and Pascal's triangle | none | — |
00.13.01 | Plane geometry (distance, area, pi) | none | — |
00.13.02 | Solid geometry (volume) | none | — |
Algebra & linear algebra
23 units · full: 2 · partial: 7 · none: 14 · stub: 0
| id | title | status | module |
|---|---|---|---|
01.01.01 | Field | full | Codex.Foundations.LinearAlgebra.Field |
01.01.03 | Vector space | full | Codex.Foundations.LinearAlgebra.VectorSpace |
01.01.04 | Subspace, basis, dimension | partial | Codex.Foundations.LinearAlgebra.Basis |
01.01.05 | Linear transformation: kernel, image, rank-nullity | partial | Codex.Foundations.LinearAlgebra.RankNullity |
01.01.07 | Determinant: axiomatic + expansion + properties | partial | Codex.Foundations.LinearAlgebra.Determinant |
01.01.08 | Eigenvalue, eigenvector, characteristic polynomial | partial | Codex.Foundations.LinearAlgebra.Eigen |
01.01.11 | Jordan canonical form and minimal polynomial | partial | Codex.Foundations.LinearAlgebra.JordanForm |
01.01.12 | Singular value decomposition (finite-dim) | partial | Codex.Foundations.LinearAlgebra.SVD |
01.01.15 | Bilinear form / quadratic form | partial | Codex.Foundations.LinearAlgebra.BilinearQuadraticForm |
01.02.01 | Group | none | — |
01.02.02 | Subgroup, coset, quotient group, isomorphism theorems | none | — |
01.02.03 | Group action, orbit-stabiliser, class equation | none | — |
01.02.04 | Sylow theorems | none | — |
01.02.05 | Solvable group, nilpotent group, Jordan-Holder theorem | none | — |
01.02.08 | Localisation of a commutative ring | none | — |
01.02.10 | Tensor product of modules (commutative case) | none | — |
01.02.11 | Exact sequence, short five lemma, snake lemma | none | — |
01.02.12 | Algebraic field extension, degree, splitting field | none | — |
01.02.16 | Nakayama's lemma | none | — |
01.02.17 | Hilbert basis theorem; Noetherian rings and modules | none | — |
01.02.19 | Tensor algebra, exterior algebra, symmetric algebra | none | — |
01.02.20 | Free group, free product, group presentation | none | — |
01.02.22 | Krull dimension; Krull's principal ideal theorem | none | — |
Analysis
43 units · full: 2 · partial: 10 · none: 31 · stub: 0
| id | title | status | module |
|---|---|---|---|
02.01.01 | Topological space | full | Codex.Analysis.Topology.TopologicalSpace |
02.01.02 | Continuous map | none | — |
02.01.05 | Metric space | full | Codex.Analysis.Topology.MetricSpace |
02.01.06 | Quotient and identification topology | none | — |
02.01.07 | Fibration (Hurewicz and Serre) | none | — |
02.01.08 | Cofibration and homotopy extension property | none | — |
02.01.09 | Compact-open topology and function spaces | none | — |
02.02.01 | Real-number axioms (ordered field) | partial | Codex.Analysis.RealNumbers.Axioms |
02.03.02 | Cauchy sequences and Bolzano-Weierstrass | partial | Codex.Analysis.Sequences.CauchyBolzanoWeierstrass |
02.03.03 | Infinite series: convergence and the standard tests | none | — |
02.04.01 | Step-function integral and the Darboux integral | none | — |
02.04.03 | Integrability of continuous functions on [a,b] | none | — |
02.04.04 | Fundamental theorems of calculus (FTC1 and FTC2) | none | — |
02.04.06 | Improper integrals and the comparison test | none | — |
02.05.01 | Multi-variable limit and continuity | partial | Codex.Analysis.MultiVariable.LimitContinuity |
02.05.02 | Mean value theorem (Rolle, Lagrange, Cauchy) | none | — |
02.05.03 | Chain rule for multi-variable functions | partial | Codex.Analysis.MultiVariable.ChainRule |
02.05.04 | Implicit and inverse function theorems | partial | Codex.Analysis.MultiVariable.ImplicitInverse |
02.05.05 | Taylor's theorem and extrema in several variables | partial | Codex.Analysis.MultiVariable.TaylorExtrema |
02.06.01 | Logarithm as an integral | none | — |
02.06.02 | n-th-order linear ODE with constant coefficients | none | — |
02.06.03 | Systems of linear ODEs and the matrix exponential | none | — |
02.06.04 | Hyperbolic functions | none | — |
02.08.01 | First-order linear and separable ODEs | none | — |
02.08.02 | Second-order linear ODEs with constant coefficients | none | — |
02.09.01 | Complex numbers and Euler's formula | none | — |
02.10.05 | Surface integral and parametric surfaces | none | — |
02.11.01 | Bounded linear operators | partial | Codex.Analysis.FunctionalAnalysis.BoundedOperators |
02.11.03 | Unbounded self-adjoint operators | none | — |
02.11.04 | Banach space fundamentals | partial | Codex.Analysis.FunctionalAnalysis.BanachSpaces |
02.11.05 | Compact operators | partial | Codex.Analysis.FunctionalAnalysis.CompactOperators |
02.11.06 | Normed vector space | none | — |
02.11.07 | Inner product space | none | — |
02.11.08 | Hilbert space | partial | Codex.Analysis.FunctionalAnalysis.HilbertSpace |
02.12.01 | Phase space, vector field, integral curve | none | — |
02.12.02 | Phase flow / one-parameter group | none | — |
02.12.05 | Rectification (straightening) of a vector field | none | — |
02.12.08 | Lyapunov stability (direct method) | none | — |
02.12.10 | Poincaré-Bendixson theorem | none | — |
02.12.12 | First integrals / conserved quantities | none | — |
02.12.13 | Inhomogeneous linear ODE / variation of constants | none | — |
02.12.14 | Limit cycle and Liénard / Van der Pol systems | none | — |
02.12.17 | Bifurcation theory pointer | none | — |
Number theory
10 units · full: 0 · partial: 10 · none: 0 · stub: 0
| id | title | status | module |
|---|---|---|---|
21.03.01 | Riemann Zeta Function | partial | Codex.NumberTheory.LFunctions.RiemannZeta |
21.03.02 | Dirichlet -functions | partial | Codex.NumberTheory.LFunctions.DirichletL |
21.03.03 | Dedekind Zeta Function, Hecke -Functions, Artin -Functions | partial | Codex.NumberTheory.LFunctions.DedekindHeckeArtin |
21.04.01 | Modular Forms on | partial | Codex.NumberTheory.ModularForms.SL2Z |
21.04.02 | Hecke Operators and Hecke Algebra | partial | Codex.NumberTheory.ModularForms.HeckeOperators |
21.04.03 | Eichler-Shimura Correspondence | partial | Codex.NumberTheory.ModularForms.EichlerShimura |
21.05.01 | -adic Galois Representations | partial | Codex.NumberTheory.GaloisReps.EllAdic |
21.06.01 | Modularity Theorem (Statement) and BSD Conjecture | partial | Codex.NumberTheory.Modularity.ModularityBSD |
21.07.01 | -extensions and Iwasawa Theory | partial | Codex.NumberTheory.Iwasawa.ZpExtensions |
21.07.02 | -adic -functions and the Iwasawa Main Conjecture | partial | Codex.NumberTheory.Iwasawa.PadicL |
Differential geometry
39 units · full: 0 · partial: 0 · none: 39 · stub: 0
| id | title | status | module |
|---|---|---|---|
03.02.01 | Topological manifold | none | — |
03.02.02 | Smooth structure and atlases | none | — |
03.02.03 | Smooth maps between manifolds | none | — |
03.02.04 | Frobenius theorem | none | — |
03.02.05 | Sectional curvature, Ricci tensor, scalar curvature | none | — |
03.02.06 | Constant-curvature spaces and Killing-Hopf | none | — |
03.02.07 | Killing fields and infinitesimal isometries | none | — |
03.02.08 | Myers-Steenrod theorem | none | — |
03.02.09 | Almost-complex structure (manifold-level) | none | — |
03.02.10 | Complex manifold and the Dolbeault complex | none | — |
03.02.11 | Hermitian manifold and the Kahler form | none | — |
03.02.12 | Kahler identities and the Hodge decomposition (Kahler version) | none | — |
