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 in prose only.
13
full
195
partial
2439
none
2500
Mathlib gaps
Precalculus foundations
33 units · full: 1 · partial: 6 · none: 26 · 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 |
00.01.E1 | Algebra and number-systems exercise pack (Lang Basic Mathematics Part I-II supplement) | none | — |
00.02.05 | Function | full | Codex.Precalc.SetAndFunction.Function |
00.02.06 | Sets, relations, and functions — the language of mathematics | none | — |
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 |
00.03.E1 | Functions, trigonometry, and coordinate-geometry exercise pack (Lang Basic Mathematics Part III-V supplement) | none | — |
00.04.01 | Inequalities (linear and quadratic) | partial | Codex.Precalc.Inequalities.Basic |
00.04.02 | Inequalities and systems of equations | none | — |
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 | — |
00.09.02 | Coordinate geometry — distance, lines, and circles | none | — |
00.10.01 | Conic sections (parabola, ellipse, hyperbola) | none | — |
00.10.02 | 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 | — |
00.14.01 | Complex numbers — rectangular and polar form | none | — |
00.15.01 | Sequences and series — arithmetic, geometric, and the bridge to calculus | none | — |
Algebra & linear algebra
52 units · full: 2 · partial: 7 · none: 43 · stub: 0
| id | title | status | module |
|---|---|---|---|
01.01.01 | Field | full | Codex.Foundations.LinearAlgebra.Field |
01.01.02 | Dual space and double dual | none | — |
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 |
01.01.06 | Systems of linear equations and the Kronecker-Capelli theorem | none | — |
01.01.07 | Determinant: axiomatic + expansion + properties | partial | Codex.Foundations.LinearAlgebra.Determinant |
01.01.08 | Eigenvalue, eigenvector, characteristic polynomial | partial | Codex.Foundations.LinearAlgebra.Eigen |
01.01.09 | Gram-Schmidt orthonormalisation and finite-dim inner-product space | none | — |
01.01.10 | Adjoint operator and isometry on a finite-dimensional inner-product space | none | — |
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 |
01.01.13 | Spectral theorem for normal operators on a finite-dim inner-product space (principal-axes theorem) | none | — |
01.01.14 | Rayleigh quotient and the Courant-Fischer min-max characterisation of eigenvalues | none | — |
01.01.15 | Bilinear form / quadratic form | partial | Codex.Foundations.LinearAlgebra.BilinearQuadraticForm |
01.01.16 | Invariant subspaces and the primary decomposition | none | — |
01.01.17 | Change of basis and the transformation laws | none | — |
01.01.18 | Linear manifolds, hyperplanes, and affine subspaces | none | — |
01.01.19 | Simultaneous diagonalisation of two quadratic forms and the generalised eigenvalue problem | none | — |
01.01.E1 | Linear algebra exercise pack (Apostol Vol. 2 Ch. 1-5 supplement) | none | — |
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 | — |
01.02.06 | Ring, ring homomorphism, and ideal | none | — |
01.02.07 | Polynomial rings, PIDs, UFDs, and Euclidean domains | none | — |
01.02.08 | Localisation of a commutative ring | none | — |
01.02.09 | Category, functor, natural transformation, the Yoneda lemma, and adjunction | 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 | — |
01.02.13 | Fundamental Theorem of Galois Theory (finite case) | none | — |
01.02.14 | Semisimple rings, the Artin-Wedderburn structure theorem, the Jacobson radical | none | — |
01.02.15 | Galois cohomology, Hilbert's Theorem 90, and the Brauer group of a field | none | — |
01.02.16 | Nakayama's lemma | none | — |
01.02.17 | Hilbert basis theorem; Noetherian rings and modules | none | — |
01.02.18 | Finite fields F_q and the Frobenius automorphism | 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 | — |
01.02.30 | Chain complex in an abelian category | none | — |
01.02.31 | Chain homotopy and the homotopy category | none | — |
01.02.32 | Mapping cone of a chain map and the distinguished triangle | none | — |
01.02.33 | Abelian category and Grothendieck axioms AB1-AB5 | none | — |
01.02.35 | Dold-Kan correspondence | none | — |
01.03.01 | Rings and modules | none | — |
01.03.02 | Modules over a PID and the structure theorem | none | — |
01.04.01 | Fields and Galois theory | none | — |
01.05.01 | Commutative algebra — Noetherian rings, localisation, and primary decomposition | none | — |
01.06.01 | Homological algebra — chain complexes, exact sequences, and derived functors | none | — |
01.06.02 | Syzygies — Hilbert's syzygy theorem, projective dimension, and Auslander-Buchsbaum | none | — |
Analysis
129 units · full: 7 · partial: 17 · none: 105 · 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 | — |
02.01.10 | Fibre Homotopy Equivalence and Dold's Theorem | none | — |
02.01.E1 | Point-set topology and the fundamental groupoid exercise pack (Brown, Topology and Groupoids supplement) | none | — |
02.02.01 | Real-number axioms (ordered field) | partial | Codex.Analysis.RealNumbers.Axioms |
02.02.02 | The real numbers — Dedekind cuts, Cauchy completeness, and uniqueness | none | — |
02.03.01 | Sequences — convergence and limits | none | — |
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 | — |
02.04.02 | The Riemann 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 |
02.05.E1 | Multivariable calculus exercise pack (Apostol Vol. 2 Ch. 8-9 supplement) | none | — |
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 | — |
02.06.E1 | Ordinary differential equations exercise pack (Apostol Vol. 2 Ch. 6-7 supplement) | none | — |
02.07.01 | Sigma-algebra, Measurable Space, and the Borel Sigma-algebra | partial | Codex.Analysis.MeasureTheory.SigmaAlgebraBorel |
02.07.02 | Lebesgue Outer Measure and the Carathéodory Construction | full | Codex.Analysis.MeasureTheory.OuterMeasureCaratheodory |
02.07.03 | Measurable Functions, Simple Functions, Egorov's Theorem, and Lusin's Theorem | full | Codex.Analysis.MeasureTheory.MeasurableFunctionsEgorovLusin |
02.07.04 | Lebesgue Integral Construction and the Monotone Convergence Theorem | full | Codex.Analysis.MeasureTheory.LebesgueIntegralMCT |
02.07.05 | Fatou's Lemma and the Dominated Convergence Theorem | full | Codex.Analysis.MeasureTheory.FatouDCT |
02.07.06 | L^p Spaces: Hölder, Minkowski, and Riesz-Fischer Completeness | full | Codex.Analysis.MeasureTheory.LpSpaces |
02.07.07 | Fubini-Tonelli Theorem and Product Measures | partial | Codex.Analysis.MeasureTheory.FubiniTonelli |
02.07.08 | Absolute Continuity and the Radon-Nikodym Theorem | partial | Codex.Analysis.MeasureTheory.AbsoluteContinuityRadonNikodym |
02.07.09 | The Whitney Extension Theorem and the Whitney Cube Decomposition | none | — |
02.07.10 | Rademacher's theorem | none | — |
02.07.11 | The Area and Coarea Formulas | none | — |
02.07.12 | The Lebesgue differentiation theorem and the Hardy-Littlewood maximal function | none | — |
02.07.E1 | Geometric measure theory exercise pack (Whitney / Federer Ch. 2-3 supplement) | none | — |
02.08.01 | First-order linear and separable ODEs | none | — |
02.08.02 | Second-order linear ODEs with constant coefficients | none | — |
02.08.03 | Picard-Lindelof existence and uniqueness for ODEs | none | — |
02.09.01 | Complex numbers and Euler's formula | none | — |
02.10.01 | Fourier Series and the Riemann-Lebesgue Lemma | partial | Codex.Analysis.Harmonic.FourierSeries |
02.10.04 | Fourier Transform on R^n and the Plancherel Theorem | partial | Codex.Analysis.Harmonic.FourierTransform |
02.10.05 | Surface integral and parametric surfaces | none | — |
02.10.06 | The Bochner-Minlos theorem and characteristic functionals on nuclear spaces | none | — |
02.10.07 | The Radon transform: inversion, Plancherel, and the range theorem | none | — |
02.10.E1 | Vector calculus exercise pack (Apostol Vol. 2 Ch. 10-12 supplement) | none | — |
02.11.01 | Bounded linear operators | partial | Codex.Analysis.FunctionalAnalysis.BoundedOperators |
02.11.02 | Hahn-Banach theorem (analytic and geometric forms) | partial | Codex.Analysis.FunctionalAnalysis.HahnBanach |
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 |
02.11.09 | Open mapping and closed graph theorems | partial | Codex.Analysis.FunctionalAnalysis.OpenMapping |
02.11.10 | Banach-Steinhaus — the uniform boundedness principle | none | — |
02.11.11 | The spectral theorem | none | — |
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 | — |
02.12.E1 | Qualitative theory of ODEs exercise pack (Arnold Ch. 2-3 supplement) | none | — |
02.13.01 | Laplace Equation, Harmonic Functions, Mean-Value Property, and Maximum Principle | none | — |
02.13.02 | Poisson Equation, Fundamental Solution, and Newtonian Potential | none | — |
02.13.03 | Heat Equation, Heat Kernel, and Duhamel's Principle | none | — |
02.13.04 | Wave Equation, d'Alembert Solution, Spherical Means, and Huygens Principle | none | — |
02.13.05 | Whitney deformation theorem | none | — |
02.13.06 | The Cauchy-Kovalevskaya Theorem and Holmgren Uniqueness | none | — |
02.13.07 | Rectifiable currents | none | — |
02.13.11 | Slicing of currents | none | — |
02.13.E1 | Integration and currents exercise pack (Whitney Ch. I, IX, XI supplement) | none | — |
02.14.01 | Wave-front set of a distribution | none | — |
02.14.02 | Pseudo-differential operators on a manifold | none | — |
02.14.03 | Propagation of singularities along Hamiltonian flow | none | — |
02.14.04 | The theory of distributions and the Schwartz kernel theorem | none | — |
02.14.05 | Rigged Hilbert space (Gel'fand triple) and the nuclear spectral theorem | none | — |
02.15.01 | Brownian motion and the Wiener process | none | — |
02.15.02 | The Itô integral and Itô's formula | none | — |
02.15.03 | Stochastic differential equations, diffusions, and the infinitesimal generator | none | — |
02.15.04 | The Feynman-Kac formula | none | — |
02.15.05 | The Stratonovich Integral and Stratonovich Calculus | none | — |
02.16.01 | Sobolev Inequalities: the Gagliardo-Nirenberg-Sobolev and Morrey Inequalities | none | — |
02.16.02 | Trace and Extension Theorems for Sobolev Functions | none | — |
02.16.03 | The Rellich-Kondrachov Compactness Theorem and the Poincaré Inequalities | none | — |
02.16.04 | Lax-Milgram and Existence of Weak Solutions of Elliptic Boundary-Value Problems | none | — |
02.16.05 | The Fredholm Alternative and Eigenvalues for Second-Order Elliptic Operators | none | — |
02.17.01 | Interior and Boundary H^2 Regularity of Weak Elliptic Solutions | none | — |
02.17.02 | Maximum Principles for General Second-Order Elliptic Operators | none | — |
02.17.03 | The Alexandrov-Bakelman-Pucci Estimate | none | — |
02.17.04 | Schauder Theory: Interior and Boundary C^{2,alpha} Estimates | none | — |
02.17.05 | The Classical Dirichlet Problem via the Method of Continuity | none | — |
02.17.06 | Lp (Calderón-Zygmund) W^{2,p} Estimates for Elliptic Equations | none | — |
02.17.07 | De Giorgi-Nash-Moser Theory: Local Boundedness and Holder Continuity of Weak Solutions | none | — |
02.17.08 | The Harnack Inequality for Elliptic Equations (Moser and Krylov-Safonov) | none | — |
02.17.09 | Quasilinear Elliptic Equations: Gradient Estimates and Existence by Leray-Schauder | none | — |
02.18.01 | Galerkin Existence and Energy Estimates for Second-Order Parabolic Equations | none | — |
02.18.02 | Galerkin Existence and Finite Propagation Speed for Second-Order Hyperbolic Equations | none | — |
02.18.03 | C0-Semigroups and the Hille-Yosida Theorem | none | — |
02.18.04 | The Direct Method of the Calculus of Variations | none | — |
02.18.05 | Viscosity Solutions of Hamilton-Jacobi Equations | none | — |
02.18.06 | Scalar Conservation Laws: Shocks, Rankine-Hugoniot, and Entropy Solutions | none | — |
02.19.01 | The Hardy-Littlewood Maximal Function and the Vitali Covering Lemma | none | — |
02.19.02 | The Calderón-Zygmund Decomposition | none | — |
02.19.03 | Calderón-Zygmund Singular Integral Operators: Lp Boundedness | none | — |
02.19.04 | The Riesz Transforms | none | — |
02.19.05 | Riesz and Bessel Potentials and the Hardy-Littlewood-Sobolev Inequality | none | — |
02.20.01 | BMO and the John-Nirenberg Inequality | none | — |
02.20.02 | Real-Variable Hardy Spaces H^p and the Atomic Decomposition | none | — |
02.20.03 | Littlewood-Paley Theory and the Square Function | none | — |
02.20.04 | Fourier Multipliers and the Hörmander-Mikhlin Theorem | none | — |
02.20.05 | Oscillatory Integrals and the Method of Stationary Phase | none | — |
02.21.01 | Dispersive Decay Estimates for the Schrödinger and Wave Propagators | none | — |
02.21.02 | Strichartz Estimates via the TT* Method | none | — |
02.21.03 | Local Well-Posedness for Semilinear NLS/NLW via Strichartz Contraction | none | — |
02.21.04 | Conservation Laws, Global Well-Posedness, and the Energy-Critical Problem | none | — |
02.21.05 | Bourgain X^{s,b} Spaces and Low-Regularity Well-Posedness | none | — |
02.21.06 | Virial Identities, Blowup, and the Soliton-Stability Outlook | none | — |
Probability & stochastics
46 units · full: 0 · partial: 0 · none: 46 · stub: 0
| id | title | status | module |
|---|---|---|---|
37.01.01 | Probability Spaces and the Kolmogorov Extension Theorem | none | — |
37.02.01 | The Borel-Cantelli Lemmas and the Kolmogorov 0-1 Law | none | — |
37.02.02 | The Strong Law of Large Numbers | none | — |
37.02.03 | The Ergodic Theorems: Birkhoff, von Neumann, and Kingman | none | — |
37.03.01 | Characteristic Functions, Inversion, and the Lévy Continuity Theorem | none | — |
37.03.02 | The Lindeberg–Feller Central Limit Theorem | none | — |
37.03.03 | Donsker's Invariance Principle and the Functional Central Limit Theorem | none | — |
37.04.01 | Discrete-Time Martingales, Stopping Times, and Optional Stopping | none | — |
37.04.02 | Doob's Upcrossing Inequality and the Almost-Sure Martingale Convergence Theorem | none | — |
37.04.03 | Doob's Maximal and L^p Inequalities, Uniform Integrability, and L^p-Bounded Martingales | none | — |
37.04.04 | Kakutani's Theorem on Product Martingales and Absolute Continuity of Product Measures | none | — |
37.05.01 | The Markov Property, Transition Matrices, and the Chapman–Kolmogorov Equations | none | — |
37.05.02 | Class Structure, Irreducibility, and Periodicity | none | — |
37.05.03 | Hitting Probabilities and Expected Hitting Times | none | — |
37.05.04 | The Strong Markov Property and the Recurrence/Transience Dichotomy | none | — |
37.05.05 | Invariant Measures and Distributions; Positive and Null Recurrence | none | — |
37.05.06 | Convergence to Equilibrium via Coupling | none | — |
37.05.07 | The Ergodic Theorem for Markov Chains and Detailed Balance | none | — |
37.05.08 | Continuous-Time Markov Chains I: Q-Matrices, Jump Chains, and Holding Times | none | — |
37.05.09 | Continuous-Time Markov Chains II: The Kolmogorov Backward and Forward Equations | none | — |
37.05.10 | Recurrence, Invariant Distributions, and Convergence for Continuous-Time Chains | none | — |
37.05.11 | The Poisson Process: Equivalent Characterizations | none | — |
37.05.12 | Birth–Death Processes and Queueing Chains | none | — |
37.06.01 | Continuous Local Martingales, Quadratic Variation, and the Doob–Meyer Decomposition | none | — |
37.06.02 | The Brownian Martingale Representation Theorem | none | — |
37.06.03 | Brownian Local Time and Tanaka's Formula | none | — |
37.07.01 | The Large Deviation Principle: Rate Functions, Bounds, and Goodness | none | — |
37.07.02 | Cramér's Theorem and the Legendre-Fenchel Rate Function | none | — |
37.07.03 | The Legendre-Fenchel Transform and Convex Duality of Rate Functions | none | — |
37.07.04 | The Gärtner-Ellis Theorem | none | — |
37.07.05 | Sanov's Theorem and the Large Deviation Principle for Empirical Measures | none | — |
37.07.06 | Relative Entropy as a Rate Function and the Donsker-Varadhan Variational Formula | none | — |
37.07.07 | Varadhan's Integral Lemma and the Laplace Principle | none | — |
37.07.08 | The Contraction Principle and the Inverse Contraction Principle | none | — |
37.07.09 | Exponential Tightness, Exponential Approximation, and the Dawson–Gärtner Projective Limit | none | — |
37.07.10 | Schilder's Theorem: Small-Noise Large Deviations for Brownian Motion | none | — |
37.07.11 | Freidlin–Wentzell Theory: Large Deviations for Small-Noise Diffusions | none | — |
37.08.01 | The Wigner Semicircle Law and the Moment Method | none | — |
37.08.02 | The Stieltjes Transform and the Semicircle Law via the Resolvent | none | — |
37.08.03 | Gaussian Ensembles GOE/GUE/GSE and the Joint Eigenvalue Density | none | — |
37.08.04 | Determinantal Point Processes and Sine-Kernel Bulk Universality | none | — |
37.08.05 | The Airy Kernel and the Tracy-Widom Edge Law | none | — |
37.08.06 | The Largest Eigenvalue and the Operator-Norm Bound | none | — |