03.02.13 | Isometric immersion and the second fundamental form | none | — |
03.02.14 | Gauss, Codazzi, and Ricci equations | none | — |
03.02.15 | Bochner technique and curvature vanishing theorems | none | — |
03.03.04 | Formal group law | none | — |
03.03.05 | p-adic Lie group and the p-adic exponential | none | — |
03.03.06 | Lie's third theorem (statement, simply-connected case) | none | — |
03.05.00 | General fibre bundle | none | — |
03.05.06 | Vertical subbundle and fundamental vector fields | none | — |
03.05.11 | Horizontal lift and parallel transport | none | — |
03.05.12 | Reduction of structure group; reduction of a connection | none | — |
03.05.13 | Associated bundle and induced connection | none | — |
03.05.14 | Torsion tensor and the two Cartan structural equations | none | — |
03.05.15 | Linear connection via the frame bundle; soldering form | none | — |
03.05.16 | Holonomy group and restricted holonomy | none | — |
03.05.17 | Ambrose-Singer holonomy theorem | none | — |
03.05.18 | Holonomy reduction theorem | none | — |
03.05.19 | Holomorphic vector bundle | none | — |
03.05.20 | Hermitian metric on a complex bundle; Chern connection | none | — |
03.12.24 | Simplicial set and the simplicial category Delta | none | — |
03.12.26 | Functorial CW approximation Gamma X = |S_*X| | none | — |
03.12.27 | Puppe cofiber sequence | none | — |
03.12.28 | Puppe fiber sequence | none | — |
03.12.29 | Thom space and Thom isomorphism | none | — |
03.12.30 | Minimal complex and minimal fibration | none | — |
03.12.36 | Bisimplicial set, diagonal, and the realisation lemma | none | — |
03.12.39 | Simplicial group and the W-bar classifying functor | none | — |
03.12.49 | Bialgebra, Hopf algebra, and the Milnor-Moore theorem | none | — |
Modern geometry
141 units · full: 3 · partial: 53 · none: 85 · stub: 0
| id | title | status | module |
|---|---|---|---|
03.01.01 | Tensor product | full | Codex.ModernGeometry.TensorAlgebra.TensorProduct |
03.01.02 | Associative algebra | full | Codex.ModernGeometry.TensorAlgebra.AssociativeAlgebra |
03.01.03 | Ideal in an algebra | full | Codex.ModernGeometry.TensorAlgebra.Ideal |
03.01.04 | Tensor algebra | none | — |
03.01.05 | Quotient algebra | none | — |
03.02.01 | Smooth manifold | partial | Codex.ModernGeometry.Manifolds.SmoothManifold |
03.03.01 | Lie group | partial | Codex.ModernGeometry.Lie.LieGroup |
03.03.02 | Group action | none | — |
03.03.03 | Orthogonal group | none | — |
03.04.01 | Lie algebra | partial | Codex.ModernGeometry.DifferentialForms.LieAlgebra |
03.04.02 | Differential forms | partial | Codex.ModernGeometry.DifferentialForms.DifferentialForms |
03.04.03 | Integration on manifolds | none | — |
03.04.04 | Exterior derivative | partial | Codex.ModernGeometry.DifferentialForms.ExteriorDerivative |
03.04.05 | Stokes' theorem | partial | Codex.ModernGeometry.DifferentialForms.StokesTheorem |
03.04.06 | De Rham cohomology | partial | Codex.ModernGeometry.DifferentialForms.DeRhamCohomology |
03.04.07 | Mayer-Vietoris sequence for de Rham cohomology | partial | Codex.ModernGeometry.DifferentialForms.MayerVietoris |
03.04.08 | Variational calculus on manifolds | none | — |
03.04.09 | Compactly-supported cohomology, integration along the fiber, and the de Rham Thom isomorphism | partial | Codex.ModernGeometry.DifferentialForms.ThomGlobalAngularForm |
03.04.10 | Good covers, finite-dimensionality of de Rham cohomology, and the Mayer-Vietoris induction | partial | Codex.ModernGeometry.DifferentialForms.GoodCover |
03.04.11 | Čech-de Rham double complex and the tic-tac-toe principle | partial | Codex.ModernGeometry.DifferentialForms.CechDeRham |
03.04.12 | Künneth formula for de Rham cohomology — two proofs | partial | Codex.DifferentialForms.Kunneth |
03.04.13 | Singular cohomology and the de Rham theorem (with coefficients) | partial | Codex.DifferentialForms.SingularCohomology |
03.04.14 | Hypercohomology of a complex of sheaves | none | — |
03.04.E1 | Mayer-Vietoris and degree-theory exercise pack (Bott-Tu Ch. I supplement) | none | — |
03.05.01 | Principal bundle | none | — |
03.05.02 | Vector bundle | none | — |
03.05.03 | Orthogonal frame bundle | none | — |
03.05.04 | Connection on a vector bundle | none | — |
03.05.05 | Double cover | none | — |
03.05.07 | Principal bundle with connection | none | — |
03.05.08 | Complex vector bundle | none | — |
03.05.09 | Curvature of a connection | none | — |
03.05.10 | Sphere bundle, the global angular form, and the Hopf index theorem | partial | Codex.Bundles.SphereBundle |
03.06.03 | Stiefel-Whitney classes | none | — |
03.06.04 | Pontryagin and Chern classes | none | — |
03.06.05 | Invariant polynomial on a Lie algebra | none | — |
03.06.06 | Chern-Weil homomorphism | none | — |
03.06.07 | Chern-Simons forms and transgression | partial | Codex.Modern.CharClasses.ChernSimonsTransgression |
03.06.08 | Kostant-Weil isomorphism and prequantum line bundle | none | — |
03.06.09 | Dixmier-Douady class and | none | — |
03.06.10 | Stiefel-Whitney and Pontryagin numbers | partial | Codex.Modern.CharClasses.SWPontryaginNumbers |
03.06.11 | Hirzebruch signature theorem | partial | Codex.Modern.CharClasses.HirzebruchSignature |
03.06.12 | Unoriented bordism and Thom's theorem | partial | Codex.Modern.CharClasses.UnorientedBordismThom |
03.06.13 | Oriented bordism and the Pontryagin-Thom construction | partial | Codex.Modern.CharClasses.OrientedBordismPT |
03.06.14 | Steenrod squares and the Wu formula | partial | Codex.Modern.CharClasses.SteenrodSquaresWu |
03.06.15 | Multiplicative sequences and the -, -, Todd genera | partial | Codex.Modern.CharClasses.MultiplicativeSequences |
03.06.16 | Whitney duality and immersion obstructions | partial | Codex.Modern.CharClasses.WhitneyDualityImmersion |
03.06.17 | Combinatorial Pontryagin classes and exotic 7-spheres | partial | Codex.Modern.CharClasses.ExoticSpheres |
03.06.18 | Chern character as a ring homomorphism | partial | Codex.Modern.CharClasses.ChernCharacter |
03.06.19 | Signature of a 4k-manifold and the intersection form | partial | Codex.Modern.CharClasses.SignatureIntersectionForm |
03.06.20 | Borel-Hirzebruch and the cohomology of | partial | Codex.Modern.CharClasses.BorelHirzebruchGT |
03.06.23 | Modularity of the elliptic genus | none | — |
03.07.05 | Yang-Mills action | none | — |
03.07.07 | BPST instanton and the Bogomolny bound | none | — |
03.07.09 | Moduli space of ASD connections | none | — |
03.07.10 | ADHM construction (Atiyah-Drinfeld-Hitchin-Manin) | none | — |
03.07.11 | Penrose twistor space and the Ward correspondence | none | — |
03.07.14 | Penrose transform at linear level | none | — |
03.07.16 | -field as a gerbe connection | none | — |
03.08.01 | Topological K-theory | none | — |
03.08.02 | Adams operations | none | — |
03.08.03 | Thom isomorphism in K-theory | none | — |
03.08.04 | Classifying space | none | — |
03.08.05 | Universal bundle, , and the Borel presentation of flag-manifold cohomology | partial | Codex.KTheory.UniversalBundle |
03.08.06 | Stable homotopy | none | — |
03.08.07 | Bott periodicity | none | — |
03.08.10 | Equivariant K-theory and | none | — |
03.08.11 | The group and the -homomorphism | none | — |
03.08.12 | KR-theory (K-theory with reality) | partial | Codex.Modern.KTheory.KRTheory |
03.09.02 | Clifford algebra | partial | Codex.SpinGeometry.CliffordAlgebra |
03.09.03 | Spin group | none | — |
03.09.04 | Spin structure on an oriented Riemannian manifold | none | — |
03.09.05 | Spinor bundle | none | — |
03.09.06 | Fredholm operators | none | — |
03.09.07 | Symbol of a differential operator | none | — |
03.09.08 | Dirac operator | none | — |
03.09.09 | Elliptic operators on a manifold | none | — |
03.09.10 | Atiyah-Singer index theorem | none | — |
03.09.11 | Clifford algebra classification — the 8×8 chessboard | partial | Codex.SpinGeometry.CliffordChessboard |
03.09.12 | KR-theory and the (1,1)-periodicity theorem | partial | Codex.SpinGeometry.KRTheory |
03.09.13 | Triality on Spin(8) and exceptional Lie groups via spinors | partial | Codex.SpinGeometry.Triality |
03.09.14 | Generalised Dirac bundles and the Bochner-Weitzenböck identity | partial | Codex.SpinGeometry.DiracBundle |
03.09.15 | Cl_k-linear Dirac operators and the KO-valued index | partial | Codex.SpinGeometry.ClkDirac |
03.09.16 | Positive scalar curvature obstruction theory | partial | Codex.SpinGeometry.PSCObstruction |
03.09.17 | Witten positive-mass theorem via spinors | partial | Codex.SpinGeometry.WittenPositiveMass |
03.09.18 | Berger holonomy classification and parallel spinors | partial | Codex.SpinGeometry.BergerHolonomy |
03.09.19 | Calibrated geometries — Special Lagrangian, associative, coassociative, Cayley | partial | Codex.SpinGeometry.CalibratedGeometries |
03.09.20 | Heat-kernel proof of the Atiyah-Singer index theorem | none | — |
03.09.21 | Family, equivariant, and Lefschetz fixed-point index theorems | none | — |
03.09.22 | Sobolev spaces, pseudodifferential operators, and elliptic parametrices | partial | Codex.SpinGeometry.Pseudodifferential |
03.09.23 | Bismut superconnection | none | — |
03.09.24 | Eta invariant and Atiyah-Patodi-Singer index theorem | none | — |
03.09.25 | Kirillov character formula via the equivariant index | none | — |
03.09.26 | Mathai-Quillen formalism and universal Thom forms | none | — |
03.09.E1 | Clifford and spin algebra exercise pack (Lawson-Michelsohn Ch. I supplement) | none | — |
03.09.E2 | Chapter IV applications exercise pack (Lawson-Michelsohn Ch. IV supplement) | none | — |
03.10.02 | CFT basics | none | — |
03.11.01 | Central extension of a Lie algebra | none | — |
03.11.02 | Infinite-dimensional Lie algebra representations | none | — |
03.11.03 | Virasoro algebra | none | — |
03.12.00 | Fundamental group | none | — |
03.12.01 | Homotopy and homotopy group | partial | Codex.ModernGeometry.Homotopy.Homotopy |
03.12.02 | Covering space | partial | Codex.ModernGeometry.Homotopy.CoveringSpace |
03.12.03 | Suspension | partial | Codex.ModernGeometry.Homotopy.Suspension |
03.12.04 | Spectrum | none | — |
03.12.05 | Eilenberg-MacLane space | none | — |
03.12.06 | Sullivan minimal models and rational homotopy theory | partial | Codex.Homotopy.SullivanMinimalModels |
03.12.07 | Whitehead tower, rational Hurewicz theorem, and Serre's finiteness | partial | Codex.Homotopy.WhiteheadTower |
03.12.08 | Fundamental groupoid | none | — |
03.12.09 | Seifert-van Kampen theorem | none | — |
03.12.10 | CW complex | none | — |
03.12.11 | Singular homology | none | — |
03.12.12 | Simplicial and -complex homology | none | — |
03.12.13 | Cellular homology and cellular approximation | none | — |
03.12.14 | Excision theorem | none | — |
03.12.15 | Eilenberg-Steenrod axioms | none | — |
03.12.16 | Poincaré duality | none | — |
03.12.17 | Cap product | none | — |
03.12.18 | Universal coefficient theorem | none | — |
03.12.19 | Hurewicz theorem | none | — |
03.12.20 | Whitehead's theorem | none | — |
03.12.21 | Blakers-Massey theorem | none | — |
03.12.22 | -complex / semi-simplicial set | none | — |
03.12.23 | Euler characteristic | none | — |
03.12.25 | Simplicial sets and geometric realization | none | — |
03.12.31 | Quillen model category | none | — |
03.12.32 | Quillen functor and Quillen equivalence | none | — |
03.12.33 | Kan-Quillen model structure on sSet | partial | Codex.Modern.Homotopy.KanQuillenModelStructure |
03.12.35 | Simplicial model category and the function complex | partial | Codex.Modern.Homotopy.SimplicialModelCategory |
03.12.37 | Homotopy colimit via the Bousfield-Kan construction | partial | Codex.Modern.Homotopy.HomotopyColimitBK |
03.12.38 | Bousfield-Kan spectral sequence | partial | Codex.Modern.Homotopy.BousfieldKanSpectralSequence |
03.12.40 | Postnikov tower of a Kan complex | partial | Codex.Modern.Homotopy.PostnikovTowerKan |
03.12.45 | Arithmetic square and integral fracture theorems | partial | Codex.Modern.Homotopy.ArithmeticSquare |
03.12.E1 | Rational homotopy and Sullivan minimal-model exercise pack (Bott-Tu Ch. III §19 supplement) | none | — |
03.13.01 | Spectral sequences — exact couples, filtered complexes, double complexes | partial | Codex.SpectralSequences.SpectralSequence |
03.13.02 | Leray-Serre spectral sequence and the Gysin sequence | partial | Codex.SpectralSequences.LeraySerre |
03.13.03 | Leray-Hirsch theorem and the splitting principle for vector bundles | partial | Codex.SpectralSequences.LerayHirschSplitting |
03.13.04 | Atiyah-Hirzebruch spectral sequence | none | — |
03.13.E1 | Spectral-sequence computation exercise pack (Bott-Tu Ch. III supplement) | none | — |
03.14.01 | Quantum free particle as a representation of E(3) | none | — |
03.14.02 | Complex structures and quantization: squeezed states | none | — |
Algebraic geometry
89 units · full: 0 · partial: 58 · none: 31 · stub: 0
| id | title | status | module |
|---|---|---|---|
04.01.01 | Sheaf | partial | Codex.AlgebraicGeometry.Sheaves.Sheaf |
04.01.02 | Stalk of a sheaf | partial | Codex.AlgebraicGeometry.Sheaves.Stalk |
04.01.03 | Sheafification | partial | Codex.AlgebraicGeometry.Sheaves.Sheafification |
04.01.04 | Direct and inverse image of sheaves | partial | Codex.AlgebraicGeometry.Sheaves.DirectInverseImage |
04.02.01 | Scheme | partial | Codex.AlgebraicGeometry.Schemes.Scheme |
04.02.02 | Affine scheme | partial | Codex.AlgebraicGeometry.Schemes.AffineScheme |
04.02.03 | Projective scheme | partial | Codex.AlgebraicGeometry.Schemes.ProjectiveScheme |
04.02.04 | Morphism of schemes | partial | Codex.AlgebraicGeometry.Schemes.Morphism |
04.02.05 | Smooth, étale, and unramified morphisms | none | — |
04.02.07 | Nullstellensatz and dimension theory | none | — |
04.03.01 | Sheaf cohomology | partial | Codex.AlgebraicGeometry.Cohomology.SheafCohomology |
04.03.02 | Local systems, monodromy, and twisted cohomology | none | — |
04.03.03 | Čech cohomology of sheaves on schemes | none | — |
04.03.04 | Cohomology of line bundles on projective space | none | — |
04.03.05 | Serre's vanishing and finiteness theorems | none | — |
04.03.06 | Derived functors and Ext | none | — |
04.03.07 | Higher direct images and base change | none | — |
04.04.01 | Riemann-Roch theorem for curves | partial | Codex.AlgebraicGeometry.RiemannRoch.Curves |
04.04.02 | Hurwitz formula | none | — |
04.04.03 | Elliptic curves | none | — |
04.04.08 | Petri map mu_0 and Gieseker-Petri theorem | none | — |
04.04.09 | Clifford's theorem with equality | none | — |
04.04.10 | Martens' theorem and Mumford's strengthening | none | — |
04.04.11 | Gonality of a curve | none | — |