37.08.07 | Spectral Concentration: Log-Sobolev and the Herbst Argument | none | — |
37.08.08 | Free Probability: Freeness, Free Convolution, and the R-Transform | none | — |
37.08.09 | The Ben Arous–Guionnet Large Deviation Principle for the Empirical Spectral Measure | none | — |
Number theory
64 units · full: 0 · partial: 10 · none: 54 · stub: 0
| id | title | status | module |
|---|---|---|---|
21.01.01 | Divisibility, GCD, Bézout's identity, and the Euclidean algorithm | none | — |
21.01.02 | Primes, the fundamental theorem of arithmetic, and the infinitude of primes | none | — |
21.01.03 | Congruences, the Chinese remainder theorem, and the ring structure of ℤ/nℤ | none | — |
21.01.04 | Fermat's little theorem, Euler's theorem, and Wilson's theorem | none | — |
21.01.05 | Primitive roots and the structure of | none | — |
21.01.06 | Quadratic residues, the Legendre symbol, and Euler's criterion | none | — |
21.01.07 | Quadratic reciprocity (Gauss's theorema aureum) | none | — |
21.01.08 | Pell equation and continued fractions | none | — |
21.02.01 | Finite fields — structure and squares | none | — |
21.02.02 | Quadratic reciprocity via Gauss sums | none | — |
21.02.03 | -adic numbers and | none | — |
21.02.04 | Hensel's lemma | none | — |
21.02.05 | Hilbert symbol and the product formula | none | — |
21.02.06 | Witt's theorem: cancellation and the Witt decomposition | none | — |
21.02.07 | Number fields: ring of integers, ideal class group, and the Dirichlet unit theorem | none | — |
21.02.08 | Hasse-Minkowski theorem | none | — |
21.02.09 | The Brauer-Manin obstruction | none | — |
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 |
21.03.04 | Dirichlet density | none | — |
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 |
21.04.04 | Theta Series of Quadratic Forms and Sums of Squares | none | — |
21.04.05 | Ramanujan -function and Ramanujan conjectures | none | — |
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 |
21.06.02 | Sato-Tate conjecture | none | — |
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 |
21.09.01 | Arakelov geometry and arithmetic surfaces (survey) | none | — |
21.09.02 | Faltings / Mordell theorem | none | — |
21.09.03 | Heights and the Néron–Tate canonical height | none | — |
21.10.01 | Langlands Philosophy Survey | none | — |
21.11.01 | Arithmetic Functions, Dirichlet Convolution, and Möbius Inversion | none | — |
21.11.02 | Average Orders of Arithmetic Functions and the Summation Toolkit | none | — |
21.11.03 | Chebyshev's Bounds, Bertrand's Postulate, and Mertens' Theorems | none | — |
21.11.04 | Perron's Formula and Mellin Inversion | none | — |
21.11.05 | The Selberg-Delange Method | none | — |
21.12.01 | The von Mangoldt Function, the Chebyshev Psi Function, and the Logarithmic Derivative of Zeta | none | — |
21.12.02 | The Prime Number Theorem via Contour Integration | none | — |
21.12.03 | Effective Zero-Free Regions for Zeta and the Prime Number Theorem Error Term | none | — |
21.12.04 | The Riemann-von Mangoldt Explicit Formula | none | — |
21.13.01 | Zero-Free Regions for Dirichlet L-Functions and Exceptional (Siegel) Zeros | none | — |
21.13.02 | Siegel's Theorem on the Exceptional Zero | none | — |
21.13.03 | The Prime Number Theorem in Arithmetic Progressions and Siegel-Walfisz | none | — |
21.13.04 | The Polya-Vinogradov Inequality | none | — |
21.13.05 | The Approximate Functional Equation, Analytic Conductor, and Convexity Bound | none | — |
21.14.01 | The Large Sieve Inequality and Brun-Titchmarsh | none | — |
21.14.02 | The Bombieri-Vinogradov Theorem | none | — |
21.14.03 | Mean Values of Multiplicative Functions: Halász's Theorem | none | — |
21.14.04 | Combinatorial Sieve Methods: Brun and Selberg | none | — |
21.14.05 | Kloosterman Sums and the Kuznetsov Spectral Formula | none | — |
21.15.01 | Poisson and Voronoi Summation | none | — |
21.15.02 | Weyl Sums, Weyl Differencing, and Equidistribution | none | — |
21.15.03 | van der Corput's Method for Exponential Sums | none | — |
21.15.04 | Gauss, Jacobi, Kloosterman, and Salié Sums; the Weil Bound | none | — |
21.15.05 | The Vinogradov Mean Value Theorem | none | — |
21.16.01 | The Partition Function, Generating Functions, and the Pentagonal Number Theorem | none | — |
21.16.02 | The Hardy-Ramanujan-Rademacher Asymptotics via the Circle Method | none | — |
21.17.01 | Algebraic number theory — number fields, Dedekind domains, and the class group | none | — |
21.17.02 | Ramification, decomposition and inertia groups, the discriminant and the different | none | — |
21.18.01 | Class field theory — abelian extensions and Artin reciprocity | none | — |
Differential geometry
75 units · full: 0 · partial: 0 · none: 75 · stub: 0
| id | title | status | module |
|---|---|---|---|
03.04.24 | Lie bialgebroids and Poisson groupoids: the Mackenzie-Xu duality | none | — |
03.06.21 | Godbillon-Vey class and secondary characteristic classes of foliations | none | — |
48.01.01 | Riemannian geometry — metric, connection, and curvature | none | — |
48.01.02 | Comparison theorems in Riemannian geometry — Cartan-Hadamard, Bonnet-Myers, Bishop-Gromov | none | — |
48.01.03 | Geodesics, the exponential map, and completeness | none | — |
48.02.01 | Topological manifold | none | — |
48.02.02 | Smooth structure and atlases | none | — |
48.02.03 | Smooth maps between manifolds | none | — |
48.02.04 | Frobenius theorem | none | — |
48.02.05 | Sectional curvature, Ricci tensor, scalar curvature | none | — |
48.02.06 | Constant-curvature spaces and Killing-Hopf | none | — |
48.02.07 | Killing fields and infinitesimal isometries | none | — |
48.02.08 | Myers-Steenrod theorem | none | — |
48.02.09 | Almost-complex structure (manifold-level) | none | — |
48.02.10 | Complex manifold and the Dolbeault complex | none | — |
48.02.11 | Hermitian manifold and the Kahler form | none | — |
48.02.12 | Kahler identities and the Hodge decomposition (Kahler version) | none | — |
48.02.13 | Isometric immersion and the second fundamental form | none | — |
48.02.14 | Gauss, Codazzi, and Ricci equations | none | — |
48.02.15 | Bochner technique and curvature vanishing theorems | none | — |
48.02.16 | Weyl tensor and conformally flat metrics | none | — |
48.02.17 | Lorentzian Hopf-Rinow and global hyperbolicity (introductory pseudo-Riemannian geometry) | none | — |
48.02.18 | Petrov classification of Lorentzian 4-curvature | none | — |
48.02.19 | Jacobi fields, conjugate points, and the Morse Index Theorem | none | — |
48.02.20 | Handles, surgery, and the cobordism category | none | — |
48.02.21 | Rearrangement and self-indexing Morse functions | none | — |
48.02.22 | The Whitney trick and handle cancellation | none | — |
48.02.23 | The h-cobordism theorem | none | — |
48.02.24 | The generalised Poincaré conjecture in high dimensions | none | — |
48.02.25 | The Lefschetz hyperplane theorem via Morse theory | none | — |
48.02.26 | Harmonic maps: energy, tension field, and the harmonic-map equation | none | — |
48.02.27 | Levi-Civita connection, exponential map, and gradient-like vector fields on a cobordism | none | — |
48.02.28 | Pointer: surgery theory and the surgery exact sequence | none | — |
48.02.29 | The harmonic-map heat flow and the Eells–Sampson theorem | none | — |
48.02.30 | Morse functions, the Morse lemma, and the Morse index | none | — |
48.02.31 | Handle attachment, CW homotopy type, and the Morse inequalities | none | — |
48.02.32 | The Riemannian Hopf–Rinow theorem | none | — |
48.02.33 | The Yamabe problem and the conformal Laplacian | none | — |
48.02.34 | The Toponogov triangle comparison theorem | none | — |
48.02.35 | Synge's theorem and the second variation of arc length | none | — |
48.02.36 | Einstein metrics as critical points of the total scalar curvature | none | — |
48.02.37 | Homogeneous Einstein metrics on G/H | none | — |
48.02.38 | Infinitesimal Einstein deformations, the Lichnerowicz Laplacian on 2-tensors, and Koiso rigidity | none | — |
48.02.39 | The Cartan–Ambrose–Hicks theorem (intrinsic rigidity) | none | — |
48.02.40 | The path space as a CW complex: the fundamental theorem of Morse theory | none | — |
48.02.41 | The two-spinor calculus: -spinors, abstract indices, and the spinor–tensor dictionary | none | — |
48.02.42 | Zero-rest-mass field equations and the spinor form of Maxwell, Weyl, and Dirac fields | none | — |
48.02.43 | The Newman-Penrose spin-coefficient formalism | none | — |
48.02.44 | Gauss-Bonnet and Chern-Gauss-Bonnet: from angle excess to the Euler class | none | — |
48.02.45 | Brownian motion on a Riemannian manifold | none | — |
48.03.04 | Formal group law | none | — |
48.03.05 | p-adic Lie group and the p-adic exponential | none | — |
48.03.06 | Lie's third theorem (statement, simply-connected case) | none | — |
48.03.07 | Invariant affine connections on a reductive homogeneous space and the canonical connection | none | — |
48.05.00 | General fibre bundle | none | — |
48.05.06 | Vertical subbundle and fundamental vector fields | none | — |
48.05.11 | Horizontal lift and parallel transport | none | — |
48.05.12 | Reduction of structure group; reduction of a connection | none | — |
48.05.13 | Associated bundle and induced connection | none | — |
48.05.14 | Torsion tensor and the two Cartan structural equations | none | — |
48.05.15 | Linear connection via the frame bundle; soldering form | none | — |
48.05.16 | Holonomy group and restricted holonomy | none | — |
48.05.17 | Ambrose-Singer holonomy theorem | none | — |
48.05.18 | Holonomy reduction theorem | none | — |
48.05.19 | Holomorphic vector bundle | none | — |
48.05.20 | Hermitian metric on a complex bundle; Chern connection | none | — |
48.12.24 | Simplicial set and the simplicial category Delta | none | — |
48.12.26 | Functorial CW approximation Gamma X = |S_*X| | none | — |
48.12.27 | Puppe cofiber sequence | none | — |
48.12.28 | Puppe fiber sequence | none | — |
48.12.29 | Thom space and Thom isomorphism | none | — |
48.12.30 | Minimal complex and minimal fibration | none | — |
48.12.36 | Bisimplicial set, diagonal, and the realisation lemma | none | — |
48.12.39 | Simplicial group and the W-bar classifying functor | none | — |
48.12.49 | Bialgebra, Hopf algebra, and the Milnor-Moore theorem | none | — |
Modern geometry
243 units · full: 3 · partial: 53 · none: 187 · 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 |
03.02.02 | Sard's theorem, transversality, and genericity | none | — |
03.03.01 | Lie group | partial | Codex.ModernGeometry.Lie.LieGroup |
03.03.02 | Group action | none | — |
03.03.03 | Orthogonal group | none | — |
03.03.10 | Lie groupoid: source, target, smooth composition | none | — |
03.03.11 | Action Lie groupoid and action Lie algebroid | none | — |
03.03.12 | Bisection group of a Lie groupoid; gauge transformations as bisections | none | — |
03.03.13 | Groupoid as a small category with all morphisms invertible | 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 | — |
03.04.15 | Hodge Laplacian on a Riemannian manifold | none | — |
03.04.16 | Lie algebroid: anchor, bracket, Leibniz law | none | — |
03.04.17 | Lie functor: differentiating a Lie groupoid to its Lie algebroid | none | — |
03.04.18 | Pradines integration theorem and Mackenzie transitive integrability | none | — |
03.04.19 | Cotangent algebroid of a Poisson manifold; pointer to symplectic groupoids | none | — |
03.04.20 | Surface integrals of 2-forms; flux of a vector field through an oriented surface | none | — |
03.04.21 | Closed and exact forms; the Poincaré lemma; the angle 1-form | none | — |
03.04.22 | Lie algebroid cohomology and the Chevalley-Eilenberg differential | none | — |
03.04.23 | Representations of a Lie algebroid and flat A-connections | none | — |
03.04.25 | Double Lie groupoids, double Lie algebroids, and VB-groupoids | none | — |
03.04.E1 | Mayer-Vietoris and degree-theory exercise pack (Bott-Tu Ch. I supplement) | none | — |
03.04.E2 | Differential forms and Stokes exercise pack (Shifrin / Arnold 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 |
03.05.21 | Gauge groupoid of a principal bundle | none | — |
03.05.22 | Atiyah algebroid of a principal bundle | none | — |
03.05.23 | Connection on a principal bundle as splitting of the Atiyah algebroid | none | — |
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 | — |
03.06.24 | Bott-Taubes rigidity theorem | none | — |
03.06.26 | Pointer: elliptic cohomology | none | — |
03.07.05 | Yang-Mills action | none | — |
03.07.06 | Anti-self-dual (ASD) equation on a 4-manifold | none | — |
03.07.07 | BPST instanton and the Bogomolny bound | none | — |
03.07.08 | Conformal compactification and finite-action instantons | 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 | — |
03.07.12 | The Geometry of Twistors: Null Planes, the Twistor Norm, and the Robinson Congruence | none | — |
03.07.14 | Penrose transform at linear level | none | — |
03.07.16 | -field as a gerbe connection | none | — |
03.07.17 | Chern-Simons functional on a 3-manifold | none | — |
03.07.18 | Configuration space and slice theorem on | none | — |
03.07.19 | Spectral flow and the Floer grading mod 8 | none | — |
03.07.20 | Uhlenbeck compactness for ASD equations on cylinders | none | — |
03.07.21 | Gluing theorem for instanton trajectories | none | — |
03.07.22 | Orientations on instanton trajectory moduli | none | — |
03.07.23 | Instanton Floer homology | none | — |
03.07.24 | Relative Donaldson invariants for 4-manifolds with boundary | none | — |
03.07.25 | Donaldson-Floer surgery exact triangle | none | — |
03.07.26 | Atiyah-Floer conjecture | none | — |
03.07.27 | Polyfolds (Hofer-Wysocki-Zehnder) | none | — |
03.07.28 | Monopole-instanton Floer equivalence (Kronheimer-Mrowka) | none | — |
03.07.29 | Electromagnetism as a U(1) Yang-Mills theory — the geometric dictionary | none | — |
03.07.30 | Aharonov-Bohm effect and holonomy of U(1) connections | none | — |
03.07.31 | BRST cohomology and Faddeev-Popov-ghost quantisation of gauge theories | none | — |
03.07.32 | Anomalies via descent equations and the Atiyah-Singer index theorem | none | — |
03.07.33 | Casson's invariant and the Euler characteristic of instanton Floer homology | none | — |
03.07.34 | Simple type, basic classes, and the structure of Donaldson invariants | 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 | — |
03.08.08 | Bott periodicity for U via Morse theory | none | — |
03.08.09 | Worked K-theory computations: spheres, projective spaces, and tori | 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 |
03.08.13 | Bott periodicity for O via iterated minimal geodesics | none | — |
03.08.20 | Whitehead torsion and the s-cobordism theorem | none | — |
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 | — |
03.09.27 | Pure spinors and the spinor variety | none | — |
03.09.28 | The Cartan Model of Equivariant de Rham Cohomology | none | — |
03.09.29 | The probabilistic heat kernel and Bismut's formula | 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 | — |
03.10.03 | The Wess-Zumino-Witten action and the level-k extension | none | — |
03.10.04 | Minimal Models, the Kac Formula, and Null Vectors | none | — |
03.10.05 | The Coulomb gas, screening charges, and the conformal bootstrap | 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 | — |
03.11.04 | The free loop space LM and transgression | none | — |
03.11.05 | The geometric central extension of the loop group LG | 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 |
03.12.34 | Acyclic models and the Eilenberg-Zilber theorem | none | — |
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 |
03.12.41 | Twisted cartesian products and simplicial fibre bundles | none | — |
03.12.42 | Combinatorial simplicial homotopy groups and the Kan-fibration long exact sequence | none | — |
03.12.43 | Quasi-categories and the Joyal model structure | none | — |
03.12.44 | Mixed Hodge structures on rational homotopy theory (Morgan's theorem) | none | — |
03.12.45 | Arithmetic square and integral fracture theorems | partial | Codex.Modern.Homotopy.ArithmeticSquare |
03.12.46 | The periodicity and thick subcategory theorems | none | — |
03.12.47 | HELP and the unified Whitehead / cellular approximation theorem | none | — |
03.12.48 | Bousfield localisation of a model category | none | — |
03.12.50 | The Cartan model for the minimal model of a homogeneous space | none | — |
03.12.51 | Massey products and the formality condition | none | — |
03.12.52 | Relative homotopy group | none | — |
03.12.53 | Whitehead's crossed module of a pair | none | — |
03.12.54 | Filtered space | none | — |
03.12.55 | Crossed complex of a filtered space | none | — |
03.12.56 | Higher Homotopy Seifert-van Kampen theorem | none | — |
03.12.57 | Cubical -groupoid | none | — |
03.12.58 | Free crossed resolution of a group | none | — |