04.04.13 | Determinantal varieties and the Porteous formula | none | — |
04.04.15 | Fulton-Lazarsfeld connectedness theorem | none | — |
04.04.16 | Lazarsfeld's K3-vector-bundle proof of Petri | none | — |
04.05.01 | Weil divisor | partial | Codex.AlgebraicGeometry.Divisors.WeilDivisor |
04.05.02 | Picard group | partial | Codex.AlgebraicGeometry.Divisors.PicardGroup |
04.05.03 | Line bundle on a scheme | partial | Codex.AlgebraicGeometry.Divisors.LineBundle |
04.05.04 | Cartier divisor | partial | Codex.AlgebraicGeometry.Divisors.CartierDivisor |
04.05.05 | Ample and very ample line bundle | partial | Codex.AlgebraicGeometry.Divisors.Ample |
04.05.06 | Intersection pairing on a surface | none | — |
04.05.07 | Adjunction formula on a surface | none | — |
04.05.08 | Riemann-Roch theorem for surfaces | none | — |
04.05.09 | Hodge index theorem | none | — |
04.05.10 | Hirzebruch-Riemann-Roch theorem (general dimension) | none | — |
04.06.01 | Quasi-coherent sheaf | partial | Codex.AlgebraicGeometry.Coherent.QuasiCoherent |
04.06.02 | Coherent sheaf | partial | Codex.AlgebraicGeometry.Coherent.Coherent |
04.07.01 | Projective space | partial | Codex.AlgebraicGeometry.Projective.ProjectiveSpace |
04.07.02 | Blowup | partial | Codex.AlgebraicGeometry.Projective.Blowup |
04.08.01 | Sheaf of differentials | partial | Codex.AlgebraicGeometry.Differentials.Differentials |
04.08.02 | Canonical sheaf | partial | Codex.AlgebraicGeometry.Differentials.CanonicalSheaf |
04.08.03 | Serre duality | partial | Codex.AlgebraicGeometry.Differentials.SerreDuality |
04.09.01 | Hodge decomposition | partial | Codex.AlgebraicGeometry.Hodge.HodgeDecomposition |
04.09.02 | Kodaira vanishing theorem | partial | Codex.AlgebraicGeometry.Hodge.KodairaVanishing |
04.09.05 | The ddbar-lemma | partial | Codex.AlgGeom.Hodge.DdbarLemma |
04.09.07 | Hard Lefschetz theorem | partial | Codex.AlgGeom.Hodge.HardLefschetz |
04.09.08 | Hodge-Riemann bilinear relations | partial | Codex.AlgGeom.Hodge.HodgeRiemannBilinear |
04.09.09 | Lefschetz (1,1)-theorem | partial | Codex.AlgGeom.Hodge.LefschetzOneOne |
04.09.10 | Akizuki-Nakano vanishing theorem | partial | Codex.AlgGeom.Hodge.AkizukiNakanoVanishing |
04.09.11 | Kodaira embedding theorem | partial | Codex.AlgGeom.Hodge.KodairaEmbedding |
04.10.01 | Moduli of curves | partial | Codex.AlgebraicGeometry.Moduli.ModuliOfCurves |
04.10.02 | Geometric invariant theory | partial | Codex.AlgebraicGeometry.Moduli.GIT |
04.10.03 | Hilbert-Mumford numerical criterion | partial | Codex.AlgGeom.Moduli.HilbertMumford |
04.10.04 | Kempf-Ness theorem and the GIT-symplectic dictionary | partial | Codex.AlgGeom.Moduli.KempfNess |
04.10.05 | Hilbert scheme Hilb^P(X) | partial | Codex.AlgGeom.Moduli.HilbertScheme |
04.10.06 | Moduli of vector bundles on a curve and slope stability | partial | Codex.AlgGeom.Moduli.VectorBundlesCurveSlopeStability |
04.10.08 | Kirwan stratification of the unstable locus | partial | Codex.AlgGeom.Moduli.KirwanStratification |
04.10.09 | Variation of GIT (VGIT) | partial | Codex.AlgGeom.Moduli.VariationOfGIT |
04.11.01 | Algebraic torus and character/cocharacter lattices | none | — |
04.11.02 | Rational polyhedral cone and dual cone | none | — |
04.11.03 | Affine toric variety | none | — |
04.11.04 | Fan and the toric variety | none | — |
04.11.05 | Smoothness and completeness via fans | partial | Codex.AlgGeom.Toric.SmoothnessCompletenessFans |
04.11.06 | Orbit-cone correspondence | partial | Codex.AlgGeom.Toric.OrbitConeCorrespondence |
04.11.07 | Toric resolution of singularities | partial | Codex.AlgGeom.Toric.ResolutionSingularities |
04.11.08 | Toric divisor and support function | partial | Codex.AlgGeom.Toric.DivisorSupportFunction |
04.11.09 | Toric Picard group | partial | Codex.AlgGeom.Toric.PicardGroup |
04.11.10 | Polytope-fan dictionary; the line bundle | partial | Codex.AlgGeom.Toric.PolytopeFanDictionary |
04.11.11 | Algebraic moment map and the polytope | none | — |
04.11.12 | Cohomology of a smooth complete toric variety | none | — |
04.11.13 | Toric intersection theory and mixed volumes | none | — |
04.11.14 | Bernstein-Kushnirenko theorem | none | — |
04.11.16 | Reflexive polytope and Batyrev mirror duality (pointer) | none | — |
04.12.01 | Tropical semiring and tropical polynomial | partial | Codex.AlgGeom.Tropical.TropicalSemiringPolynomial |
04.12.02 | Tropical curve as balanced rational metric graph | partial | Codex.AlgGeom.Tropical.TropicalCurve |
04.12.03 | Kapranov's theorem (fundamental theorem of tropical geometry) | partial | Codex.AlgGeom.Tropical.KapranovTheorem |
04.12.04 | Newton polytope and non-archimedean amoeba | partial | Codex.AlgGeom.Tropical.NewtonPolytopeAmoeba |
04.12.05 | Mikhalkin's correspondence theorem | partial | Codex.AlgGeom.Tropical.MikhalkinCorrespondence |
04.12.06 | Nishinou-Siebert correspondence theorem | partial | Codex.AlgGeom.Tropical.NishinouSiebertCorrespondence |
04.12.07 | Toric degeneration of a Calabi-Yau variety | partial | Codex.AlgGeom.Tropical.ToricDegeneration |
04.12.08 | Dual Intersection Complex; Tropical Manifold B | partial | Codex.AlgGeom.Tropical.DualIntersectionComplex |
04.12.09 | Gross-Siebert Reconstruction Theorem (Statement) | partial | Codex.AlgGeom.Tropical.GrossSiebertReconstruction |
04.12.10 | Strominger-Yau-Zaslow (SYZ) Conjecture | partial | Codex.AlgGeom.Tropical.StromingerYauZaslow |
04.12.11 | Slab function and structure of a tropical manifold | partial | Codex.AlgGeom.Tropical.SlabFunction |
04.12.12 | Theta function of a polarised tropical manifold | partial | Codex.AlgGeom.Tropical.ThetaFunction |
04.12.13 | Period integral and the mirror map (pointer) | partial | Codex.AlgGeom.Tropical.PeriodIntegral |
04.12.15 | Log Gromov-Witten Invariants (pointer) | partial | Codex.AlgGeom.Tropical.LogGromovWitten |
Symplectic geometry
61 units · full: 0 · partial: 0 · none: 61 · stub: 0
| id | title | status | module |
|---|---|---|---|
05.00.01 | Lagrangian mechanics on the tangent bundle | none | — |
05.00.02 | Hamilton's principle of least action | none | — |
05.00.03 | Legendre transform | none | — |
05.00.04 | Noether's theorem | none | — |
05.00.06 | Galilean group and Newtonian mechanics | none | — |
05.00.07 | Galilei group and Bargmann central extension | none | — |
05.00.08 | Mechanical similarity / virial theorem | none | — |
05.00.09 | Worked Lagrangian examples | none | — |
05.00.10 | Scattering and Rutherford formula | none | — |
05.00.11 | Small oscillations and normal modes | none | — |
05.00.14 | Motion in a non-inertial frame / Coriolis force | none | — |
05.01.01 | Symplectic vector space | none | — |
05.01.02 | Symplectic manifold | none | — |
05.01.03 | Symplectic group | none | — |
05.01.04 | Darboux's theorem | none | — |
05.01.05 | Moser's trick | none | — |
05.02.01 | Hamiltonian vector field | none | — |
05.02.02 | Poisson bracket and Poisson manifold | none | — |
05.02.03 | Integrable system | none | — |
05.02.04 | Action-angle coordinates | none | — |
05.02.05 | Cotangent bundle as canonical symplectic manifold | none | — |
05.02.06 | Geodesic flow as a Hamiltonian flow | none | — |
05.02.07 | Liouville's volume theorem | none | — |
05.02.08 | Poincaré recurrence theorem | none | — |
05.02.09 | Poincaré-Cartan integral invariants | none | — |
05.02.10 | The Routhian | none | — |
05.02.11 | Maupertuis' principle and abbreviated action | none | — |
05.03.01 | Coadjoint orbit | none | — |
05.03.02 | Souriau Gibbs state on a symplectic G-space | none | — |
05.04.01 | Moment map | none | — |
05.04.02 | Marsden-Weinstein symplectic reduction | none | — |
05.04.03 | Atiyah-Guillemin-Sternberg convexity theorem | none | — |
05.04.04 | Delzant theorem (symplectic toric classification) | none | — |
05.04.05 | Duistermaat-Heckman theorem | none | — |
05.04.06 | Symplectic blow-up and symplectic cut | none | — |
05.04.07 | Souriau cocycle and non-equivariant moment maps | none | — |
05.05.01 | Lagrangian submanifold | none | — |
05.05.02 | Weinstein Lagrangian neighbourhood theorem | none | — |
05.05.03 | Generating functions for symplectomorphisms | none | — |
05.05.04 | Hamilton-Jacobi equation | none | — |
05.06.01 | Almost-complex structure on a symplectic manifold | none | — |
05.06.02 | Pseudoholomorphic curve | none | — |
05.06.03 | Newlander-Nirenberg integrability theorem | none | — |
05.07.01 | Gromov non-squeezing theorem | none | — |
05.07.02 | Symplectic capacity | none | — |
05.08.01 | Arnold conjecture and Floer homology setup | none | — |
05.08.02 | Floer homology | none | — |
05.08.03 | Maslov index | none | — |
05.08.04 | Conley-Zehnder index | none | — |
05.09.01 | Kolmogorov-Arnold-Moser theorem | none | — |
05.09.02 | Adiabatic invariants | none | — |
05.09.03 | Birkhoff normal form | none | — |
05.09.04 | Williamson normal form for quadratic Hamiltonians | none | — |
05.09.05 | Euler-Arnold equations | none | — |
05.09.06 | Nekhoroshev estimates | none | — |
05.09.07 | Exponential accuracy of the adiabatic invariant | none | — |
05.10.01 | Contact manifold | none | — |
05.10.02 | Symplectisation of a contact manifold | none | — |
05.10.03 | Gray's stability theorem | none | — |
05.10.04 | Contact topology and Reeb dynamics | none | — |
05.11.02 | Prequantisation of the spin coadjoint orbit | none | — |
Statistical field theory
24 units · full: 0 · partial: 1 · none: 23 · stub: 0
| id | title | status | module |
|---|---|---|---|
08.01.01 | Partition function (statistical mechanics) | none | — |
08.01.02 | Ising model | none | — |
08.01.03 | Boltzmann distribution and canonical ensemble | none | — |
08.01.04 | Free energy | none | — |
08.02.01 | Mean-field theory and Curie-Weiss model | none | — |
08.02.02 | Spontaneous symmetry breaking | none | — |
08.02.03 | Mermin-Wagner theorem | none | — |
08.03.01 | Onsager solution of the 2D Ising model (transfer matrix) | none | — |
08.03.02 | Transfer matrix | none | — |
08.04.01 | Renormalisation group (real-space block decimation) | none | — |
08.04.02 | Wilson-Fisher fixed point and universality | none | — |
08.04.03 | Beta function (renormalisation group) | none | — |
08.04.04 | Block-spin decimation | none | — |
08.05.01 | Critical exponents and scaling laws | none | — |
08.05.02 | Correlation functions (statistical mechanics) | none | — |
08.06.01 | Gaussian field theory and free boson | none | — |
08.06.02 | Conformal symmetry at criticality | none | — |
08.07.01 | Path integral formulation of statistical mechanics | none | — |
08.08.01 | Wilson's lattice gauge theory | none | — |
08.08.02 | Wilson action | none | — |
08.08.03 | Effective field theory | none | — |
08.09.01 | Quantum-classical correspondence (Wick rotation) | none | — |
08.10.01 | Bosonic Fock space and second quantisation | none | — |
08.10.03 | φ⁴ theory and the Dyson series | partial | Codex.StatMech.QFT.PhiFourDyson |
Riemann surfaces
65 units · full: 0 · partial: 22 · none: 43 · stub: 0
| id | title | status | module |
|---|---|---|---|
06.01.01 | Holomorphic function | partial | Codex.RiemannSurfaces.ComplexAnalysis.Holomorphic |
06.01.02 | Cauchy integral formula | partial | Codex.RiemannSurfaces.ComplexAnalysis.CauchyIntegralFormula |
06.01.03 | Residue theorem | partial | Codex.RiemannSurfaces.ComplexAnalysis.ResidueTheorem |
06.01.04 | Analytic continuation | partial | Codex.RiemannSurfaces.ComplexAnalysis.AnalyticContinuation |
06.01.05 | Meromorphic function | partial | Codex.RiemannSurfaces.ComplexAnalysis.MeromorphicFunction |
06.01.06 | Riemann mapping theorem | partial | Codex.RiemannSurfaces.ComplexAnalysis.RiemannMappingTheorem |
06.01.07 | Riemann sphere | none | — |
06.01.08 | Möbius (linear-fractional) transformations | none | — |
06.01.10 | Cauchy-Riemann equations and harmonic conjugate | partial | Codex.RiemannSurfaces.ComplexAnalysis.CauchyRiemann |
06.01.11 | Harmonic functions on the plane | none | — |
06.01.12 | Maximum modulus + Schwarz lemma | none | — |
06.01.13 | Argument principle and Rouché's theorem | none | — |
06.01.14 | Normal families and Montel's theorem | none | — |
06.01.15 | Gamma function Gamma(z) | none | — |
06.01.16 | Riemann zeta function zeta(s) | none | — |
06.01.17 | Weierstrass factorization theorem | none | — |
06.01.18 | Mittag-Leffler theorem on C | none | — |
06.01.19 | Schwarz-Christoffel formula | none | — |
06.01.20 | Picard's little theorem | none | — |
06.01.21 | Picard's great theorem | none | — |
06.01.22 | Phragmen-Lindelof principle | none | — |
06.01.23 | Schwarz reflection principle | none | — |
06.01.24 | Dirichlet problem on the disc + Perron's method | none | — |
06.01.25 | Weierstrass p-function | none | — |
06.01.26 | Modular function and j-invariant | none | — |
06.01.27 | Power series and Laurent series | none | — |
06.01.28 | Index / winding number of a closed curve | none | — |
06.01.29 | Schottky's and Bloch's theorems | none | — |
06.01.30 | Riemann-Hurwitz for plane meromorphic / sphere maps | none | — |
06.02.01 | Branch point and ramification | partial | Codex.RiemannSurfaces.BranchPoints.BranchPointRamification |
06.02.02 | Branched coverings of Riemann surfaces | none | — |
06.02.03 | Riemann's existence theorem for algebraic curves | none | — |
06.03.01 | Riemann surface | partial | Codex.RiemannSurfaces.Surfaces.RiemannSurface |
06.03.02 | Genus of a Riemann surface | partial | Codex.RiemannSurfaces.Surfaces.GenusRiemannSurface |
06.03.03 | Uniformization theorem | partial | Codex.RiemannSurfaces.Surfaces.UniformizationTheorem |
06.04.01 | Riemann-Roch theorem for compact Riemann surfaces | partial | Codex.RiemannSurfaces.RiemannRoch.CompactSurface |
06.04.02 | Čech cohomology of holomorphic line bundles | none | — |
06.04.03 | Hodge decomposition on a compact Riemann surface | none | — |
06.04.04 | Serre duality on a curve | none | — |
06.04.05 | Hilbert-space PDE for | none | — |
06.04.07 | Survey of sheaf cohomology on Riemann surfaces | none | — |
06.05.01 | Divisor on a Riemann surface | partial | Codex.RiemannSurfaces.DivisorsBundles.DivisorRiemannSurface |
06.05.02 | Holomorphic line bundle on a Riemann surface | partial | Codex.RiemannSurfaces.DivisorsBundles.HolomorphicLineBundle |
06.05.03 | Riemann-Hurwitz formula | partial | Codex.RiemannSurfaces.DivisorsBundles.RiemannHurwitzFormula |
06.06.01 | Holomorphic 1-form / abelian differential | partial | Codex.RiemannSurfaces.Jacobians.HolomorphicOneForm |
06.06.02 | Period matrix | partial | Codex.RiemannSurfaces.Jacobians.PeriodMatrix |
06.06.03 | Jacobian variety | partial | Codex.RiemannSurfaces.Jacobians.JacobianVariety |
06.06.04 | Abel-Jacobi map | partial | Codex.RiemannSurfaces.Jacobians.AbelJacobiMap |