03.12.59 | Classifying space of a crossed complex | none | — |
03.12.60 | Localisation of nilpotent spaces at a set of primes | none | — |
03.12.61 | Nilpotent groups and nilpotent spaces | none | — |
03.12.E1 | Rational homotopy and Sullivan minimal-model exercise pack (Bott-Tu Ch. III §19 supplement) | none | — |
03.12.E2 | Singular and cellular homology exercise pack (Hatcher Ch. 2 supplement) | none | — |
03.12.E3 | Simplicial homotopy theory exercise pack (Goerss-Jardine supplement) | none | — |
03.12.E4 | Localization and completion exercise pack (May-Ponto supplement) | none | — |
03.12.E5 | Simplicial objects and Dold-Kan exercise pack (May 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 | — |
03.13.05 | The Brown-Peterson spectrum BP and its Hopf algebroid | none | — |
03.13.06 | The chromatic spectral sequence | none | — |
03.13.07 | Greek-letter elements in the stable homotopy of spheres | none | — |
03.13.08 | The telescope conjecture and its disproof | 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 | — |
03.15.01 | Gradient flow, stable/unstable manifolds, and the Morse-Smale condition | none | — |
03.15.02 | Trajectory spaces, the Fredholm setup, and transversality | none | — |
03.15.03 | Compactness: broken trajectories | none | — |
03.15.04 | Gluing of trajectories | none | — |
03.15.05 | Coherent orientations and characteristic signs | none | — |
03.15.06 | The Morse complex and | none | — |
03.15.07 | Continuation maps and invariance of | none | — |
03.15.08 | The Morse Homology Theorem | none | — |
03.15.09 | Morse cohomology, cup product, and the ring structure | none | — |
03.15.10 | Poincaré duality via flow reversal and the filtered Morse spectral sequence | none | — |
03.15.11 | From finite-dimensional Morse homology to Floer homology | none | — |
03.15.12 | Witten's supersymmetric Morse theory (survey/pointer) | none | — |
03.15.E1 | Morse homology exercise pack (Schwarz Morse Homology supplement) | none | — |
03.16.01 | The Atiyah–Segal axioms for topological quantum field theory | none | — |
03.16.02 | Classification of 2d oriented TQFTs: the Frobenius-algebra theorem | none | — |
03.16.03 | Extended TQFT and the cobordism hypothesis | none | — |
03.16.04 | Invertible field theories and the Freed–Hopkins classification | none | — |
03.16.05 | Anomalies as invertible field theories in one dimension higher | none | — |
03.16.06 | Chern-Simons theory as a quantum TQFT, the Jones polynomial, and Reshetikhin-Turaev | none | — |
03.17.01 | Homology and cohomology — the Eilenberg-Steenrod axioms and de Rham | none | — |
Algebraic geometry
142 units · full: 0 · partial: 58 · none: 84 · 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 | — |
04.03.08 | Étale cohomology and -adic cohomology of varieties | none | — |
04.03.10 | Triangulated category — Verdier axioms TR1-TR4 and the octahedral axiom | none | — |
04.03.11 | Derived category — localisation at quasi-isomorphisms | none | — |
04.03.12 | Derived functors and via derived categories | none | — |
04.03.13 | Grothendieck spectral sequence | none | — |
04.03.14 | Spectral sequence of a filtered complex | none | — |
04.03.15 | Sheaf cohomology - Leray spectral sequence (general form) | none | — |
04.03.16 | Six-functor formalism — adjunctions and base change | none | — |
04.03.17 | Derived tensor product ⊗^L and Tor in derived categories | none | — |
04.03.18 | t-Structure on a triangulated category — heart and truncations | none | — |
04.03.19 | Perverse sheaves Perv(X) — pointer + foundations | none | — |
04.03.20 | Hochschild homology and cohomology | none | — |
04.03.21 | Hochschild-Kostant-Rosenberg theorem | none | — |
04.03.22 | Cyclic homology and Connes' long exact sequence | none | — |
04.03.23 | The Verdier quotient of a triangulated category | none | — |
04.03.E1 | Cohomology of schemes exercise pack (Hartshorne Ch. III supplement) | 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 | — |
04.04.04 | Castelnuovo's Genus Bound for Space Curves and Extremal 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 | — |
04.04.14 | The Enriques-Babbage-Petri Theorem on Canonical Curves | none | — |
04.04.15 | Fulton-Lazarsfeld connectedness theorem | none | — |
04.04.16 | Lazarsfeld's K3-vector-bundle proof of Petri | none | — |
04.04.E1 | Curves exercise pack (Hartshorne Ch. IV supplement) | 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 | — |
04.05.11 | Worked Hirzebruch-Riemann-Roch computations | none | — |
04.05.12 | Pointer: Grothendieck-Riemann-Roch (GRR) | none | — |
04.05.E1 | Surfaces exercise pack (Hartshorne Ch. V supplement) | 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 |
04.07.03 | Monads on projective space and the Beilinson resolution | none | — |
04.07.04 | Stable rank-2 bundles on projective space and Barth's theorem | none | — |
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 |
04.09.03 | Serre's GAGA comparison theorem | none | — |
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 |
04.10.07 | Linear algebraic groups, reductivity, and finite generation of invariants | none | — |
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 |
04.10.11 | Gieseker stability and moduli of sheaves | none | — |
04.10.12 | Bridgeland stability conditions | none | — |
04.10.13 | K-stability and the Yau-Tian-Donaldson conjecture | none | — |
04.10.14 | Non-reductive GIT | none | — |
04.10.15 | Derived GIT and magic windows | none | — |
04.10.16 | Abelian varieties: group law, polarizations, and the dual | none | — |
04.10.20 | Deformation theory of smooth curves | none | — |
04.10.22 | Stable curve and Deligne-Mumford stability | none | — |
04.10.26 | Forgetful and gluing morphisms on | none | — |
04.10.29 | Limit linear series (Eisenbud-Harris) | none | — |
04.10.30 | Hurwitz numbers and the Hurwitz scheme | none | — |
04.10.31 | Severi varieties of nodal plane curves and the Harris irreducibility theorem | none | — |
04.10.32 | ELSV formula: Hurwitz numbers as Hodge integrals | none | — |
04.10.33 | Explicit low-genus moduli: Igusa-Clebsch invariants for M_2 and the plane-quartic model of M_3 | none | — |
04.10.34 | Torelli morphism and Torelli theorem | none | — |
04.10.35 | Moduli of stable maps and Gromov-Witten invariants | none | — |
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 | — |
04.11.15 | The Cox homogeneous coordinate ring and the toric GIT quotient | 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 |
04.12.14 | Logarithmic structures and log smooth morphisms | none | — |
04.12.15 | Log Gromov-Witten Invariants (pointer) | partial | Codex.AlgGeom.Tropical.LogGromovWitten |
04.12.16 | The A-model, the B-model, and the mirror symmetry conjecture | none | — |
04.12.17 | Special Lagrangian fibrations and McLean's theorem | none | — |
04.15.01 | Étale cohomology — the Weil conjectures and ℓ-adic methods | none | — |
04.16.01 | Abelian varieties — group objects in algebraic geometry | none | — |
04.17.01 | Birational geometry and the minimal model program | none | — |
04.18.01 | Intersection theory — Chow rings and enumerative geometry | none | — |
04.18.02 | Schubert calculus and the cohomology of Grassmannians | none | — |
04.19.01 | Deformation theory — infinitesimal families and moduli | none | — |
04.19.02 | Cotangent complex and deformation-obstruction theory | none | — |
Symplectic geometry
98 units · full: 0 · partial: 0 · none: 98 · 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 | — |
05.00.05 | The charged particle and the twisted symplectic form | 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 | — |
05.00.E1 | Lagrangian and variational mechanics exercise pack (Arnold Part II supplement) | 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 | — |
05.02.12 | Hamiltonian monodromy and the spherical pendulum | none | — |
05.02.E1 | Hamiltonian mechanics and canonical transformations exercise pack (Arnold Part III supplement) | none | — |
05.03.01 | Coadjoint orbit | none | — |
05.03.02 | Souriau Gibbs state on a symplectic G-space | none | — |
05.03.03 | Classification of homogeneous symplectic manifolds (Kirillov-Kostant-Souriau) | 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 | — |
05.05.05 | Jet bundle and total derivative | none | — |
05.05.06 | Prolongation of vector fields and the infinitesimal symmetry criterion | none | — |
05.05.07 | Group-invariant solutions and symmetry reduction | none | — |
05.05.08 | Noether's second theorem and the Bianchi identity | none | — |
05.05.09 | Generalised symmetries (Lie-Bäcklund) and recursion operators | none | — |
05.05.10 | Lie's classification of second-order ODEs and the symmetry algorithm for ODEs | none | — |
05.05.11 | Differential invariants and the moving-frame method | 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 | — |
05.07.04 | Eliashberg-Gromov -rigidity of | 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 | — |
05.09.08 | Infinite-dimensional Poisson manifolds and Hamiltonian evolution equations | none | — |
05.09.09 | Finite-gap integration and theta-function solutions | none | — |
05.09.10 | KP hierarchy, Sato Grassmannian, and tau-functions | none | — |
05.09.11 | Master symmetries and the Fuchssteiner construction | none | — |
05.09.12 | Casimir functions of degenerate Poisson structures | none | — |
05.09.E1 | Symplectic geometry and integrable systems exercise pack (Arnold Part III appendices supplement) | 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 | — |
05.11.01 | Prequantum line bundle and the integrality condition | none | — |
05.11.02 | Prequantisation of the spin coadjoint orbit | none | — |
05.11.03 | Polarisation, half-densities, and metaplectic correction | none | — |
05.11.04 | The Groenewold–van Hove no-go theorem | none | — |
05.11.05 | Geometric quantization of coadjoint orbits and the Borel–Weil theorem | none | — |
05.11.09 | Quantization of the relativistic particle | none | — |
05.12.01 | Lagrangian Grassmannian and the universal Maslov class | none | — |
05.12.03 | Legendrian singularities and wave-front evolution | none | — |
05.12.04 | Lagrangian and Legendrian cobordism | none | — |
05.14.01 | Helicity as a Casimir invariant of the ideal fluid | none | — |
05.14.02 | Helicity as Asymptotic Linking Number (Arnold's Theorem) | none | — |
05.14.03 | The Hopf invariant and the vortex-unknotting obstruction | none | — |
05.14.04 | Ideal magnetohydrodynamics: frozen flux and magnetic helicity | none | — |
05.14.05 | Arnold's energy-Casimir stability theorem | none | — |
05.14.06 | KdV and Camassa-Holm as geodesics on the Bott-Virasoro group | none | — |
05.14.07 | Beltrami fields, ABC flows, and chaotic streamlines | none | — |
05.14.08 | Fast dynamo problem and Arnold cat-map dynamo | none | — |
05.15.01 | Wasserstein metric and Otto's formal Riemannian calculus | none | — |
05.15.02 | Korteweg / Madelung quantum hydrodynamics | none | — |
Statistical field theory
63 units · full: 0 · partial: 1 · none: 62 · 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 | — |
08.04.05 | Momentum-shell (Wilson) renormalization group | 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 | — |
08.08.04 | The roughening transition and the confining string | none | — |
08.09.01 | Quantum-classical correspondence (Wick rotation) | none | — |
08.10.01 | Bosonic Fock space and second quantisation | none | — |
08.10.02 | Fokker-Planck equation and equilibrium distribution | none | — |
08.10.03 | theory and the Dyson series | partial | Codex.StatMech.QFT.PhiFourDyson |
08.10.04 | Wick's theorem for operator products | none | — |
08.10.05 | Feynman propagator and the contour-integral representation | none | — |
08.10.06 | One-loop renormalisation in | none | — |
08.10.07 | Wightman axioms (W1–W7) | none | — |
08.10.08 | Langevin updates and lattice numerics | none | — |
08.10.09 | Fermionic Fock space, Pauli exclusion, anticommutators | none | — |
08.10.10 | Dirac field and the Dirac adjoint | none | — |
08.10.11 | Supersymmetric quantum mechanics: superpotential, supercharges, and the Witten index | none | — |
08.10.12 | The Nicolai map and stochastic quantisation of supersymmetric theories | none | — |
08.10.13 | Parisi-Sourlas dimensional reduction and random-field supersymmetry | none | — |
08.10.14 | The Martin-Siggia-Rose / Janssen-De Dominicis response-field formalism | none | — |
08.10.15 | Stochastic perturbation theory and the tree expansion | none | — |
08.11.02 | Debye theory of specific heats of solids | none | — |
08.11.03 | Real gases — virial expansion and van der Waals | none | — |
08.11.04 | Real gases — the virial expansion and van der Waals | none | — |
08.12.01 | Fluctuation-dissipation theorem (Landau-Callen-Welton) | none | — |
08.12.02 | Equilibrium fluctuations of thermodynamic quantities | none | — |
08.12.03 | Jarzynski equality and Crooks fluctuation theorem | none | — |
08.13.01 | The Yang–Baxter equation and the star–triangle relation | none | — |
08.13.02 | The six-vertex (ice-type) model and the Bethe ansatz | none | — |
08.13.03 | The eight-vertex model (Baxter 1971) | none | — |
08.13.04 | The corner transfer matrix | none | — |
08.13.05 | The hard-hexagon model (Baxter 1980) | none | — |
08.13.07 | The spherical model (Berlin-Kac) | none | — |
08.13.08 | The Ising model on the Bethe lattice | none | — |
08.14.01 | Brownian motion, the Wiener measure, and the path integral | none | — |
08.14.02 | Grassmann integration and the 2D Ising model as free fermions | none | — |
08.14.03 | The large-N limit | none | — |
08.14.04 | Lattice fermions and the doubling problem | none | — |
08.14.05 | The Pfaffian and the dimer model | none | — |
08.14.06 | Pointer: matrix models and the topological expansion | none | — |
08.14.07 | The Kardar-Parisi-Zhang equation and dynamic scaling | none | — |
08.14.08 | Liouville field theory and 2D quantum gravity (KPZ-DDK scaling) | none | — |
08.15.01 | The Kosterlitz-Thouless transition (2D XY model) | none | — |
08.15.02 | The nonlinear σ-model and the O(n) renormalization group | none | — |
08.15.03 | Topological defects in ordered media | none | — |
Riemann surfaces
100 units · full: 0 · partial: 17 · none: 83 · stub: 0
| id | title | status | module |
|---|---|---|---|
06.01.01 | Holomorphic function | partial | Codex.RiemannSurfaces.ComplexAnalysis.Holomorphic |
06.01.02 | Cauchy integral formula | none | — |
06.01.03 | Residue theorem | none | — |
06.01.04 | Analytic continuation | none | — |
06.01.05 | Meromorphic function | none | — |
06.01.06 | Riemann mapping theorem | none | — |
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 | — |
06.01.31 | Jacobi theta functions and the triple product | none | — |
06.01.E1 | Complex analysis exercise pack I (Ahlfors Ch. 1-4 supplement) | none | — |
06.01.E2 | Complex analysis exercise pack II (Ahlfors Ch. 5-8 supplement) | 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 |
06.03.04 | Uniformization via constant-curvature conformal metrics and Ricci flow on surfaces | none | — |
06.03.05 | The prescribed-Gaussian-curvature equation on a surface (Kazdan-Warner) | none | — |
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 | — |
06.06.09 | Weierstrass points and gap sequences | 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 | — |
06.10.01 | Domains of holomorphy and holomorphic convexity | none | — |
06.10.02 | Plurisubharmonic functions | none | — |
06.10.03 | Pseudoconvexity and the Levi form | none | — |
06.10.04 | The ∂̄-equation and Hörmander's L² estimates | none | — |
06.10.05 | Solution of the Levi problem | none | — |
06.10.06 | Bochner-Martinelli kernel and formula | none | — |
06.10.07 | Cauchy-Fantappiè and Henkin-Ramirez kernels | none | — |
06.10.08 | Bergman kernel and Bergman metric | none | — |
06.10.09 | Szegő kernel and Fefferman boundary asymptotics | none | — |
06.10.10 | The ∂̄-Neumann problem and subelliptic estimates | none | — |
06.10.11 | Cousin I/II and the Levi problem in | none | — |
06.10.12 | Invariant metrics: Carathéodory, Kobayashi, Bergman | none | — |
06.10.13 | Automorphism groups and the Fefferman mapping theorem | none | — |
06.10.14 | Weierstrass preparation and division | none | — |
06.10.15 | Tangential CR complex, ∂̄_b, and the Lewy example | none | — |
06.10.16 | Wong-Rosay theorem and boundary rigidity | none | — |
06.10.17 | Local analytic Nullstellensatz and the ideal–germ correspondence | none | — |
06.10.18 | Analytic sets: local parametrisation, dimension, and irreducible components | none | — |
06.10.19 | The local ring of an analytic set; regular points, singular locus, Remmert–Stein | none | — |
06.10.20 | Coherent analytic sheaves and Oka's coherence theorem | none | — |
06.10.21 | Cartan Theorems A and B in (with proof) | none | — |
06.10.22 | Complex spaces and coherence on them | none | — |
06.10.E1 | Several complex variables exercise pack (Krantz supplement) | none | — |
06.11.01 | Ideal boundary and exhaustions of an open Riemann surface | none | — |
06.11.02 | Hilbert space of differentials; orthogonal decomposition on an open surface | none | — |
06.11.03 | Green's function on a Riemann surface and the type problem (parabolic vs. hyperbolic) | none | — |
06.11.04 | Null-classes O_G, O_HB, O_HD, O_AD and the classification of open surfaces | none | — |
06.11.05 | Capacity and harmonic measure of the ideal boundary | none | — |
06.11.06 | Extremal length and the modulus of curve families | none | — |
Representation theory
91 units · full: 0 · partial: 26 · none: 65 · 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 | — |
07.01.13 | The irreducible representations of GL₂(𝔽_q) | none | — |
07.02.01 | Maschke's theorem | partial | Codex.RepresentationTheory.Character.Maschke |
07.02.02 | The Fong-Swan theorem | none | — |
07.02.03 | Grothendieck groups and the cde-triangle | none | — |
07.02.04 | Brauer character | none | — |
07.02.06 | Block theory of kG | none | — |
07.02.E1 | Finite-group representation exercise pack (Serre Linear Representations supplement) | 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 | — |
07.04.05 | Real forms of a complex semisimple Lie algebra | 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 | — |
07.04.11 | Invariant differential operators on G/K and the Harish-Chandra isomorphism | none | — |
07.04.12 | Spherical function on G/K | none | — |
07.04.13 | Classification tables of irreducible Riemannian symmetric spaces (Cartan's list) | 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 | — |
07.05.06 | Association schemes, the Bose-Mesner algebra, and Krawtchouk/Hahn polynomials | 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 | — |
07.05.15 | The Bernoulli-Laplace and Ehrenfest urn diffusion models | none | — |
07.05.16 | Wreath products and the representations of the hyperoctahedral group | none | — |
07.05.E1 | Lie-group and Lie-algebra representation exercise pack (Fulton-Harris supplement) | 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 | — |
07.06.13 | Free Lie algebras, the Hall basis, and Magnus's theorem | none | — |
07.06.14 | Engel's theorem + Lie's theorem | none | — |
07.06.15 | The Campbell–Baker–Hausdorff formula | 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 | The Killing form and the trace form | none | — |
07.06.22 | Weyl complete-reducibility theorem | none | — |
07.06.23 | Lie algebra cohomology and Whitehead's lemmas | none | — |
07.06.24 | The Hochschild-Serre spectral sequence for a Lie-algebra ideal | none | — |
07.06.25 | Weyl construction of the classical-group irreducibles | none | — |
07.06.26 | via the octonions | none | — |
07.06.27 | Lie superalgebras: the graded bracket, the super-Jacobi identity, and basic classification | none | — |
07.06.28 | Supermanifolds as ringed spaces, the functor of points, and super Lie groups | none | — |
07.06.29 | Composition algebras and the octonions | none | — |
07.06.E2 | Lie algebra structure exercise pack (Serre Lie Algebras and Lie Groups supplement) | 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 | — |
07.07.05 | Representations of SU(2) and SO(3): the double cover, spin, and projective representations | none | — |
07.07.06 | Wigner's classification of the unitary irreducible representations of the Poincaré group | none | — |
07.07.07 | Mackey theory of induced representations and systems of imprimitivity | none | — |
07.07.08 | Crystallographic point groups, space groups, and the crystallographic restriction theorem | none | — |
07.07.09 | Representations of the Lorentz group: , the reps, and Wigner's theorem | none | — |
07.07.10 | The unitary dual of : principal, discrete, and complementary series | none | — |
07.08.01 | Representations of finite groups — Maschke, characters, and orthogonality | none | — |
Logic
8 units · full: 0 · partial: 0 · none: 8 · stub: 0
| id | title | status | module |
|---|---|---|---|
49.01.01 | Propositional logic and truth tables | none | — |
49.02.01 | Predicate logic and quantifiers | none | — |
49.03.01 | Informal fallacies and argument analysis | none | — |
49.04.01 | Deductive reasoning and syllogisms | none | — |
49.05.01 | Inductive reasoning, analogy, and causation | none | — |
49.06.01 | Decision theory and Bayesian reasoning | none | — |
49.07.01 | Cognitive biases and rationality | none | — |
49.08.01 | Critical thinking in media, science, and everyday life | none | — |
Mathematical logic
39 units · full: 0 · partial: 0 · none: 39 · stub: 0
| id | title | status | module |
|---|---|---|---|
42.01.01 | Propositional Logic as a Formal System | none | — |
42.01.02 | The Compactness Theorem for Propositional Logic | none | — |
42.01.03 | First-Order Languages: Syntax and Unique Readability | none | — |
42.01.04 | Structures and Tarski's Definition of Truth | none | — |
42.01.05 | A Deductive Calculus for First-Order Logic and Soundness | none | — |
42.01.06 | Gödel's Completeness Theorem and the Henkin Construction | none | — |
42.01.07 | Compactness and the Löwenheim-Skolem Theorems for First-Order Logic | none | — |
42.01.08 | Representability of Recursive Functions in Arithmetic | none | — |
42.01.09 | Gödel Numbering, the Fixed-Point Lemma, and the Incompleteness Theorems | none | — |
42.01.10 | The Entscheidungsproblem, Church's Theorem, and Decidable Theories | none | — |
42.02.01 | Structures, Embeddings, and Elementary Equivalence | none | — |
42.02.02 | The Compactness Theorem and the Method of Diagrams | none | — |
42.02.03 | Types and the Omitting Types Theorem | none | — |
42.02.04 | Saturation, Homogeneity, and Monster Models | none | — |
42.02.05 | Quantifier Elimination and Model-Completeness | none | — |
42.02.06 | Categoricity: Ryll-Nardzewski, Morley, and Baldwin-Lachlan | none | — |
42.02.07 | Strongly Minimal Sets, Morley Rank, and Stability | none | — |
42.02.08 | O-Minimality and the Cell Decomposition Theorem | none | — |
42.02.09 | Indiscernibles and Ehrenfeucht-Mostowski Models | none | — |
42.03.01 | The ZFC Axioms and the Cumulative Hierarchy | none | — |
42.03.02 | Ordinals, Transfinite Induction, and Recursion | none | — |
42.03.03 | Cardinals and the Arithmetic of the Infinite | none | — |
42.03.04 | Cofinality, Cardinal Exponentiation, and the Singular Cardinals Hypothesis | none | — |
42.03.05 | The Axiom of Choice and Its Equivalents | none | — |
42.03.06 | The Constructible Universe L and the Consistency of GCH | none | — |
42.03.07 | Forcing I: Posets, Generic Filters, Names, and the Fundamental Theorem | none | — |
42.03.08 | Forcing II: Cohen and the Independence of the Continuum Hypothesis | none | — |
42.03.09 | Martin's Axiom, Iterated Forcing, and Large Cardinals | none | — |
42.03.10 | Club Sets, Stationary Sets, and Fodor's Lemma | none | — |
42.04.01 | Models of Computation and the Church-Turing Thesis | none | — |
42.04.02 | The Halting Problem, Undecidability, and the Recursion Theorem | none | — |
42.04.03 | Computably Enumerable Sets: Creative and Simple Sets | none | — |
42.04.04 | Turing Reducibility, Oracles, and the Jump | none | — |
42.04.05 | The Arithmetical Hierarchy and Post's Theorem | none | — |
42.04.06 | The Turing Degrees and the Priority Method | none | — |
42.04.07 | Unsolvable Problems: the Word Problem and Hilbert's Tenth | none | — |
42.04.08 | Kolmogorov Complexity and Algorithmic Randomness | none | — |
42.05.01 | Sequent Calculus, Cut-Elimination, and the Consistency of Arithmetic | none | — |
42.05.02 | Proof theory — cut-elimination, Gentzen, and ordinal analysis | none | — |
Category theory
17 units · full: 0 · partial: 0 · none: 17 · stub: 0
| id | title | status | module |
|---|---|---|---|
41.01.01 | Categories, Functors, and the Duality Principle | none | — |
41.01.02 | Natural Transformations, Functor Categories, and Equivalence of Categories | none | — |
41.02.01 | Limits and Colimits as Universal Cones | none | — |
41.02.02 | Constructing Limits: Products, Equalizers, Preservation, and Filtered Colimits | none | — |
41.02.03 | Limits — creation, preservation, completeness, and presentability | none | — |
41.03.01 | Adjunctions: Hom-Set and Unit-Counit Definitions | none | — |
41.03.02 | RAPL, Reflective Subcategories, and the Adjoint Functor Theorems | none | — |
41.03.03 | Adjunctions — the unit-counit calculus and the adjoint functor theorems | none | — |
41.04.01 | Representable Functors and Universal Elements | none | — |
41.04.02 | The Yoneda Lemma, the Yoneda Embedding, and Density | none | — |
41.04.03 | Yoneda, representability, and density — the depth | none | — |
41.05.01 | Monads, Eilenberg-Moore Algebras, and the Kleisli Category | none | — |
41.05.02 | Beck's Monadicity Theorem and Lawvere Theories | none | — |
41.06.01 | Ends, Coends, and the Calculus of (Co)ends | none | — |
41.06.02 | Kan Extensions: All Concepts Are Kan Extensions | none | — |
41.07.01 | Monoidal Categories and Mac Lane's Coherence Theorem | none | — |
41.07.02 | Braided, Symmetric, and Ribbon Monoidal Categories | none | — |
Statistics
10 units · full: 0 · partial: 0 · none: 10 · stub: 0
| id | title | status | module |
|---|---|---|---|
26.01.01 | Descriptive statistics: central tendency and variability | none | — |
26.02.01 | Probability theory: rules and distributions | none | — |
26.03.01 | Random variables and expected value | none | — |
26.04.01 | Sampling distributions and the Central Limit Theorem | none | — |
26.05.01 | Hypothesis testing, p-values, and confidence intervals | none | — |
26.06.01 | Correlation and regression analysis | none | — |
26.07.01 | Bayesian statistics: prior and posterior | none | — |
26.08.01 | Nonparametric methods and resampling | none | — |
26.09.01 | Experimental design and ANOVA | none | — |
26.10.01 | Statistical literacy, misuse, and data ethics | none | — |
Numerical analysis & PDE
22 units · full: 0 · partial: 0 · none: 22 · stub: 0
| id | title | status | module |
|---|---|---|---|
24.01.00 | Numerical-PDE chapter README and notation crosswalk | none | — |
24.01.01 | Sobolev spaces and | none | — |
24.01.02 | Sobolev spaces of differential forms | none | — |
24.01.03 | Weak / variational formulation of elliptic PDE | none | — |
24.01.04 | Babuška-Brezzi (inf-sup) condition for saddle-point problems | none | — |
24.02.01 | Classical conforming FEM — Galerkin, Céa, Bramble-Hilbert | none | — |
24.02.02 | Mixed FEM for the Poisson equation (Raviart-Thomas) | none | — |
24.03.01 | Whitney forms | none | — |
24.03.02 | Nédélec first-kind edge elements and | none | — |
24.03.03 | Polynomial differential form spaces and | none | — |
24.03.04 | Discrete de Rham complex and the FEEC subcomplex axiom | none | — |
24.03.05 | Bounded cochain projection and the commuting diagram | none | — |
24.03.06 | FEEC convergence theorem (Arnold-Falk-Winther) | none | — |
24.03.07 | Abstract Hilbert complexes, the abstract Hodge decomposition, and abstract Galerkin stability | none | — |
24.04.01 | Mixed FEM for the Hodge Laplacian | none | — |
24.04.02 | Maxwell equations and FEEC edge elements | none | — |
24.04.03 | Linearised elasticity via AFW symmetric-tensor mixed elements | none | — |
24.04.04 | Smooth FEEC pointer (Falk-Neilan) | none | — |
24.04.05 | Isogeometric exterior calculus pointer | none | — |
24.04.06 | Virtual element exterior calculus pointer | none | — |
24.04.07 | Eigenvalue approximation and discrete compactness in FEEC | none | — |
24.04.E1 | Finite element exterior calculus exercise pack (Arnold-Falk-Winther supplement) | none | — |
Dynamical systems & ergodic theory
24 units · full: 0 · partial: 0 · none: 24 · stub: 0
| id | title | status | module |
|---|---|---|---|
38.01.01 | Dynamical Systems, Orbits, and Limit Sets | none | — |
38.01.02 | Minimality and Recurrence | none | — |
38.01.03 | Topological Transitivity, Topological Mixing, and Devaney Chaos | none | — |
38.01.04 | Circle Rotations and Unique Ergodicity | none | — |
38.01.05 | The Poincaré Rotation Number and Denjoy's Theorem | none | — |
38.02.01 | Shift Spaces and Subshifts | none | — |
38.02.02 | Shifts of Finite Type, Transition Matrices, and Coding | none | — |
38.02.03 | Perron-Frobenius Theory, SFT Growth Rate, and Subshift Entropy | none | — |
38.03.01 | Hyperbolic Sets, Anosov and Axiom-A Systems, and the Smale Spectral Decomposition | none | — |
38.03.02 | The Smale Horseshoe and the Smale-Birkhoff Homoclinic Theorem | none | — |
38.03.03 | The Hadamard-Perron Stable and Unstable Manifold Theorem | none | — |
38.03.04 | Shadowing and Structural Stability | none | — |
38.04.01 | Measure-Preserving Systems, Poincaré Recurrence, and the Kac Formula | none | — |
38.04.02 | Ergodicity, Unique Ergodicity, and Equidistribution | none | — |
38.05.01 | The Mixing Hierarchy: Mixing, Weak Mixing, and Ergodicity | none | — |
38.05.02 | Spectral Theory of Dynamical Systems and the Halmos-von Neumann Theorem | none | — |
38.06.01 | Topological Entropy | none | — |
38.06.02 | Kolmogorov-Sinai Entropy and the Generator Theorem | none | — |
38.06.03 | The Shannon-McMillan-Breiman Theorem | none | — |
38.06.04 | Topological Pressure, the Variational Principle, and Equilibrium States | none | — |
38.07.01 | The Oseledets Multiplicative Ergodic Theorem and Lyapunov Exponents | none | — |
38.07.02 | Pesin Theory and the Entropy Formula | none | — |
38.07.03 | The Hopf Argument for Ergodicity of Geodesic and Anosov Flows | none | — |
38.07.04 | The Livšic Cohomological Rigidity Theorem | none | — |
Operator algebras & NCG
34 units · full: 0 · partial: 0 · none: 34 · stub: 0
| id | title | status | module |
|---|---|---|---|
39.01.01 | C*-Algebras: Axioms, Spectrum, and the Continuous Functional Calculus | none | — |
39.01.02 | Commutative C*-Algebras and Gelfand Duality | none | — |
39.01.03 | States, the GNS Construction, and the Gelfand-Naimark Representation Theorem | none | — |
39.01.04 | The Toeplitz Algebra, Cuntz Algebras, and Extensions | none | — |
39.02.01 | AF Algebras, Bratteli Diagrams, and the Irrational Rotation Algebra | none | — |
39.02.02 | Operator K-Theory: K_0 and K_1 of C*-Algebras | none | — |
39.02.03 | The Six-Term Exact Sequence, Bott Periodicity, and AF Classification | none | — |
39.03.01 | Von Neumann Algebras and the Bicommutant Theorem | none | — |
39.03.02 | The Predual, Normal States, and the σ-Weak Topology | none | — |
39.03.03 | The Kaplansky Density Theorem | none | — |
39.03.04 | Comparison of Projections and the Murray-von Neumann Type Classification | none | — |
39.03.05 | Traces, Continuous Dimension, and the II_1 Factor | none | — |
39.04.01 | Cyclic and Separating Vectors and the Standard Form | none | — |
39.04.02 | Tomita's Theorem: the Modular Operator and Modular Conjugation | none | — |
39.04.03 | The Modular Automorphism Group and the KMS Condition | none | — |
39.04.04 | The Connes Classification of Type III Factors | none | — |
39.05.01 | Completely Positive Maps and the Stinespring Dilation Theorem | none | — |
39.05.02 | Operator Systems, Arveson's Extension Theorem, and the Choi-Effros Theorem | none | — |
39.05.03 | Tensor Products of C*-Algebras: the Minimal and Maximal Norms | none | — |
39.05.04 | Nuclear C*-Algebras and the Completely Positive Approximation Property | none | — |
39.05.05 | Exact C*-Algebras and Nuclear Embeddability | none | — |
39.05.06 | Amenable Groups: Invariant Means, the Følner Condition, and Paradoxical Decompositions | none | — |
39.05.07 | Group C*-Algebras: Amenability and Nuclearity | none | — |
39.05.08 | Quasidiagonality | none | — |
39.05.09 | Group Approximation Properties and the Connes Embedding Problem | none | — |
39.05.10 | Exact Groups, Amenable Actions, and Property A | none | — |
39.06.01 | Spectral Triples and the Reconstruction Theorem | none | — |
39.06.02 | The Connes Distance Formula | none | — |
39.06.03 | Fredholm Modules and the K-Theory/K-Homology Index Pairing | none | — |
39.06.04 | The Noncommutative Torus and Its Geometry | none | — |
39.06.05 | The Dixmier Trace and the Noncommutative Integral | none | — |
39.06.06 | The Connes-Moscovici Local Index Formula | none | — |
39.07.01 | Cyclic Cohomology and the Pairing with K-Theory | none | — |
39.07.02 | The Chern Character in K-Homology | none | — |
Combinatorics & graph theory
59 units · full: 0 · partial: 0 · none: 59 · stub: 0
| id | title | status | module |
|---|---|---|---|
40.01.01 | Basic Counting and the Twelvefold Way | none | — |
40.01.02 | Inclusion-Exclusion and the Sieve | none | — |
40.01.03 | Generating Functions: Ordinary, Exponential, and the Exponential Formula | none | — |
40.01.04 | Rational Generating Functions, the Transfer-Matrix Method, and P-Partitions | none | — |