06.06.05 | Theta function | partial | Codex.RiemannSurfaces.Jacobians.ThetaFunction |
06.06.06 | Jacobi inversion theorem | none | — |
06.06.07 | Riemann's bilinear relations | none | — |
06.06.08 | Schottky problem | none | — |
06.07.01 | Holomorphic functions of several variables | partial | Codex.RiemannSurfaces.SeveralVariables.HolomorphicSeveralVariables |
06.07.02 | Hartogs phenomenon | partial | Codex.RiemannSurfaces.SeveralVariables.HartogsPhenomenon |
06.08.01 | Gauss-Manin connection | none | — |
06.08.02 | Variation of Hodge structure on the Jacobian | none | — |
06.08.03 | Moduli of Riemann surfaces | none | — |
06.09.01 | Stein Riemann surfaces | none | — |
06.09.02 | Cartan's Theorems A and B for Stein Riemann surfaces | none | — |
06.09.03 | Behnke-Stein theorem | none | — |
06.09.04 | Cousin I (additive) | none | — |
06.09.05 | Cousin II (multiplicative) | none | — |
06.09.06 | Mittag-Leffler on RS | none | — |
06.09.07 | Runge approximation on RS | none | — |
06.09.08 | Survey: Cartan-Serre Stein theory in higher dim | none | — |
Representation theory
64 units · full: 0 · partial: 26 · none: 38 · stub: 0
| id | title | status | module |
|---|---|---|---|
07.01.01 | Group representation | partial | Codex.RepresentationTheory.Foundations.GroupRepresentation |
07.01.02 | Schur's lemma | partial | Codex.RepresentationTheory.Foundations.SchurLemma |
07.01.03 | Character of a representation | partial | Codex.RepresentationTheory.Foundations.Character |
07.01.04 | Character orthogonality | partial | Codex.RepresentationTheory.Foundations.CharacterOrthogonality |
07.01.05 | Regular representation | partial | Codex.RepresentationTheory.Foundations.RegularRepresentation |
07.01.06 | Tensor product of representations | partial | Codex.RepresentationTheory.Foundations.TensorProduct |
07.01.07 | Induced representation | partial | Codex.RepresentationTheory.Foundations.InducedRepresentation |
07.01.08 | Frobenius reciprocity | partial | Codex.RepresentationTheory.Foundations.FrobeniusReciprocity |
07.01.09 | Non-abelian Fourier transform on a finite group | none | — |
07.01.10 | Artin's induction theorem | none | — |
07.01.11 | Brauer's induction theorem | none | — |
07.01.12 | Frobenius-Schur indicator | none | — |
07.02.01 | Maschke's theorem | partial | Codex.RepresentationTheory.Character.Maschke |
07.02.03 | Grothendieck groups and the cde-triangle | none | — |
07.02.04 | Brauer character | none | — |
07.02.06 | Block theory of kG | none | — |
07.03.01 | Highest weight representation | partial | Codex.RepresentationTheory.HighestWeight.HighestWeight |
07.04.01 | Cartan-Weyl classification | partial | Codex.RepresentationTheory.Classification.CartanWeyl |
07.04.02 | Compact real form of a complex semisimple Lie algebra | none | — |
07.04.03 | Cartan involution | none | — |
07.04.06 | Orthogonal symmetric Lie algebra | none | — |
07.04.07 | Riemannian symmetric space | none | — |
07.04.08 | Restricted root system | none | — |
07.04.09 | Iwasawa decomposition G=KAN | none | — |
07.04.10 | Bruhat decomposition | none | — |
07.04.12 | Spherical function on G/K | none | — |
07.04.14 | Hermitian symmetric space | none | — |
07.05.01 | Symmetric group representation | partial | Codex.RepresentationTheory.Symmetric.SymmetricGroupRepresentation |
07.05.02 | Young diagram and tableau | partial | Codex.RepresentationTheory.Symmetric.YoungDiagram |
07.05.03 | Specht module | partial | Codex.RepresentationTheory.Symmetric.SpechtModule |
07.05.04 | Schur-Weyl duality | none | — |
07.05.05 | Random walk on a finite group; Upper Bound Lemma | none | — |
07.05.07 | Riffle shuffle and the 7-shuffle theorem | none | — |
07.05.08 | Cutoff phenomenon | none | — |
07.05.09 | Strong stationary time; coupling argument | none | — |
07.05.10 | Murnaghan-Nakayama rule | none | — |
07.05.11 | Spectral analysis of permutation-valued data | none | — |
07.05.12 | Metrics on S_n | none | — |
07.05.13 | Models for partially ranked data on S_n/S_{n-k} | none | — |
07.05.14 | De Finetti / exchangeability and the symmetric group | none | — |
07.06.01 | Lie algebra representation | partial | Codex.RepresentationTheory.LieAlgebraic.LieAlgebraRepresentation |
07.06.02 | Universal enveloping algebra | partial | Codex.RepresentationTheory.LieAlgebraic.UniversalEnvelopingAlgebra |
07.06.03 | Root system | partial | Codex.RepresentationTheory.LieAlgebraic.RootSystem |
07.06.04 | Weyl group | partial | Codex.RepresentationTheory.LieAlgebraic.WeylGroup |
07.06.05 | Dynkin diagram | partial | Codex.RepresentationTheory.LieAlgebraic.DynkinDiagram |
07.06.06 | Verma module | partial | Codex.RepresentationTheory.LieAlgebraic.VermaModule |
07.06.07 | Weyl character formula | partial | Codex.RepresentationTheory.LieAlgebraic.WeylCharacterFormula |
07.06.08 | Weyl dimension formula | partial | Codex.RepresentationTheory.LieAlgebraic.WeylDimensionFormula |
07.06.09 | Borel-Weil theorem | partial | Codex.RepresentationTheory.LieAlgebraic.BorelWeilTheorem |
07.06.10 | Casimir element | none | — |
07.06.11 | Representations of | none | — |
07.06.12 | Representations of | none | — |
07.06.14 | Engel's theorem + Lie's theorem | none | — |
07.06.16 | Cartan's criterion for solvability and semisimplicity | none | — |
07.06.17 | Cartan subalgebra | none | — |
07.06.18 | Root-space decomposition | none | — |
07.06.19 | Cartan matrix | none | — |
07.06.20 | Serre relations and Serre's theorem | none | — |
07.06.21 | Casimir element | none | — |
07.06.22 | Weyl complete-reducibility theorem | none | — |
07.07.01 | Compact Lie group representation | partial | Codex.RepresentationTheory.CompactLie.CompactLieGroupRepresentation |
07.07.02 | Peter-Weyl theorem | partial | Codex.RepresentationTheory.CompactLie.PeterWeylTheorem |
07.07.03 | Haar measure | partial | Codex.RepresentationTheory.CompactLie.HaarMeasure |
07.07.04 | Weyl integration formula | none | — |
Classical mechanics
13 units · full: 0 · partial: 0 · none: 13 · stub: 0
| id | title | status | module |
|---|---|---|---|
09.01.01 | Kinematics — position, velocity, acceleration | none | — |
09.01.02 | Newton's laws of motion | none | — |
09.01.03 | Conservation laws — energy, momentum, angular momentum | none | — |
09.02.01 | The action principle and variational calculus | none | — |
09.02.02 | Euler-Lagrange equations | none | — |
09.03.01 | Noether's theorem — symmetries and conservation laws | none | — |
09.04.01 | Legendre transform — from Lagrangian to Hamiltonian | none | — |
09.04.02 | Hamilton's equations | none | — |
09.05.01 | Canonical transformations | none | — |
09.05.02 | Hamilton-Jacobi equation | none | — |
09.06.01 | Action-angle variables | none | — |
09.07.01 | Continuum Mechanics and Field Theory | none | — |
09.08.01 | KAM theorem and chaos | none | — |
Electromagnetism & special relativity
10 units · full: 0 · partial: 0 · none: 10 · stub: 0
| id | title | status | module |
|---|---|---|---|
10.01.01 | Coulomb's law and Gauss's law | none | — |
10.01.02 | Laplace equation and boundary value problems | none | — |
10.02.01 | Biot-Savart law and Ampere's law | none | — |
10.03.01 | Faraday's law and electromagnetic induction | none | — |
10.04.01 | Maxwell's equations in differential form | none | — |
10.04.02 | EM waves and the wave equation | none | — |
10.05.01 | Special relativity — postulates and Lorentz transformations | none | — |
10.05.02 | Relativistic kinematics and dynamics | none | — |
10.06.01 | Covariant electrodynamics — Faraday tensor | none | — |
10.07.01 | Radiation from accelerating charges — Larmor formula | none | — |
Statistical mechanics
9 units · full: 0 · partial: 0 · none: 9 · stub: 0
| id | title | status | module |
|---|---|---|---|
11.01.01 | First and second laws of thermodynamics | none | — |
11.01.02 | Thermodynamic potentials and Legendre transforms | none | — |
11.02.01 | Maxwell-Boltzmann distribution from kinetic theory | none | — |
11.03.01 | Microcanonical ensemble | none | — |
11.04.01 | Canonical ensemble and partition function | none | — |
11.05.01 | Bose-Einstein distribution | none | — |
11.05.02 | Fermi-Dirac distribution and electron gas | none | — |
11.06.01 | Ising model and phase transitions | none | — |
11.07.01 | Critical phenomena and renormalization group | none | — |
Quantum mechanics & QFT
18 units · full: 0 · partial: 0 · none: 18 · stub: 0
| id | title | status | module |
|---|---|---|---|
12.01.01 | Wave-particle duality and the double-slit | none | — |
12.01.02 | Stern-Gerlach and spin-1/2 | none | — |
12.02.01 | Hilbert-space formalism of quantum mechanics | none | — |
12.02.02 | Operators, observables, and Hermiticity | none | — |
12.03.01 | Schrödinger and Heisenberg pictures | none | — |
12.04.01 | Particle in a box | none | — |
12.04.02 | Quantum harmonic oscillator | none | — |
12.05.01 | Angular momentum operators and SU(2) representations | none | — |
12.06.01 | Hydrogen atom bound states | none | — |
12.07.01 | Time-independent perturbation theory | none | — |
12.07.02 | Time-dependent perturbation theory and Fermi's golden rule | none | — |
12.08.01 | Scattering Theory | none | — |
12.09.01 | Identical Particles and Many-Body Quantum Mechanics | none | — |
12.10.01 | Path integral formulation of quantum mechanics | none | — |
12.11.01 | Dirac equation and relativistic spin | none | — |
12.12.01 | Canonical Quantum Field Theory | none | — |
12.13.01 | Bosonic Fock space and second quantisation | none | — |
12.13.02 | Fermionic Fock space, Pauli exclusion, and anticommutators | none | — |
General relativity & cosmology
10 units · full: 0 · partial: 0 · none: 10 · stub: 0
| id | title | status | module |
|---|---|---|---|
13.01.01 | The equivalence principle | none | — |
13.02.01 | Tensors on smooth manifolds | none | — |
13.02.02 | Geodesics and parallel transport | none | — |
13.03.01 | Riemann curvature tensor | none | — |
13.04.01 | Einstein field equations | none | — |
13.05.01 | Schwarzschild solution | none | — |
13.05.02 | Orbits in Schwarzschild geometry | none | — |
13.06.01 | Black Holes | none | — |
13.07.01 | Linearized GR and gravitational waves | none | — |
13.08.01 | FLRW cosmology and Friedmann equations | none | — |
General & physical chemistry
13 units · full: 0 · partial: 0 · none: 13 · stub: 0
| id | title | status | module |
|---|---|---|---|
14.01.01 | Atomic structure and electron configurations | none | — |
14.02.01 | Lewis structures and VSEPR | none | — |
14.02.02 | Hybridization and valence bond theory | none | — |
14.03.01 | Stoichiometry and gas laws | none | — |
14.04.01 | Hydrogen atom quantum chemistry | none | — |
14.05.02 | Molecular orbital theory for homonuclear diatomics | none | — |
14.06.01 | Chemical thermodynamics: free energies and equilibrium | none | — |
14.07.01 | Statistical Mechanics for Chemistry | none | — |
14.08.01 | Chemical kinetics: rate laws and the Arrhenius equation | none | — |
14.09.01 | Solutions and Phase Equilibria | none | — |
14.10.01 | Acid-base chemistry: Bronsted-Lowry, Lewis, and pKa | none | — |
14.11.01 | Electrochemistry: the Nernst equation and electrochemical cells | none | — |
14.12.01 | UV-Vis, IR, and NMR — fundamentals of molecular spectroscopy | none | — |
Organic chemistry
14 units · full: 0 · partial: 0 · none: 14 · stub: 0
| id | title | status | module |
|---|---|---|---|
15.01.01 | Structure of organic molecules — stereochemistry | none | — |
15.02.01 | Functional groups and nomenclature | none | — |
15.03.01 | Acids and bases in organic chemistry | none | — |
15.04.02 | SN1 vs SN2 substitution mechanisms | none | — |
15.05.01 | Electrophilic addition to alkenes | none | — |
15.06.01 | Aromatic chemistry — EAS, Huckel | none | — |
15.07.01 | Carbonyl chemistry — nucleophilic addition | none | — |
15.08.01 | Radical and Pericyclic Reactions | none | — |
15.09.01 | Organometallic Methods in Synthesis | none | — |
15.10.01 | Retrosynthetic analysis | none | — |
15.11.01 | NMR spectroscopy of organic molecules | none | — |
15.12.01 | Amino acids and protein chemistry | none | — |
15.13.01 | Nucleic acid chemistry | none | — |
15.14.01 | Enzyme mechanism | none | — |
Inorganic chemistry
9 units · full: 0 · partial: 0 · none: 9 · stub: 0
| id | title | status | module |
|---|---|---|---|
16.01.01 | Periodic trends quantified | none | — |
16.02.01 | Symmetry and group theory in chemistry | none | — |
16.03.01 | Crystal field theory fundamentals | none | — |
16.03.02 | Crystal field splitting in octahedral complexes | none | — |
16.04.01 | Coordination chemistry | none | — |
16.04.02 | Crystal field stabilization energy and the spectrochemical series | none | — |
16.05.01 | Organometallic chemistry | none | — |
16.06.01 | Bioinorganic chemistry | none | — |
16.07.01 | Solid-state chemistry | none | — |
Molecular & cellular biology
18 units · full: 0 · partial: 0 · none: 18 · stub: 0
| id | title | status | module |
|---|---|---|---|
17.01.01 | Biomolecules in cells — overview | none | — |
17.02.01 | Cell membranes: structure | none | — |
17.02.02 | Membrane transport — passive and active | none | — |
17.03.01 | Cellular organization: organelles | none | — |
17.03.02 | Cytoskeleton and contractile proteins | none | — |
17.04.01 | Cellular respiration: glycolysis and CAC | none | — |
17.04.02 | Oxidative phosphorylation and ATP synthesis | none | — |
17.04.03 | Photosynthesis: light and dark reactions | none | — |
17.05.01 | DNA replication | none | — |
17.05.02 | Transcription | none | — |
17.05.03 | Translation | none | — |
17.06.01 | Mutation and repair | none | — |
17.07.01 | Cell signaling: receptors and GPCRs | none | — |
17.07.02 | Receptor tyrosine kinases and the MAPK signaling cascade | none | — |
17.08.01 | Cell cycle and mitosis | none | — |
17.09.01 | Resting membrane potential and ion channels | none | — |
17.09.02 | The action potential — ionic basis | none | — |
17.10.01 | Innate immunity at the molecular level | none | — |
Organismal biology
9 units · full: 0 · partial: 0 · none: 9 · stub: 0
| id | title | status | module |
|---|---|---|---|
18.02.01 | Cardiovascular physiology — the heart | none | — |
18.02.02 | Cardiac action potentials, pacemaker physiology, and the ECG | none | — |
18.03.01 | Respiratory physiology — gas exchange and transport | none | — |
18.04.01 | Skeletal muscle physiology | none | — |
18.04.02 | Muscle contraction — the actin-myosin cycle | none | — |
18.05.01 | Nervous system — gross anatomy and systems | none | — |
18.07.01 | Endocrine system — hormones and regulation | none | — |
18.08.01 | Renal physiology — homeostasis and the nephron | none | — |