40.01.05 | Permutation Statistics: Descents, the Major Index, and Eulerian Polynomials | none | — |
40.01.06 | q-Analogues, Gaussian Binomial Coefficients, and the Combinatorics of Partitions | none | — |
40.01.07 | Trees, Cayley's Formula, the Matrix-Tree Theorem, and Lagrange Inversion | none | — |
40.02.01 | Posets, Lattices, and Birkhoff's Representation Theorem | none | — |
40.02.02 | The Incidence Algebra, the Möbius Function of a Poset, and Möbius Inversion | none | — |
40.02.03 | Eulerian Posets, Face Lattices, and the Characteristic Polynomial | none | — |
40.03.01 | The Ring of Symmetric Functions and Its Bases | none | — |
40.03.02 | Schur Functions: the Combinatorial Definition and the Jacobi-Trudi Determinant | none | — |
40.03.03 | The Cauchy Identity, Dual Bases, and the Hall Inner Product | none | — |
40.03.04 | The Robinson-Schensted-Knuth Correspondence | none | — |
40.03.05 | The Littlewood-Richardson Rule, Skew Schur Functions, and Jeu de Taquin | none | — |
40.03.06 | The Frobenius Characteristic Map and the Symmetric-Function Dictionary | none | — |
40.03.07 | Plane Partitions and the MacMahon Box Formula | none | — |
40.03.08 | Quasisymmetric Functions and Gessel's Fundamental Basis | none | — |
40.04.01 | Graphs, Basic Invariants, and the Foundational Lemmas | none | — |
40.04.02 | Matchings I: König's Theorem and Hall's Marriage Theorem | none | — |
40.04.03 | Matchings II: Tutte's 1-Factor Theorem and the Tutte-Berge Formula | none | — |
40.04.04 | Connectivity and Menger's Theorem | none | — |
40.04.05 | Planar Graphs: Euler's Formula, Kuratowski's and Wagner's Theorems | none | — |
40.04.06 | Vertex Colouring: Brooks' Theorem and the Chromatic Polynomial | none | — |
40.04.07 | Edge Colouring and List Colouring: Vizing's Theorem and Choosability | none | — |
40.04.08 | Map Colouring: the Five-Colour Theorem and the Four-Colour Theorem | none | — |
40.04.09 | Network Flows: Max-Flow Min-Cut and Nowhere-Zero Flows | none | — |
40.04.10 | Graph Minors and the Robertson-Seymour Theorem | none | — |
40.04.11 | Hamilton Cycles: Dirac, Ore, and Chvátal-Erdős | none | — |
40.05.01 | Extremal Graph Theory: Turán's Theorem and Erdős-Stone-Simonovits | none | — |
40.05.02 | Bipartite Extremal Problems: the Kővári-Sós-Turán Theorem | none | — |
40.05.03 | The Szemerédi Regularity Lemma and the Triangle Removal Lemma | none | — |
40.05.04 | Ramsey's Theorem and Ramsey Numbers | none | — |
40.06.01 | Balanced Incomplete Block Designs and Fisher's Inequality | none | — |
40.06.02 | Symmetric Designs and the Bruck-Ryser-Chowla Theorem | none | — |
40.06.03 | Finite Projective and Affine Planes, MOLS, and the 36-Officers Problem | none | — |
40.06.04 | Steiner Systems and the Existence of Steiner Triple Systems | none | — |
40.06.05 | Hadamard Matrices and the Paley Construction | none | — |
40.06.06 | Linear Codes and the Hamming, Singleton, and Gilbert-Varshamov Bounds | none | — |
40.06.07 | Perfect Codes: the Hamming and Golay Codes | none | — |
40.06.08 | Cyclic Codes: BCH, Reed-Solomon, Reed-Muller, and Assmus-Mattson | none | — |
40.06.09 | Strongly Regular Graphs and the Design-Graph Dictionary | none | — |
40.06.10 | Pólya-Redfield Enumeration and the Cycle Index | none | — |
40.07.01 | The Probabilistic Method: First-Moment and Counting Arguments | none | — |
40.07.02 | Linearity of Expectation and the Deletion Method | none | — |
40.07.03 | The Second-Moment Method and Thresholds for Random Graphs | none | — |
40.07.04 | The Lovász Local Lemma and the Moser-Tardos Algorithm | none | — |
40.07.05 | Concentration for Combinatorial Functionals: Azuma and the Bounded-Differences Method | none | — |
40.07.06 | The Entropy Method and Shearer's Lemma | none | — |
40.07.07 | Correlation Inequalities: FKG, Harris, and the Janson Inequalities | none | — |
40.07.08 | Combinatorial Discrepancy: Spencer's Theorem and the Beck-Fiala Bound | none | — |
40.07.09 | The Rödl Nibble and the Semi-Random Method | none | — |
40.08.01 | The Symbolic Method for Unlabelled Structures | none | — |
40.08.02 | The Symbolic Method for Labelled Structures | none | — |
40.08.03 | Meromorphic Coefficient Asymptotics | none | — |
40.08.04 | Singularity Analysis and the Transfer Theorems | none | — |
40.08.05 | Asymptotics of Tree Families and Simple Varieties of Trees | none | — |
40.08.06 | The Saddle-Point Method for Asymptotic Enumeration | none | — |
40.08.07 | Limit Laws and the Quasi-Powers Theorem | none | — |
Numerical analysis & scientific computing
45 units · full: 0 · partial: 0 · none: 45 · stub: 0
| id | title | status | module |
|---|---|---|---|
43.01.01 | Floating-point arithmetic and the IEEE model | none | — |
43.01.02 | Conditioning and condition numbers of problems | none | — |
43.01.03 | Backward stability and backward-error analysis of algorithms | none | — |
43.02.01 | The Bisection Method and the Scalar Root-Finding Problem | none | — |
43.02.02 | Fixed-Point Iteration, Contraction Convergence, and Order of Convergence | none | — |
43.02.03 | Newton's Method and the Secant Method: Superlinear and Quadratic Convergence | none | — |
43.03.01 | Gaussian elimination, LU factorization, and its stability | none | — |
43.03.02 | Cholesky factorization and the symmetric positive-definite solve | none | — |
43.03.03 | Perturbation theory and a posteriori error for linear systems | none | — |
43.03.08 | Symmetric-indefinite factorisation: the Bunch-Kaufman LDLᵀ algorithm | none | — |
43.04.01 | Least squares: normal equations vs QR vs SVD conditioning | none | — |
43.04.08 | Updating and downdating matrix factorisations | none | — |
43.05.08 | Total least squares and the generalised SVD | none | — |
43.05.09 | Randomized SVD and sketching | none | — |
43.06.01 | Power iteration, inverse iteration, and Rayleigh quotient iteration | none | — |
43.06.02 | Reduction to Hessenberg/tridiagonal form | none | — |
43.06.03 | The QR algorithm for eigenvalues, with shifts | none | — |
43.06.04 | Bauer-Fike and the conditioning of eigenvalues | none | — |
43.06.10 | The generalised eigenvalue problem Ax=λBx and the QZ algorithm | none | — |
43.06.11 | Computing matrix functions: the matrix exponential | none | — |
43.06.12 | Sylvester and Lyapunov matrix equations: the Bartels-Stewart algorithm | none | — |
43.07.01 | Stationary iterative methods: Jacobi, Gauss-Seidel, SOR | none | — |
43.07.02 | Arnoldi and Lanczos iterations | none | — |
43.07.03 | GMRES | none | — |
43.07.04 | The conjugate gradient method | none | — |
43.07.05 | Preconditioning | none | — |
43.08.01 | Polynomial interpolation: existence, uniqueness, and the Lagrange form | none | — |
43.08.02 | Interpolation error and the Runge phenomenon | none | — |
43.08.03 | Hermite interpolation and piecewise / cubic spline interpolation | none | — |
43.08.04 | Best uniform approximation, minimax, and Chebyshev polynomials | none | — |
43.08.05 | Least-squares approximation and orthogonal polynomials | none | — |
43.09.01 | Newton-Cotes rules and their error via the Peano kernel | none | — |
43.09.02 | Composite rules, Euler-Maclaurin, and Romberg / adaptive quadrature | none | — |
43.09.03 | Gauss quadrature via orthogonal polynomials | none | — |
43.10.01 | One-step methods: Euler, trapezoidal, Runge-Kutta; consistency and order | none | — |
43.10.02 | Linear multistep methods: Adams and BDF families, order via the characteristic polynomials | none | — |
43.10.03 | Zero-stability, the root condition, and the Dahlquist equivalence theorem | none | — |
43.10.04 | Absolute stability, stability regions, and the linear test equation | none | — |
43.10.05 | Stiff equations, A-stability, and the Dahlquist second barrier | none | — |
43.10.06 | Finite-difference methods for two-point boundary-value problems | none | — |
43.11.01 | Finite differences for the elliptic BVP: the 5-point Laplacian and its convergence | none | — |
43.11.02 | The method of lines and stability for parabolic problems | none | — |
43.11.03 | Von Neumann stability analysis and the CFL condition | none | — |
43.11.04 | Hyperbolic finite differences: upwind, Lax-Friedrichs, Lax-Wendroff; numerical diffusion and dispersion | none | — |
43.11.05 | The Lax-Richtmyer equivalence theorem for finite-difference schemes | none | — |
Optimization & control
47 units · full: 0 · partial: 0 · none: 47 · stub: 0
| id | title | status | module |
|---|---|---|---|
44.01.01 | Convex Sets, Convex Functions, and Convexity-Preserving Operations | none | — |
44.01.02 | Carathéodory's Theorem and the Relative Interior of Convex Sets | none | — |
44.01.03 | Recession Cones and Polyhedral Convexity (Minkowski-Weyl) | none | — |
44.01.04 | Quasiconvex Optimization and the Bisection Method | none | — |
44.01.05 | Multicriteria and Vector Optimization: Pareto Optimality and Scalarization | none | — |
44.02.01 | Lagrangian Duality, Weak and Strong Duality, and Slater's Condition | none | — |
44.02.02 | Karush-Kuhn-Tucker Conditions: Constraint Qualifications and Second-Order Conditions | none | — |
44.02.03 | Fenchel Duality and the Conjugate as a Modeling Tool | none | — |
44.02.04 | Farkas' Lemma and the Theorems of the Alternative | none | — |
44.02.05 | Subdifferential Calculus: Sum Rule, Chain Rule, and Danskin's Theorem | none | — |
44.03.01 | Line-Search Methods: Wolfe Conditions and Global Convergence | none | — |
44.03.02 | Trust-Region Methods: The Cauchy Point and Dogleg Step | none | — |
44.03.03 | Newton's Method for Optimization: Unconstrained and Equality-Constrained | none | — |
44.03.04 | Quasi-Newton Methods: BFGS, DFP, and SR1 | none | — |
44.03.05 | Limited-Memory L-BFGS and Inexact (Truncated) Newton-CG | none | — |
44.03.06 | Nonlinear Conjugate Gradient Methods: Fletcher-Reeves and Polak-Ribière | none | — |
44.03.07 | Gauss-Newton and Levenberg-Marquardt for Nonlinear Least Squares | none | — |
44.03.08 | Derivative Computation: Finite Differences and Automatic Differentiation | none | — |
44.03.09 | Newton and Quasi-Newton Methods for Nonlinear Systems: Broyden's Method | none | — |
44.03.10 | Derivative-Free Optimization: Nelder-Mead and Model-Based Trust-Region Methods | none | — |
44.04.01 | Active-Set Methods for Quadratic Programming | none | — |
44.04.02 | Penalty and Augmented-Lagrangian Methods | none | — |
44.04.03 | Sequential Quadratic Programming | none | — |
44.04.04 | Interior-Point Methods for Nonlinear Programming | none | — |
44.05.01 | Conic Optimization: Linear, Second-Order Cone, and Semidefinite Programs | none | — |
44.05.02 | Linear-Programming Duality and the Simplex Method | none | — |
44.05.03 | Geometric Programming | none | — |
44.05.04 | Interior-Point Methods: The Barrier Method, Central Path, and Primal-Dual Algorithms | none | — |
44.06.01 | Subgradient Methods for Nonsmooth Convex Optimization | none | — |
44.06.02 | The Proximal Operator, Moreau Envelope, and Proximal-Gradient Method | none | — |
44.06.03 | Accelerated Gradient Methods: Nesterov Acceleration and FISTA | none | — |
44.06.04 | Operator Splitting: ADMM and Douglas-Rachford | none | — |
44.06.05 | Stochastic Gradient and Mirror Descent | none | — |
44.06.06 | The Frank-Wolfe (Conditional Gradient) Method | none | — |
44.06.07 | Coordinate Descent and Block-Coordinate Methods | none | — |
44.06.08 | Nesterov Smoothing of Structured Nonsmooth Functions | none | — |
44.06.09 | The Chambolle-Pock Primal-Dual Hybrid Gradient Method | none | — |
44.07.01 | Pontryagin's Maximum Principle for Constrained Optimal Control | none | — |
44.07.02 | The Linear-Quadratic Regulator: Differential and Algebraic Riccati Equations | none | — |
44.07.03 | Bang-Bang and Minimum-Time Optimal Control | none | — |
44.07.04 | The Hamilton-Jacobi-Bellman Equation and the Verification Theorem | none | — |
44.08.01 | Finite-Horizon Dynamic Programming and the Principle of Optimality | none | — |
44.08.02 | Infinite-Horizon Discounted Dynamic Programming: Bellman Contraction, Value and Policy Iteration | none | — |
44.08.03 | Markov Decision Processes and Stochastic Dynamic Programming | none | — |
44.08.04 | Imperfect State Information and Partially Observed Markov Decision Processes | none | — |
44.08.05 | The Kalman Filter, Linear-Quadratic-Gaussian Control, and the Separation Principle | none | — |
44.08.06 | Stochastic Shortest Path and Total-Cost Dynamic Programming | none | — |
Mathematical statistics & learning theory
68 units · full: 0 · partial: 0 · none: 68 · stub: 0
| id | title | status | module |
|---|---|---|---|
45.01.01 | Statistical Decision Theory: Loss, Risk, Admissibility, Minimax, and Bayes Rules | none | — |
45.01.02 | Sufficiency, the Factorization Theorem, and Exponential Families | none | — |
45.01.03 | Ancillarity, Completeness, and Basu's Theorem | none | — |
45.01.04 | Point Estimation: The Method of Moments and Maximum Likelihood | none | — |
45.01.05 | Fisher Information and the Cramér-Rao Lower Bound | none | — |
45.01.06 | Rao-Blackwell and Lehmann-Scheffé: UMVU Estimation | none | — |
45.01.07 | Equivariance and the Pitman Best-Equivariant Estimator | none | — |
45.02.01 | The Neyman-Pearson Lemma and Most-Powerful Tests | none | — |
45.02.02 | Monotone Likelihood Ratio and Uniformly Most Powerful Tests | none | — |
45.02.03 | Likelihood-Ratio, Wald, and Score Tests | none | — |
45.02.04 | Confidence Sets, Pivotal Quantities, and Test-Interval Duality | none | — |
45.02.05 | Multiple Testing: Familywise Error, the False Discovery Rate, and Benjamini-Hochberg | none | — |
45.03.01 | Bayes Estimation Under Loss: Posterior Mean, Median, and Mode | none | — |
45.03.02 | Conjugate Priors and Exponential-Family Bayesian Inference | none | — |
45.03.03 | Hierarchical and Empirical Bayes | none | — |
45.03.04 | The Bernstein-von Mises Theorem | none | — |
45.03.05 | Markov Chain Monte Carlo: Metropolis-Hastings and Gibbs Sampling | none | — |
45.04.01 | Consistency of Estimators and the Modes of Stochastic Convergence in Statistics | none | — |
45.04.02 | The Delta Method and the Second-Order Delta Method | none | — |
45.04.03 | Asymptotic Normality and Efficiency of the Maximum-Likelihood Estimator | none | — |
45.04.04 | M-Estimators and Z-Estimators: The Argmax Theorem and the Master Theorem | none | — |
45.04.05 | Asymptotic Relative Efficiency | none | — |
45.04.06 | Local Asymptotic Normality and the Local-Experiment Limit | none | — |
45.04.07 | Contiguity and Le Cam's Three Lemmas | none | — |
45.04.08 | The Convolution Theorem and the Local Asymptotic Minimax Bound | none | — |
45.04.09 | Asymptotics of the Likelihood-Ratio, Wald, and Score Tests: The Chi-Squared Trinity | none | — |
45.04.10 | Asymptotic Consistency of the Bootstrap | none | — |
45.04.11 | Cube-Root Asymptotics and Chernoff's Distribution | none | — |
45.05.01 | Bernstein's Inequality and the Sub-Exponential / Bennett Regime | none | — |
45.05.02 | The Empirical Distribution and Glivenko-Cantelli Classes | none | — |
45.05.03 | Donsker Classes and the Empirical-Process Weak Limit | none | — |
45.05.04 | Entropy and VC Conditions for the Donsker Property: Maximal Inequalities | none | — |
45.05.05 | U-Statistics and Their Asymptotics | none | — |
45.05.06 | The Functional Delta Method and Hadamard Differentiability | none | — |
45.05.07 | Linear Rank Statistics and Locally Most Powerful Rank Tests | none | — |
45.05.08 | Orthogonal-Series and Wavelet Density Estimation | none | — |
45.05.09 | Minimax Lower Bounds for Nonparametric Estimation: Le Cam, Fano, and Assouad | none | — |
45.06.01 | Linear Model Theory: Gauss-Markov, the Hat Matrix, and BLUE | none | — |
45.06.02 | The Bias-Variance Decomposition and the Prediction-Error Trade-off | none | — |
45.06.03 | Ridge Regression and Shrinkage Estimation | none | — |
45.06.04 | The James-Stein Estimator and Inadmissibility of the MLE | none | — |
45.06.05 | Subset Selection and the Curse of Dimensionality | none | — |
45.06.06 | The LASSO: Sparsity, Oracle Inequalities, and the Irrepresentable Condition | none | — |
45.06.07 | Elastic Net and Least Angle Regression | none | — |
45.06.08 | Principal Components Regression and Partial Least Squares | none | — |
45.06.09 | Model Assessment and Selection: Cp, AIC, BIC, Cross-Validation, and the Bootstrap | none | — |
45.06.10 | Gaussian Graphical Models and the Graphical LASSO | none | — |
45.07.01 | PAC Learning, Empirical Risk Minimization, and the Finite-Class Bound | none | — |
45.07.02 | Uniform Convergence, the Bias-Complexity Tradeoff, and the No-Free-Lunch Theorem | none | — |
45.07.03 | VC Dimension, the Growth Function, and the Sauer-Shelah Lemma | none | — |
45.07.04 | The Fundamental Theorem of Statistical Learning | none | — |
45.07.05 | Rademacher Complexity and Generalization Bounds | none | — |
45.07.06 | Margin-Based Generalization Bounds and Covering Numbers | none | — |