18.11.01 | Embryology and morphogenesis | none | — |
Ecology & evolution
13 units · full: 0 · partial: 0 · none: 13 · stub: 0
| id | title | status | module |
|---|---|---|---|
19.01.01 | Mendelian genetics — segregation and dominance | none | — |
19.02.01 | Hardy-Weinberg equilibrium | none | — |
19.02.05 | Wright-Fisher model and the diffusion approximation | none | — |
19.03.01 | Natural selection — directional, stabilizing, and disruptive | none | — |
19.03.02 | Sexual selection | none | — |
19.03.03 | Kin selection and Hamilton's rule | none | — |
19.04.01 | Genetic drift | none | — |
19.05.01 | Quantitative genetics — heritability and the breeder's equation | none | — |
19.06.01 | Speciation — allopatric and sympatric | none | — |
19.07.01 | Phylogenetics — tree reconstruction | none | — |
19.09.01 | Population ecology — Lotka-Volterra | none | — |
19.10.01 | Community ecology — interactions and food webs | none | — |
19.15.01 | Origin of life — mechanistic scenarios | none | — |
Philosophy
7 units · full: 0 · partial: 0 · none: 7 · stub: 0
| id | title | status | module |
|---|---|---|---|
20.01.01 | Epistemology: knowledge, justification, and truth | none | — |
20.02.02 | Rights: natural, human, and legal | none | — |
20.02.03 | Freedom and liberty: negative, positive, and free will | none | — |
20.02.04 | The trolley problem and moral dilemmas | none | — |
20.02.05 | The good life: eudaimonia, flourishing, and meaning | none | — |
20.03.01 | The measurement problem in quantum mechanics | none | — |
20.05.02 | The unit of selection | none | — |
Grammar
24 units · full: 0 · partial: 0 · none: 24 · stub: 0
| id | title | status | module |
|---|---|---|---|
22.01.01 | Nouns | none | — |
22.01.02 | Verbs | none | — |
22.01.03 | Sentences: subject and predicate | none | — |
22.01.04 | Pronouns | none | — |
22.01.05 | Adjectives | none | — |
22.01.06 | Adverbs | none | — |
22.01.07 | Prepositions | none | — |
22.01.08 | Conjunctions | none | — |
22.01.09 | Interjections | none | — |
22.01.10 | Noun phrases and verb phrases | none | — |
22.01.11 | Subject-verb agreement | none | — |
22.01.12 | Verb tense: present, past, future | none | — |
22.01.13 | Perfect and progressive aspects | none | — |
22.01.14 | Active and passive voice | none | — |
22.01.15 | Clauses: independent and dependent | none | — |
22.01.16 | Compound and complex sentences | none | — |
22.01.17 | Relative clauses | none | — |
22.01.18 | Punctuation: end marks and commas | none | — |
22.01.19 | Punctuation: semicolons, colons, dashes | none | — |
22.01.20 | Apostrophes and quotation marks | none | — |
22.01.21 | Common errors: fragments, run-ons, dangling modifiers | none | — |
22.01.22 | Parallel structure | none | — |
22.01.23 | Pronoun case and reference | none | — |
22.01.24 | Capitalization conventions | none | — |
Writing
11 units · full: 0 · partial: 0 · none: 11 · stub: 0
| id | title | status | module |
|---|---|---|---|
22.02.01 | Writing a clear sentence | none | — |
22.02.02 | Paragraph structure | none | — |
22.02.03 | Transitions and flow | none | — |
22.02.04 | Thesis statement | none | — |
22.02.05 | Structuring an argument | none | — |
22.02.06 | Using evidence | none | — |
22.02.07 | Counterargument and rebuttal | none | — |
22.02.08 | Introduction and conclusion | none | — |
22.02.09 | Citation and attribution | none | — |
22.02.10 | Revision and editing | none | — |
22.02.11 | Style and voice | none | — |
Literature techniques
15 units · full: 0 · partial: 0 · none: 15 · stub: 0
| id | title | status | module |
|---|---|---|---|
22.03.01 | Literal vs Figurative Language | none | — |
22.03.02 | Metaphor and Simile | none | — |
22.03.03 | Symbolism and Allegory | none | — |
22.03.04 | Irony | none | — |
22.03.05 | Foreshadowing and Suspense | none | — |
22.03.06 | Point of View | none | — |
22.03.07 | Tone and Mood | none | — |
22.03.08 | Theme | none | — |
22.03.09 | Motif and Repetition | none | — |
22.03.10 | Unreliable Narration | none | — |
22.03.11 | Satire and Parody | none | — |
22.03.12 | Imagery and Sensory Detail | none | — |
22.03.13 | Allusion | none | — |
22.03.14 | Personification | none | — |
22.03.15 | Hyperbole and Understatement | none | — |
Economics
30 units · full: 0 · partial: 0 · none: 30 · stub: 0
| id | title | status | module |
|---|---|---|---|
23.01.01 | Scarcity and choice | none | — |
23.01.02 | Opportunity Cost | none | — |
23.01.03 | Supply and Demand | none | — |
23.01.04 | Market Equilibrium | none | — |
23.01.05 | Elasticity | none | — |
23.01.06 | Price Controls | none | — |
23.01.07 | Consumer and Producer Surplus | none | — |
23.01.08 | Costs of Production | none | — |
23.01.09 | Perfect Competition | none | — |
23.01.10 | Monopoly | none | — |
23.01.11 | Oligopoly and Monopolistic Competition | none | — |
23.01.12 | Profit Maximization | none | — |
23.01.13 | Labor Markets and Wages | none | — |
23.01.14 | Money and Banking | none | — |
23.01.15 | Inflation and Deflation | none | — |
23.01.16 | GDP and economic measurement | none | — |
23.01.17 | Unemployment | none | — |
23.01.18 | Fiscal policy | none | — |
23.01.19 | Monetary policy | none | — |
23.01.20 | International trade and comparative advantage | none | — |
23.01.21 | Exchange rates | none | — |
23.01.22 | Game theory basics | none | — |
23.01.23 | Externalities and public goods | none | — |
23.01.24 | Income inequality and redistribution | none | — |
23.01.25 | Behavioral economics | none | — |
23.01.26 | Market failures | none | — |
23.01.27 | Economic systems | none | — |
23.01.28 | Development economics | none | — |
23.01.29 | Personal finance: budgeting, saving, compound interest | none | — |
23.01.30 | Personal finance: credit, debt, investing | none | — |
Civics
15 units · full: 0 · partial: 0 · none: 14 · stub: 0
| id | title | status | module |
|---|---|---|---|
23.02.01 | What is government | none | — |
23.02.02 | Types of government | none | — |
23.02.03 | What is a constitution | none | — |
23.02.04 | Separation of powers | none | — |
23.02.05 | The legislature | none | — |
23.02.06 | The executive | none | — |
23.02.07 | The judiciary | none | — |
23.02.08 | How a law is made | none | — |
23.02.09 | Electoral systems | none | — |
23.02.10 | Political parties and interest groups | none | — |
23.02.11 | Rights and civil liberties | none | — |
23.02.12 | Federalism and local government | none | — |
23.02.13 | International organizations | none | — |
23.02.14 | Treaties and international law | none | — |
23.02.15 | Citizenship and civic participation | unspecified | — |
Geography
12 units · full: 0 · partial: 0 · none: 12 · stub: 0
| id | title | status | module |
|---|---|---|---|
23.03.01 | Maps and Map Projections | none | — |
23.03.02 | Latitude, Longitude, and Coordinate Systems | none | — |
23.03.03 | Continents and Oceans | none | — |
23.03.04 | Landforms | none | — |
23.03.05 | Climate Zones and Biomes | none | — |
23.03.06 | Population Distribution and Density | none | — |
23.03.07 | Urbanization and Settlement | none | — |
23.03.08 | Natural Resources and Distribution | none | — |
23.03.09 | Cultural Geography | none | — |
23.03.10 | Political Geography | none | — |
23.03.11 | Human Migration | none | — |
23.03.12 | Environmental Geography | none | — |