45.07.07 | Algorithmic Stability and Online Learning | none | — |
45.07.08 | PAC-Bayes Generalization Bounds | none | — |
45.07.09 | Sample Compression Schemes and Compression-Based Generalization | none | — |
45.08.01 | Kernel Methods, Reproducing-Kernel Hilbert Spaces, and the Representer Theorem | none | — |
45.08.02 | Support Vector Machines: The Margin, the Dual, and the Kernel Trick | none | — |
45.08.03 | Linear and Quadratic Discriminant Analysis and Logistic Regression | none | — |
45.08.04 | Tree-Based Methods: Classification and Regression Trees (CART) | none | — |
45.08.05 | Bagging and Random Forests | none | — |
45.08.06 | Boosting: AdaBoost and Gradient Boosting | none | — |
45.08.07 | The EM Algorithm and Gaussian-Mixture / Latent-Variable Models | none | — |
45.08.08 | k-Means and Model-Based Clustering | none | — |
45.08.09 | Generalized Additive Models, Smoothing Splines, and Backfitting | none | — |
45.08.10 | Neural Networks: The Multilayer Perceptron and Backpropagation | none | — |
45.08.11 | k-Nearest-Neighbors and Local Regression | none | — |
45.08.12 | Generalized Linear Models | none | — |
Information & coding theory
40 units · full: 0 · partial: 0 · none: 40 · stub: 0
| id | title | status | module |
|---|---|---|---|
46.01.01 | Entropy, Joint Entropy, Conditional Entropy, and the Chain Rule | none | — |
46.01.02 | Mutual Information, KL Divergence, and Variational Characterizations (Donsker-Varadhan) | none | — |
46.01.03 | Information Inequalities: Log-Sum Inequality, Subadditivity, and Pinsker's Inequality | none | — |
46.01.04 | Differential Entropy, the Maximum-Entropy Gaussian, and the Entropy-Power Inequality | none | — |
46.01.05 | Entropy Rates of Stationary Processes and the Entropy of Markov Chains | none | — |
46.02.01 | The Asymptotic Equipartition Property (AEP) and Strongly Typical Sequences | none | — |
46.02.02 | Shannon's Source Coding Theorem: Optimal Lossless Rates and the Kraft Inequality | none | — |
46.02.03 | Huffman Codes: Construction, Optimality, and the Redundancy Bound | none | — |
46.02.04 | Arithmetic Coding and Lempel-Ziv Universal Source Coding | none | — |
46.02.05 | Rate-Distortion Theory: R(D) and Shannon's Lossy Source Coding Theorem | none | — |
46.02.06 | The Method of Types and Sanov's Theorem (IT formulation) | none | — |
46.03.01 | Discrete Memoryless Channels, Mutual Information Maximization, and Channel Capacity | none | — |
46.03.02 | Shannon's Noisy Channel Coding Theorem: Achievability via Random Coding | none | — |
46.03.03 | Fano's Inequality and the Converse to the Channel Coding Theorem | none | — |
46.03.04 | Binary Symmetric Channel and Binary Erasure Channel: Capacity and Sphere-Packing | none | — |
46.03.05 | The Gaussian Channel: Capacity, Bandwidth, and Water-Filling | none | — |
46.03.06 | Feedback Does Not Increase Capacity | none | — |
46.03.07 | Joint Source-Channel Coding and the Separation Theorem | none | — |
46.04.01 | Hypothesis Testing and Stein's Lemma: The KL Divergence as the Error Exponent | none | — |
46.04.02 | Chernoff Information and the Symmetric Chernoff Bound | none | — |
46.04.03 | The Method of Types: Large Deviations for Empirical Distributions (IT perspective) | none | — |
46.04.04 | The I-Projection, Pythagorean Identity, and Iterative Proportional Fitting (Blahut-Arimoto) | none | — |
46.05.01 | Slepian-Wolf Coding: Distributed Lossless Compression | none | — |
46.05.02 | Wyner-Ziv Coding: Rate-Distortion With Side Information at the Decoder | none | — |
46.05.03 | Gelfand-Pinsker and Costa's Dirty-Paper Coding | none | — |
46.05.04 | Common Information: Wyner's and Gacs-Korner Definitions | none | — |
46.06.01 | Multiple-Access Channel: Achievable Rates, Superposition, and the Capacity Region | none | — |
46.06.02 | Broadcast Channel: Degraded Broadcast Capacity and Marton's Inner Bound | none | — |
46.06.03 | Relay Channel: Decode-and-Forward, Compress-and-Forward, and the Cut-Set Bound | none | — |
46.06.04 | Interference Channel: Han-Kobayashi Bound and Gaussian Interference | none | — |
46.07.01 | Weight Enumerators, the MacWilliams Identity, and the Linear-Programming Bound | none | — |
46.07.02 | Algebraic Geometry Codes: The Goppa Construction and the TVZ Bound | none | — |
46.07.03 | List Decoding: The Sudan and Guruswami-Sudan Algorithms for Reed-Solomon Codes | none | — |
46.07.04 | Expander Codes, LDPC Bipartite Graphs, and Sipser-Spielman Linear-Time Decoding | none | — |
46.08.01 | LDPC Codes, Tanner Graphs, Belief Propagation, and Density Evolution Analysis | none | — |
46.08.02 | Turbo Codes: Parallel Concatenated Convolutional Codes and Iterative Decoding | none | — |
46.08.03 | Polar Codes: Arikan's Channel Polarization Phenomenon and Successive Cancellation | none | — |
46.08.04 | Capacity-Achieving Code Families: A Unified View Through Polarization and LDPC | none | — |
46.09.01 | Convolutional Codes: Trellis Structure, the Viterbi Algorithm, and Free Distance | none | — |
46.09.02 | Gambling, the Kelly Criterion, and the Doubling Rate: Entropy as Growth | none | — |
Theoretical computer science
36 units · full: 0 · partial: 0 · none: 36 · stub: 0
| id | title | status | module |
|---|---|---|---|
47.01.01 | Deterministic Finite Automata: States, Transitions, and Accepted Languages | none | — |
47.01.02 | Nondeterministic Finite Automata and the Subset Construction | none | — |
47.01.03 | Regular Expressions and the Kleene-Rabin-Scott Theorem | none | — |
47.01.04 | Pumping Lemma for Regular Languages: Proving Non-Regularity | none | — |
47.01.05 | Context-Free Grammars, Parse Trees, and Pushdown Automata | none | — |
47.01.06 | Pumping Lemma for Context-Free Languages and CYK Parsing | none | — |
47.02.01 | Time Complexity, P, NP, and Polynomial-Time Reductions | none | — |
47.02.02 | NP-Completeness and the Cook-Levin Theorem | none | — |
47.02.03 | Classic NP-Complete Problems: SAT, 3SAT, Clique, Vertex Cover, and Hamiltonian Cycle | none | — |
47.02.04 | Polynomial Hierarchy, Oracle Machines, and Alternating Quantifiers | none | — |
47.02.05 | Space Complexity: PSPACE, Savitch's Theorem, and PSPACE-Completeness | none | — |
47.02.06 | L, NL, and the Immerman-Szelepcsenyi Theorem | none | — |
47.02.07 | #P, Counting, the Permanent, and Toda's Theorem | none | — |
47.03.01 | Randomized Complexity: BPP, RP, ZPP, and the Schwartz-Zippel Lemma | none | — |
47.03.02 | Circuit Complexity: P/poly and the Karp-Lipton Theorem | none | — |
47.03.03 | AC0 and TC0: Parity Lower Bounds, Threshold Gates, and Circuit Separations | none | — |
47.03.04 | Interactive Proofs: IP = PSPACE | none | — |
47.03.05 | The PCP Theorem: Probabilistically Checkable Proofs and Hardness of Approximation | none | — |
47.03.06 | Hardness of Approximation: Inapproximability of MAX-3SAT and Vertex Cover | none | — |
47.04.01 | Amortized Analysis: Aggregate, Accounting, and Potential-Function Methods | none | — |
47.04.02 | Self-Balancing Trees: Red-Black Trees | none | — |
47.04.03 | Network Flow: Ford-Fulkerson, Augmenting Paths, and the Max-Flow Min-Cut Theorem | none | — |
47.04.04 | Bipartite Matching: Hall's Theorem, Augmenting Paths, and the Hungarian Algorithm | none | — |
47.04.05 | All-Pairs Shortest Paths: Floyd-Warshall and Johnson's Algorithm | none | — |
47.04.06 | Minimum Spanning Trees: Kruskal's and Prim's Algorithms | none | — |
47.04.07 | String Algorithms: KMP, Suffix Arrays, and Aho-Corasick | none | — |
47.04.08 | Approximation Algorithms: Vertex Cover, TSP, and Set Cover | none | — |
47.05.01 | Universal Hashing, Perfect Hashing, and the FKS Construction | none | — |
47.05.02 | Randomized Algorithms: Quicksort, Skip Lists, and Treaps | none | — |
47.05.03 | Miller-Rabin Primality Testing and the Solovay-Strassen Test | none | — |
47.05.04 | FPRAS for Counting Problems: DNF Counting and the Permanent | none | — |
47.05.05 | Derandomization: Pairwise Independence and the Method of Conditional Expectations | none | — |
47.06.01 | One-Way Functions and Pseudorandom Generators | none | — |
47.06.02 | Public-Key Cryptography: RSA and the Discrete Logarithm | none | — |
47.06.03 | Zero-Knowledge Proofs: Interactive Protocols and Proof Systems | none | — |
47.06.04 | Digital Signatures and Collision-Resistant Hash Functions | none | — |
Classical mechanics
44 units · full: 0 · partial: 0 · none: 44 · 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 | — |
09.01.04 | Two-body central-force problem, Kepler orbits, and Rutherford scattering | none | — |
09.01.05 | Newton's Law of Gravitation: Gravitational Field, Potential, and the Shell Theorem | none | — |
09.01.06 | Friction, Normal Forces, and Constraints: Holonomic and Non-holonomic Systems | none | — |
09.01.07 | Non-inertial Frames: Centrifugal Force, Coriolis Force, and Foucault's Pendulum | none | — |
09.01.08 | Variable-Mass Systems: The Rocket Equation (Tsiolkovsky) and Momentum Transport | none | — |
09.02.01 | The action principle and variational calculus | none | — |
09.02.02 | Euler-Lagrange equations | none | — |
09.02.03 | Constrained Lagrangian Systems: Lagrange Multipliers and D'Alembert's Principle | none | — |
09.02.04 | Small Oscillations: Normal Modes and the Eigenvalue Problem for Coupled Oscillators | none | — |
09.02.05 | Lagrangian Perturbation Theory: Secular Terms and the Method of Multiple Scales | none | — |
09.02.06 | Field Lagrangians and the Continuum Limit: Elastic Media and the Wave Equation as a Field Theory | none | — |
09.03.01 | Noether's theorem — symmetries and conservation laws | none | — |
09.03.02 | Cyclic Coordinates, Ignorable Degrees of Freedom, and Routh's Procedure | none | — |
09.03.04 | Discrete Symmetries in Mechanics: Parity, Time-Reversal, and Conservation Laws | none | — |
09.03.05 | Galilean Symmetry Group and the Ten Integrals of Motion | none | — |
09.04.01 | Legendre transform — from Lagrangian to Hamiltonian | none | — |
09.04.02 | Hamilton's equations | none | — |
09.04.03 | Poisson Brackets: Structure, the Jacobi Identity, and Bracket Algebra of Conserved Quantities | none | — |
09.04.04 | Liouville's Theorem and the Incompressibility of Phase-Space Flow | none | — |
09.04.05 | Symplectic Structure, the Symplectic Form, and Darboux's Theorem | none | — |
09.04.06 | Hamiltonian Perturbation Theory: Averaging and Adiabatic Invariants | none | — |
09.04.08 | Classical Spin: The SO(3) Poisson Bracket and Euler's Equations for Rigid Body Rotation | none | — |
09.04.09 | The Hamilton-Jacobi Equation as the Eikonal Limit: WKB and Geometric Optics | none | — |
09.05.01 | Canonical transformations | none | — |
09.05.02 | Hamilton-Jacobi equation | none | — |
09.05.03 | Generating Functions for Canonical Transformations: Four Types (F1, F2, F3, F4) | none | — |
09.05.04 | Point Transformations and Their Canonical Lifts | none | — |
09.05.05 | The Symplectic Group Sp(2n, R) and Its Role in Linear Canonical Transformations | none | — |
09.06.01 | Action-angle variables | none | — |
09.06.02 | The Liouville-Arnold Theorem: Integrability and the Existence of Action-Angle Variables | none | — |
09.06.03 | Completely Integrable Systems: The Toda Lattice and the Kepler Problem Revisited | none | — |
09.06.04 | Lax Pairs and the Inverse Scattering Method: Solitons in Classical Mechanics | none | — |
09.07.01 | Continuum Mechanics and Field Theory | none | — |
09.07.02 | Elastic Waves in Solids: Longitudinal and Transverse Modes, the Cauchy Stress Tensor | none | — |
09.07.03 | Ideal Fluid Mechanics: Euler's Equations, Bernoulli's Equation, and Vorticity | none | — |
09.07.04 | Viscous Flow: The Navier-Stokes Equations, Reynolds Number, and Laminar-Turbulent Transition | none | — |
09.08.01 | KAM theorem and chaos | none | — |
09.08.02 | Phase-Space Structure of Chaos: Poincare Sections, Homoclinic Tangles, and Horseshoe Maps | none | — |
09.08.03 | Lyapunov Exponents: Definition, Computation, and the Chaotic Double Pendulum | none | — |
09.08.04 | Period-doubling route to chaos: the logistic map and Feigenbaum universality | none | — |
09.08.05 | Strange attractors: the Lorenz system, fractal dimension, and sensitive dependence | none | — |
Quantum mechanics
2 units · full: 0 · partial: 0 · none: 2 · stub: 0
| id | title | status | module |
|---|---|---|---|
09.03.03 | Quantum free particle as a representation of | none | — |
09.04.07 | Complex structures and quantization; squeezed states | none | — |
Electromagnetism & special relativity
38 units · full: 0 · partial: 0 · none: 38 · 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 | — |
10.01.03 | Conductors, capacitance, and electrostatic energy | none | — |
10.01.04 | Dielectrics, polarization P, and the electric displacement D | none | — |
10.01.05 | Multipole expansion of the electrostatic potential | none | — |
10.02.01 | Biot-Savart law and Ampere's law | none | — |
10.02.02 | Magnetic vector potential, gauge freedom, and the Coulomb gauge | none | — |
10.02.03 | Magnetization, H-field, and magnetic materials: diamagnets, paramagnets, ferromagnets | none | — |
10.02.04 | Boundary conditions at interfaces: normal and tangential components of E, D, B, H | none | — |
10.02.05 | Magnetic scalar potential and the demagnetization factor | none | — |
10.03.01 | Faraday's law and electromagnetic induction | none | — |
10.03.02 | Faraday's law in integral and differential form: mutual and self-inductance, energy in inductors | none | — |
10.03.03 | Energy and momentum in the electromagnetic field: Poynting vector, Maxwell stress tensor, conservation laws | none | — |
10.03.04 | Displacement current, the continuity equation, and the complete Maxwell equations in matter | none | — |
10.03.05 | Energy and momentum in the electromagnetic field: the Poynting vector and radiation pressure | none | — |
10.04.01 | Maxwell's equations in differential form | none | — |
10.04.02 | EM waves and the wave equation | none | — |
10.04.03 | Plane waves in matter: dispersion relations, skin depth, and the complex refractive index | none | — |
10.04.04 | Waveguides and transmission lines: TE, TM, TEM modes and the cutoff frequency | none | — |
10.04.05 | Cavities and resonant modes: the quality factor Q and normal-mode expansion | none | — |
10.05.01 | Special relativity — postulates and Lorentz transformations | none | — |
10.05.02 | Relativistic kinematics and dynamics | none | — |
10.05.03 | Four-velocity, four-momentum, and the relativistic energy-momentum identity | none | — |
10.06.01 | Covariant electrodynamics — Faraday tensor | none | — |
10.06.02 | The four-current, charge-current density, and covariant continuity | none | — |
10.06.03 | The electromagnetic stress-energy tensor and covariant energy-momentum conservation | none | — |
10.06.04 | Transformation of E and B fields under Lorentz boosts | none | — |
10.07.01 | Radiation from accelerating charges — Larmor formula | none | — |
10.07.02 | Electric dipole radiation: angular distribution, polarization, and total radiated power | none | — |
10.07.03 | Magnetic dipole and electric quadrupole radiation: multipole expansion of radiation fields | none | — |
10.07.04 | Radiation reaction: the Abraham-Lorentz force and the self-energy problem | none | — |
10.07.05 | Synchrotron radiation: relativistic circular motion and the 1/gamma^4 power formula | none | — |
10.07.06 | Radiation — antennas, scattering, and radiation reaction | none | — |
10.08.01 | Green's function method for the scalar and vector potentials: retarded, advanced, and Feynman propagators | none | — |
10.08.02 | Jefimenko's Equations: Exact Retarded Solutions for E and B Given Arbitrary Sources | none | — |
10.08.03 | Diffraction: Kirchhoff Integral, Fraunhofer and Fresnel Regimes | none | — |
10.08.04 | Geometrical Optics as the Short-Wavelength Limit: Eikonal Equation and Ray Tracing | none | — |
10.08.05 | Macroscopic Maxwell Equations: Linear Response, Kramers-Kronig Relations, and Causality | none | — |
Statistical mechanics
36 units · full: 0 · partial: 0 · none: 36 · 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 | — |
11.01.03 | Third law of thermodynamics: Nernst heat theorem, absolute entropy, and zero-point entropy | none | — |
11.01.04 | Thermodynamic stability: convexity, Le Chatelier's principle, and the spinodal curve | none | — |
11.01.05 | Equations of state: virial expansion, van der Waals gas, and the law of corresponding states | none | — |
11.01.06 | Thermodynamic cycles: Carnot efficiency, Otto and Diesel cycles, and entropy production | none | — |
11.02.01 | Maxwell-Boltzmann distribution from kinetic theory | none | — |
11.02.02 | Transport phenomena: diffusion, viscosity, thermal conductivity, and the mean free path | none | — |
11.02.03 | The Boltzmann transport equation: H-theorem, the approach to equilibrium, and the BGK approximation | none | — |
11.02.04 | Fluctuations and Noise: The Fluctuation-Dissipation Theorem and Nyquist's Formula | none | — |
11.03.01 | Microcanonical ensemble | none | — |
11.03.02 | Grand canonical ensemble: chemical potential, fugacity, and variable particle number | none | — |
11.03.03 | Equivalence of Ensembles in the Thermodynamic Limit: Legendre Structure and Equivalence Conditions | none | — |
11.03.04 | Isothermal-Isobaric (NPT) Ensemble: Gibbs Free Energy as the Natural Potential | none | — |
11.04.01 | Canonical ensemble and partition function | none | — |
11.04.02 | Souriau Gibbs state on a symplectic -space | none | — |
11.04.03 | Factorization of the partition function: independent subsystems and the grand potential | none | — |
11.04.04 | Classical ideal gas partition function: Sackur-Tetrode entropy and the Gibbs paradox | none | — |
11.04.05 | Harmonic Oscillator in a Thermal Bath: Heat Capacity and the Einstein Solid | none | — |
11.05.01 | Bose-Einstein distribution | none | — |
11.05.02 | Fermi-Dirac distribution and electron gas | none | — |
11.05.03 | Blackbody radiation: Planck distribution, Stefan-Boltzmann law, Wien displacement law | none | — |
11.05.04 | Bose-Einstein condensation and the critical temperature | none | — |
11.05.05 | Fermi gas: heat capacity, electron specific heat, and Pauli paramagnetism | none | — |
11.05.06 | Photon gas, phonon gas, and the Debye model of solids | none | — |
11.06.01 | Ising model and phase transitions | none | — |
11.06.02 | Mean-field theory of phase transitions: the van der Waals and Curie-Weiss models | none | — |
11.06.03 | Landau Theory: Order Parameter, Symmetry Breaking, and the Free Energy Expansion | none | — |
11.06.04 | Fluctuations Beyond Mean Field: The Ginzburg Criterion and Ginzburg-Landau Theory | none | — |
11.06.05 | Scaling Hypothesis: Widom Scaling, Kadanoff Block Spins, and Data Collapse | none | — |
11.06.06 | Transfer Matrix Method: Exact Solution of the 1D Ising Model and Onsager's Legacy | none | — |
11.07.01 | Critical phenomena and renormalization group | none | — |
11.07.02 | The renormalization group — Wilson's framework | none | — |
11.07.03 | Block-Spin Renormalization: Wilson's Real-Space RG, Fixed Points, and Relevant Operators | none | — |
11.07.04 | Kosterlitz-Thouless Transition: Topological Phase Transitions in Two Dimensions | none | — |
11.07.05 | The Epsilon Expansion, RG Flows Near Four Dimensions, Critical Exponents, and Conformal Field Theory | none | — |
Quantum mechanics & QFT
94 units · full: 0 · partial: 0 · none: 94 · 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 | — |
12.01.03 | WKB approximation: tunneling, Bohr-Sommerfeld quantization, and connection formulas | none | — |
12.01.04 | Probability current, continuity equation, and the flow interpretation of the wavefunction | none | — |
12.01.05 | Spreading of the free Gaussian wave packet: group velocity and dispersion | none | — |
12.02.01 | Hilbert-space formalism of quantum mechanics | none | — |
12.02.02 | Operators, observables, and Hermiticity | none | — |
12.02.03 | Density matrix and pure / mixed states | none | — |
12.02.04 | Measurement postulates: Born rule, wavefunction collapse, and the projection postulate | none | — |
12.03.01 | Schrödinger and Heisenberg pictures | none | — |
12.03.02 | Time-dependent perturbation theory: Fermi's golden rule and transition rates | none | — |
12.03.03 | The interaction picture, Dyson series, and time-ordered exponentials | none | — |
12.04.01 | Particle in a box | none | — |
12.04.02 | Quantum harmonic oscillator | none | — |
12.04.03 | The finite square well: bound states, tunneling, and resonant scattering | none | — |
12.04.04 | The delta-function potential: exactly solvable bound state and scattering | none | — |
12.05.01 | Angular momentum operators and SU(2) representations | none | — |
12.05.02 | Spherical harmonics and Legendre polynomials | none | — |
12.05.03 | Addition of angular momenta and Clebsch-Gordan coefficients | none | — |
12.05.04 | Free Klein-Gordon scalar quantum field | none | — |
12.05.05 | Free Dirac spin-1/2 quantum field | none | — |
12.05.06 | Free Maxwell / massive vector fields; photon and Proca | none | — |
12.05.07 | Molecular vibrations and spectroscopic selection rules via symmetry | none | — |
12.06.01 | Hydrogen atom bound states | none | — |
12.06.02 | The isotropic harmonic oscillator in three dimensions: Cartesian and spherical solutions | none | — |
12.06.03 | The Runge-Lenz vector, SO(4) symmetry, and the accidental degeneracy of hydrogen | none | — |
12.06.04 | Crossing symmetry; CPT theorem at the -matrix level | none | — |
12.07.01 | Time-independent perturbation theory | none | — |
12.07.02 | Time-dependent perturbation theory and Fermi's golden rule | none | — |
12.07.03 | Variational method (Rayleigh-Ritz) in quantum mechanics | none | — |
12.07.04 | WKB approximation and Bohr-Sommerfeld quantisation | none | — |
12.07.05 | Stark and Zeeman effects in LL3 framing | none | — |
12.07.07 | Adiabatic theorem and Berry phase preview | none | — |
12.07.08 | Berry phase and the geometric phase | none | — |
12.08.01 | Scattering Theory | none | — |
12.08.02 | Born approximation and the Lippmann-Schwinger equation | none | — |
12.08.03 | Partial-wave expansion and phase shifts | none | — |
12.08.04 | Inelastic collisions and the distorted-wave Born approximation | none | — |
12.09.01 | Identical Particles and Many-Body Quantum Mechanics | none | — |
12.09.02 | Exchange interaction and the helium atom | none | — |
12.09.03 | Hartree-Fock self-consistent field method | none | — |
12.09.04 | Multi-electron atomic structure and LS coupling | none | — |
12.09.05 | Diatomic molecule and the Born-Oppenheimer approximation | none | — |
12.10.01 | Path integral formulation of quantum mechanics | none | — |
12.10.02 | Euclidean path integrals, instantons, and tunneling between vacua | none | — |
12.10.03 | The semiclassical approximation: stationary phase, van Vleck determinant, and Gutzwiller trace formula | none | — |
12.11.01 | Dirac equation and relativistic spin | none | — |
12.11.02 | Klein-Gordon equation in external EM field: Coulomb and uniform-magnetic cases | none | — |
12.11.03 | Dirac equation in a Coulomb field | none | — |
12.11.04 | Klein paradox | none | — |
12.11.05 | Furry's theorem and charge-conjugation symmetry of QED | none | — |
12.12.01 | Canonical Quantum Field Theory | none | — |
12.12.02 | Coulomb gauge vs Lorenz gauge in QED | none | — |
12.12.03 | Compton scattering and the Klein-Nishina formula | none | — |
12.12.04 | Møller scattering (electron-electron) | none | — |
12.12.05 | Bhabha scattering (electron-positron) | none | — |
12.12.06 | Bethe-Heitler bremsstrahlung and pair production | none | — |
12.13.01 | Bosonic Fock space and second quantisation | none | — |
12.13.02 | Fermionic Fock space, Pauli exclusion, and anticommutators | none | — |
12.13.03 | Cluster decomposition and the connected S-matrix | none | — |
12.14.01 | CCR algebra, Weyl algebra, and quasi-free states | none | — |
12.14.02 | The Heisenberg group, the Schrödinger representation, and Stone–von Neumann as quantization | none | — |
12.15.01 | Time-reversal symmetry and Kramers' degeneracy | none | — |
12.15.02 | Parity, discrete-symmetry groups, and the Wigner-Eckart theorem | none | — |
12.15.03 | CP violation and the CKM matrix | none | — |
12.16.01 | Electron self-energy and mass renormalization at one loop | none | — |
12.16.02 | One-loop QED vertex function and the anomalous magnetic moment | none | — |
12.16.03 | Vacuum polarization at one loop and the Uehling potential | none | — |
12.16.04 | Lamb shift from one-loop QED | none | — |
12.16.05 | Infrared divergences and the Bloch-Nordsieck cancellation in QED | none | — |
12.16.06 | Power counting, the superficial degree of divergence, and renormalizability classification | none | — |
12.17.01 | Density matrix, pure states, and mixed states | none | — |
12.17.02 | Entanglement, Schmidt decomposition, and entanglement entropy | none | — |
12.17.03 | Bell inequalities, CHSH inequality, and the Tsirelson bound | none | — |
12.17.07 | Quantum teleportation and superdense coding | none | — |
12.18.01 | The Higgs mechanism: spontaneously broken gauge symmetry | none | — |
12.18.02 | The Goldstone theorem and effective Goldstone Lagrangians | none | — |
12.18.03 | Asymptotic freedom and the running gauge coupling | none | — |
12.18.04 | Theta-vacua, the vacuum angle, and the strong-CP problem | none | — |
12.18.05 | The chiral (Adler-Bell-Jackiw) anomaly from the triangle diagram | none | — |
12.18.06 | Operator product expansion and short-distance behaviour | none | — |
12.18.13 | Vortices (Nielsen-Olesen / Abrikosov flux tubes) | none | — |
12.18.16 | Lattice gauge theory and confinement (QFT pointer) | none | — |
12.19.01 | The supersymmetry algebra: Coleman-Mandula, Haag-Lopuszanski-Sohnius, and the graded extension of Poincare | none | — |
12.19.02 | Superspace, superfields, and Berezin integration | none | — |
12.19.03 | Supermultiplets and the Wess-Zumino model: Bose-Fermi degeneracy, the superpotential, and the auxiliary F-field | none | — |
12.19.04 | Super-Yang-Mills and the non-renormalization theorem: the vector superfield, Wess-Zumino gauge, , and supergraphs | none | — |
12.19.05 | Spontaneous SUSY breaking: the Goldstino theorem, O'Raifeartaigh and Fayet-Iliopoulos, the supertrace sum rule, and the field-theory Witten index | none | — |
12.19.06 | Supersymmetric QCD and Seiberg duality: the moduli space of vacua, holomorphy, and N=1 electric-magnetic duality | none | — |
12.20.01 | Condensed-matter physics — band structure, phonons, and emergent order | none | — |
12.20.02 | Superconductivity — Cooper pairs, BCS theory, and macroscopic quantum order | none | — |
12.20.03 | Magnetism, Mott physics, and topological phases of matter | none | — |
12.21.01 | The Standard Model — electroweak unification and QCD | none | — |
12.21.02 | Electroweak theory and the Higgs mechanism | none | — |
General relativity & cosmology
38 units · full: 0 · partial: 0 · none: 38 · stub: 0
| id | title | status | module |
|---|---|---|---|
13.01.01 | The equivalence principle | none | — |
13.01.02 | The equivalence principle — weak, strong, Einstein, and experimental tests | none | — |
13.02.01 | Tensors on smooth manifolds | none | — |
13.02.02 | Geodesics and parallel transport | none | — |
13.02.03 | Cartan tetrad and spin-connection formulation of general relativity | none | — |
13.03.01 | Riemann curvature tensor | none | — |
13.03.02 | Ricci and scalar curvature, and the Einstein tensor | none | — |
13.03.03 | The ADM 3+1 formalism and the Hamiltonian constraint | none | — |
13.04.01 | Einstein field equations | none | — |
13.04.02 | Einstein-Hilbert action and variational derivation of the Einstein equations | none | — |
13.04.03 | Palatini first-order variational formulation of general relativity | none | — |
13.04.04 | Stress-energy tensor as functional derivative of the matter action | none | — |
13.05.01 | Schwarzschild solution | none | — |
13.05.02 | Orbits in Schwarzschild geometry | none | — |
13.05.03 | Solar-system tests of general relativity: perihelion precession, light bending, Shapiro time delay, gravitational redshift, frame-dragging | none | — |
13.05.04 | Kerr black hole, ergosphere, and the Penrose process | none | — |
13.06.01 | Black Holes | none | — |
13.06.03 | Black hole thermodynamics: the four laws, Bekenstein-Hawking entropy, and the area theorem | none | — |
13.06.04 | Hawking radiation: Bogoliubov derivation, thermal spectrum, and black-hole evaporation | none | — |
13.07.01 | Linearized GR and gravitational waves | none | — |
13.07.02 | Null infinity, the BMS group, and the Bondi-Sachs mass-loss formula | none | — |
13.07.03 | Interferometric gravitational-wave detectors: LIGO, Virgo, KAGRA | none | — |
13.08.01 | FLRW cosmology and Friedmann equations | none | — |
13.08.02 | Cosmology — FLRW, inflation, nucleosynthesis, CMB, and structure | none | — |
13.08.03 | Cosmological inflation: slow-roll scalar fields and the origin of structure | none | — |
13.08.04 | Gravitational lensing: strong, weak, and microlensing — Einstein rings, dark matter maps, and exoplanet detection | none | — |
13.09.01 | Globally hyperbolic Lorentzian manifolds | none | — |
13.09.02 | Klein-Gordon equation on a globally hyperbolic spacetime | none | — |
13.09.03 | Hadamard states via the wave-front-set criterion | none | — |
13.09.04 | Existence of Hadamard states via the FNW deformation argument | none | — |
13.09.05 | Hadamard states by pseudo-differential calculus (Gérard-Wrochna) | none | — |
13.09.06 | Wick polynomials in curved spacetime via Hadamard parametrix subtraction | none | — |
13.09.07 | Time-ordered products and Hollands-Wald renormalisation on curved spacetimes | none | — |
13.09.08 | Bunch-Davies state on de Sitter spacetime | none | — |
13.09.09 | Unruh effect via the Bisognano-Wichmann theorem | none | — |
13.09.10 | Hartle-Hawking and Unruh states on Schwarzschild | none | — |
13.09.11 | Quantum energy inequalities (Fewster) | none | — |
13.09.12 | The Peierls bracket and the covariant phase space of an interacting field theory | none | — |
General & physical chemistry
47 units · full: 0 · partial: 0 · none: 47 · stub: 0
| id | title | status | module |
|---|---|---|---|
14.01.01 | Atomic structure and electron configurations | none | — |
14.01.02 | Periodic trends: ionization energy, electron affinity, atomic radius, and electronegativity | none | — |
14.01.03 | Many-electron atoms: Hartree-Fock self-consistent field and the Pauli exclusion principle | none | — |
14.02.01 | Lewis structures and VSEPR | none | — |
14.02.02 | Hybridization and valence bond theory | none | — |
14.02.03 | Molecular geometry and dipole moments: polarity and intermolecular forces | none | — |
14.02.04 | Valence bond theory: resonance structures and the limitations of Lewis models | none | — |
14.03.01 | Stoichiometry and gas laws | none | — |
14.03.02 | Solution stoichiometry: molarity, titration, and gravimetric analysis | none | — |
14.03.03 | Ideal and real gas laws: van der Waals equation and compressibility factor | none | — |
14.04.01 | Hydrogen atom quantum chemistry | none | — |
14.04.02 | Variational principle in quantum chemistry: the secular determinant and basis set expansion | none | — |
14.04.03 | Perturbation theory in chemistry: Stark effect and the helium atom first-order correction | none | — |
14.04.04 | Density functional theory basics: the Hohenberg-Kohn theorems and the Kohn-Sham equations | none | — |
14.05.01 | Molecular orbital theory: LCAO, bonding and antibonding orbitals, and the H2 molecule | none | — |
14.05.02 | Molecular orbital theory for homonuclear diatomics | none | — |
14.05.03 | MO theory of heteronuclear diatomics: CO, NO, HF and the Walsh correlation diagrams | none | — |
14.05.04 | Band theory of solids: the Bloch theorem, the tight-binding model, and conductors versus insulators | none | — |
14.06.01 | Chemical thermodynamics: free energies and equilibrium | none | — |
14.06.02 | Hess's law, standard enthalpies of formation, and Born-Haber cycles | none | — |
14.06.03 | Entropy in chemistry: the third law, standard entropy changes, and the Gibbs energy | none | — |
14.06.04 | Chemical equilibrium: Le Chatelier's principle and the equilibrium constant Kp versus Kc | none | — |
14.07.01 | Statistical Mechanics for Chemistry | none | — |
14.07.02 | Partition functions for chemical systems: rotational, vibrational, and electronic contributions | none | — |
14.07.03 | Statistical thermodynamics of ideal gases: entropy, heat capacity, and the Sackur-Tetrode equation | none | — |
14.07.04 | Activated complex theory: the Eyring equation and the Arrhenius prefactor | none | — |
14.08.01 | Chemical kinetics: rate laws and the Arrhenius equation | none | — |
14.08.02 | Integrated rate laws: first, second, and zeroth order reactions and half-lives | none | — |
14.08.03 | Reaction mechanisms: elementary steps, rate-determining step, and the steady-state approximation | none | — |
14.08.04 | Catalysis mechanisms: homogeneous, heterogeneous, and enzymatic Michaelis-Menten kinetics | none | — |
14.09.01 | Solutions and Phase Equilibria | none | — |
14.09.02 | Raoult's law and vapor pressure lowering: ideal solutions and Henry's law for gases | none | — |
14.09.03 | Colligative properties: boiling-point elevation, freezing-point depression, and osmotic pressure | none | — |
14.09.04 | Phase diagrams: single-component, binary, and the lever rule | none | — |
14.10.01 | Acid-base chemistry: Bronsted-Lowry, Lewis, and pKa | none | — |
14.10.02 | Buffer solutions: the Henderson-Hasselbalch equation and buffer capacity | none | — |
14.10.03 | Acid-base titration curves: strong-strong, weak-strong, and polyprotic titrations | none | — |
14.10.04 | Solubility equilibria: Ksp, common-ion effect, and dissolution of sparingly soluble salts | none | — |
14.11.01 | Electrochemistry: the Nernst equation and electrochemical cells | none | — |
14.11.02 | Galvanic cells: half-cell potentials, cell diagrams, and standard reduction potentials | none | — |
14.11.03 | Electrolysis: Faraday's laws, overpotential, and electrodeposition | none | — |
14.11.04 | Batteries and fuel cells: lead-acid, lithium-ion, and hydrogen fuel cell thermodynamics | none | — |
14.12.01 | UV-Vis, IR, and NMR — fundamentals of molecular spectroscopy | none | — |
14.12.02 | Rotational spectroscopy: the rigid rotor, rotational constants, and microwave spectroscopy | none | — |
14.12.03 | Vibrational spectroscopy: harmonic oscillator selection rules, anharmonicity, and IR-active modes | none | — |
14.12.04 | Electronic spectroscopy: the Franck-Condon principle, chromophores, and UV-Vis transitions | none | — |
14.12.05 | Mass spectrometry: ionization methods, fragmentation patterns, and molecular mass determination | none | — |
Organic chemistry
49 units · full: 0 · partial: 0 · none: 49 · stub: 0
| id | title | status | module |
|---|---|---|---|
15.01.01 | Structure of organic molecules — stereochemistry | none | — |
15.01.02 | Conformational analysis: Newman projections, cyclohexane chair flip, and axial versus equatorial | none | — |
15.01.03 | Stereoisomerism: enantiomers, diastereomers, meso compounds, and chiral resolution | none | — |
15.01.04 | Enantioselective synthesis: chiral auxiliaries, asymmetric catalysis, and ee measurement | none | — |
15.02.01 | Functional groups and nomenclature | none | — |
15.02.02 | Alkane reactions: combustion, halogenation, and radical selectivity | none | — |
15.02.03 | Alcohol and ether chemistry: synthesis, reactions, and protection strategies | none | — |
15.03.01 | Acids and bases in organic chemistry | none | — |
15.03.02 | Enols, enolates, and keto-enol tautomerism: kinetic versus thermodynamic control | none | — |
15.04.01 | Elimination reactions: E1 and E2 mechanisms, Zaitsev's rule, and stereochemistry | none | — |
15.04.02 | SN1 vs SN2 substitution mechanisms | none | — |
15.04.03 | Stereochemistry of SN2 and E2: Walden inversion and the anti-periplanar requirement | none | — |
15.05.01 | Electrophilic addition to alkenes | none | — |
15.05.02 | Electrophilic addition to alkynes: Markovnikov's rule and anti-Markovnikov hydroboration | none | — |
15.05.03 | Diels-Alder reaction: [4+2] cycloaddition, endo/exo selectivity, and synthetic applications | none | — |
15.06.01 | Aromatic chemistry — EAS, Huckel | none | — |
15.06.02 | Nucleophilic aromatic substitution: the SNAr mechanism and activation conditions | none | — |
15.06.03 | Arene side-chain reactions: benzylic radicals and oxidation of alkylbenzenes | none | — |
15.06.04 | Synthesis of substituted benzenes: directing effects and strategic functional group manipulation | none | — |
15.07.01 | Carbonyl chemistry — nucleophilic addition | none | — |
15.07.02 | Nucleophilic addition-elimination at acyl carbons | none | — |
15.07.03 | Enolate chemistry: alkylation, aldol condensation, and Claisen condensation | none | — |
15.07.04 | Carbonyl condensation reactions: Knoevenagel, Michael addition, and Robinson annulation | none | — |
15.07.05 | Oxidation and reduction of carbonyls | none | — |
15.07.06 | The Baeyer-Villiger oxidation: Criegee intermediate, migratory aptitude, and lactone regiochemistry | none | — |
15.08.01 | Radical and Pericyclic Reactions | none | — |
15.08.02 | Sigmatropic rearrangements: Cope, Claisen, and [1,5]-H shifts | none | — |
15.08.03 | Electrocyclic reactions: orbital symmetry rules and ring opening/closing | none | — |
15.09.01 | Organometallic Methods in Synthesis | none | — |
15.09.02 | Palladium catalysis: Suzuki, Heck, Sonogashira, and Buchwald-Hartwig cross-coupling | none | — |
15.09.03 | Olefin metathesis: Grubbs catalyst, ring-closing, and cross metathesis | none | — |
15.10.01 | Retrosynthetic analysis | none | — |
15.10.02 | Multi-step synthesis design: protecting groups, convergent versus linear strategies | none | — |
15.10.03 | Total synthesis case study: landmark syntheses and the logic of route selection | none | — |
15.11.01 | NMR spectroscopy of organic molecules | none | — |
15.11.02 | 1H NMR: chemical shift, coupling constants, integration, and 2D NMR | none | — |
15.11.03 | 13C NMR, DEPT, and the complete structure elucidation workflow | none | — |
15.12.01 | Amino acids and protein chemistry | none | — |
15.12.02 | Peptide bond geometry: planarity, resonance, and the Ramachandran plot | none | — |
15.12.03 | Protein folding: Anfinsen, Levinthal, and molecular chaperones | none | — |
15.13.01 | Nucleic acid chemistry | none | — |
15.13.02 | RNA secondary structure and ribozymes | none | — |
15.13.03 | DNA topology: supercoiling, the Calugareanu-White theorem, and topoisomerases | none | — |
15.14.01 | Enzyme mechanism | none | — |
15.14.02 | Enzyme kinetics beyond Michaelis-Menten: inhibition, allosteric enzymes, and Hill coefficients | none | — |
15.15.01 | Heterocyclic chemistry — rings containing N, O, S | none | — |
15.15.02 | Named heterocyclic ring synthesis — Paal–Knorr, Fischer, Skraup, Hantzsch | none | — |
15.16.01 | Natural products and organocatalysis | none | — |
15.17.01 | Polymer chemistry and photochemistry | none | — |
Inorganic chemistry
39 units · full: 0 · partial: 0 · none: 39 · stub: 0
| id | title | status | module |
|---|---|---|---|
16.01.01 | Periodic trends quantified | none | — |
16.01.02 | Main-group chemistry: s- and p-block trends, inert-pair effect, and diagonal relationships | none | — |
16.01.03 | Acid-base theories: Lewis acids, HSAB, and Drago-Wayland parameters | none | — |
16.01.04 | Mendeleev's periodic table of 1869: prediction of eka-elements, the path to atomic-number ordering, and the modern table | none | — |
16.02.01 | Symmetry and group theory in chemistry | none | — |
16.02.02 | Character tables and reducible representations: vibrational modes and spectroscopy | none | — |
16.02.03 | Projection operators and symmetry-adapted linear combinations (SALCs) | none | — |
16.02.04 | Molecular symmetry and point groups: Schoenflies notation, character tables, and group theory in chemistry | none | — |
16.03.01 | Crystal field theory fundamentals | none | — |
16.03.02 | Crystal field splitting in octahedral complexes | none | — |
16.03.03 | Tetrahedral and square-planar crystal fields: Tanabe-Sugano diagrams and d-d transitions | none | — |
16.03.04 | Jahn-Teller distortions: orbital degeneracy, distortion modes, and Cu(II) complexes | none | — |
16.04.01 | Coordination chemistry | none | — |
16.04.02 | Crystal field stabilization energy and the spectrochemical series | none | — |
16.04.03 | Ligand field theory: molecular orbital treatment of octahedral complexes | none | — |
16.04.04 | Stability of coordination complexes: thermodynamic and kinetic stability, the chelate effect | none | — |
16.04.05 | Alfred Werner and the foundation of coordination chemistry: octahedral geometry, isomerism, and the 1893 coordination theory | none | — |
16.05.01 | Organometallic chemistry | none | — |
16.05.02 | Organometallic reaction types: oxidative addition, reductive elimination, insertion, and beta-hydride elimination | none | — |
16.05.03 | Homogeneous catalysis: Wilkinson's catalyst and Ziegler-Natta polymerization | none | — |
16.05.04 | Ferrocene and the sandwich compounds: metallocene discovery, the 18-electron rule, and Wilkinson-Fischer Nobel revolution | none | — |
16.06.01 | Bioinorganic chemistry | none | — |
16.06.02 | Oxygen transport and storage: hemoglobin, myoglobin, and cooperative binding | none | — |
16.06.03 | Metalloenzyme active sites: nitrogenase, cytochrome P450, and zinc proteases | none | — |
16.06.04 | Iron-sulfur clusters: [2Fe-2S], [4Fe-4S] cubanes, electron-transfer proteins, and the origins-of-life question | none | — |
16.07.01 | Solid-state chemistry | none | — |
16.07.02 | Crystal structures: close-packing, unit cells, and ionic radius ratios | none | — |
16.07.03 | Defects in solids: point defects, line defects, and their role in conductivity | none | — |
16.07.04 | Electronic properties of solids: insulators, semiconductors, conductors, and the band gap | none | — |
16.07.05 | Perovskite solar cells: ABX3 lead-halide crystal chemistry, defect tolerance, and the photovoltaic revolution | none | — |
16.07.06 | The lithium-ion battery: Goodenough, Yoshino, intercalation chemistry, and the 2019 Nobel Prize | none | — |
16.08.01 | Main-group descriptive chemistry — s-block, p-block, and periodic structure | none | — |
16.08.02 | Noble gas compounds: xenon fluorides and the fall of the octet rule | none | — |
16.08.03 | Boron hydrides and Wade's rules: cluster chemistry, electron counting, and the polyhedral skeletal electron pair theory | none | — |
16.09.01 | Lanthanides and actinides — the f-block | none | — |
16.09.02 | Actinide chemistry and the nuclear fuel cycle | none | — |
16.09.03 | Lanthanide luminescence, f-f transitions, and the antenna effect | none | — |
16.10.01 | Catalysis — homogeneous, heterogeneous, and enzyme | none | — |
16.10.02 | Electrocatalysis and water splitting: the oxygen-evolution reaction, the hydrogen-evolution reaction, and the search for non-precious-metal catalysts | none | — |
Molecular & cellular biology
48 units · full: 0 · partial: 0 · none: 48 · stub: 0
| id | title | status | module |
|---|---|---|---|
17.01.01 | Biomolecules in cells — overview | none | — |
17.01.02 | Protein structure: primary through quaternary, the Ramachandran plot, alpha-helix and beta-sheet | none | — |
17.01.03 | Carbohydrate chemistry: monosaccharides, glycosidic bonds, polysaccharides, and glycoconjugates | none | — |
17.01.04 | Lipid chemistry: fatty acids, glycerophospholipids, sphingolipids, sterols, and their roles | none | — |
17.02.01 | Cell membranes: structure | none | — |
17.02.02 | Membrane transport — passive and active | none | — |
17.02.03 | Membrane proteins: integral versus peripheral, topology, and lipid-protein interactions | none | — |
17.02.04 | Vesicle trafficking: SNARE proteins, clathrin-coated vesicles, and the secretory pathway | none | — |
17.03.01 | Cellular organization: organelles | none | — |
17.03.02 | Cytoskeleton and contractile proteins | none | — |
17.03.03 | Nucleus and nuclear transport: nuclear pore complex, importins, exportins, and chromatin organization | none | — |
17.03.04 | Endoplasmic reticulum and Golgi: protein folding, glycosylation, and vesicle budding | 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 | — |
17.04.04 | Fatty acid metabolism: beta-oxidation, fatty acid synthesis, and the acetyl-CoA hub | none | — |
17.04.05 | Amino acid catabolism: transamination, the urea cycle, and amino acid biosynthesis overview | none | — |
17.04.06 | Metabolic regulation: allosteric control, covalent modification, and the AMP kinase switch | none | — |
17.05.01 | DNA replication | none | — |
17.05.02 | Transcription | none | — |
17.05.03 | Translation | none | — |
17.05.04 | RNA processing: 5-prime capping, splicing, 3-prime polyadenylation, and alternative splicing | none | — |
17.05.05 | Ribosomes and the genetic code: codon-anticodon recognition and aminoacyl-tRNA synthetases | none | — |
17.06.01 | Mutation and repair | none | — |
17.06.02 | DNA repair pathways: base excision, nucleotide excision, mismatch repair, and double-strand break repair | none | — |
17.06.03 | Transposable elements: mechanisms of transposition, genome evolution, and epigenetic silencing | none | — |
17.06.04 | Epigenetics: histone modification, DNA methylation, X-inactivation, and imprinting | none | — |
17.07.01 | Cell signaling: receptors and GPCRs | none | — |
17.07.02 | Receptor tyrosine kinases and the MAPK signaling cascade | none | — |
17.07.03 | PI3K-Akt-mTOR pathway: growth factor response, nutrient sensing, and cancer connections | none | — |
17.07.04 | NF-kB and JAK-STAT pathways: cytokine signaling and transcriptional responses | none | — |
17.08.01 | Cell cycle and mitosis | none | — |
17.08.02 | Cyclin-CDK complexes: regulation of each cell cycle transition and checkpoint mechanisms | none | — |
17.08.03 | Meiosis: recombination, crossing over, and the generation of genetic diversity | none | — |
17.08.04 | Apoptosis: intrinsic and extrinsic pathways, the caspase cascade, and Bcl-2 family regulation | none | — |
17.09.01 | Resting membrane potential and ion channels | none | — |
17.09.02 | The action potential — ionic basis | none | — |
17.09.03 | Synaptic transmission: neurotransmitter release, SNARE-dependent exocytosis, and receptor gating | none | — |
17.09.04 | Ion channel pharmacology: channel types, blockers, and the molecular basis of excitability | none | — |
17.10.01 | Innate immunity at the molecular level | none | — |
17.10.02 | Adaptive immunity overview: B cells, T cells, clonal selection, and the antibody response | none | — |
17.10.03 | MHC and antigen presentation: class I and II pathways, and T cell activation | none | — |
17.10.04 | Antibody structure and diversity: V(D)J recombination, affinity maturation, and isotype switching | none | — |
17.11.01 | Cell and molecular biology methods — microscopy, PCR, sequencing, CRISPR | none | — |
17.11.02 | Optical tweezers and single-molecule force spectroscopy | none | — |
17.11.03 | CRISPR-Cas9 genome editing: PAM recognition, sgRNA guidance, and DSB repair outcomes | none | — |
17.12.01 | Cytoskeleton, molecular motors, and cell motility | none | — |
17.12.02 | Molecular motors: kinesin, dynein, and myosin mechanics | none | — |
Organismal biology
51 units · full: 0 · partial: 0 · none: 51 · stub: 0
| id | title | status | module |
|---|---|---|---|
18.01.01 | Body plans and organization | none | — |
18.01.02 | Body plan diversity: symmetry types, coelom origins, and the major animal phyla | none | — |
18.01.03 | The Ediacaran biota, the Cambrian explosion, and the origin of animals | none | — |
18.02.01 | Cardiovascular physiology — the heart | none | — |
18.02.02 | Cardiac action potentials, pacemaker physiology, and the ECG | none | — |
18.02.03 | Hemodynamics: Poiseuille's law, Laplace's law, and blood pressure regulation | none | — |
18.02.04 | The cardiac cycle: systole and diastole, the Wiggers diagram, and cardiac output regulation | none | — |
18.02.05 | Cardiac gap junctions: connexin architecture, intercellular coupling, and arrhythmia | none | — |
18.03.01 | Respiratory physiology — gas exchange and transport | none | — |
18.03.02 | Lung mechanics: compliance, surfactant, the work of breathing, and spirometry values | none | — |
18.03.03 | Gas exchange and transport: the oxygen-hemoglobin dissociation curve and CO2 transport as bicarbonate | none | — |
18.03.04 | Control of ventilation: medullary rhythmogenesis, central and peripheral chemoreceptors, and integrated ventilatory response | none | — |
18.04.01 | Skeletal muscle physiology | none | — |
18.04.02 | Muscle contraction — the actin-myosin cycle | none | — |
18.04.03 | Motor unit recruitment and fatigue: slow versus fast twitch, the size principle, and graded force | none | — |
18.04.04 | Skeletal mechanics: bone remodeling, joint biomechanics, and the lever principles | none | — |
18.04.05 | Bone remodeling and osteoporosis: RANKL-RANK-OPG signaling, bisphosphonates, and the Frost mechanostat | none | — |
18.05.01 | Nervous system — gross anatomy and systems | none | — |
18.05.02 | Spinal cord and reflex arcs: monosynaptic stretch reflex, reciprocal inhibition, and pain pathways | none | — |
18.05.03 | Brain regions: cerebral cortex functional areas, basal ganglia, cerebellum, and limbic system | none | — |
18.05.04 | The autonomic nervous system: sympathetic versus parasympathetic, neurotransmitters, and homeostasis | none | — |
18.06.01 | Digestive physiology and nutrition | none | — |
18.06.02 | Gastrointestinal motility and secretion: peristalsis, gastric acid, enzymes, and absorption sites | none | — |
18.06.03 | Nutrient absorption: monosaccharides, amino acids, lipid micelles, and the enterohepatic circulation | none | — |
18.06.04 | The gut-brain axis: vagal signaling, the microbiome, and enteric neuroscience | none | — |
18.07.01 | Endocrine system — hormones and regulation | none | — |
18.07.02 | Hypothalamic-pituitary axis: releasing hormones, tropic hormones, and feedback regulation | none | — |
18.07.03 | Glucose homeostasis: insulin and glucagon action, the fed versus fasted state, and diabetes mechanisms | none | — |
18.07.04 | Thyroid hormones and metabolic regulation: HPT axis, iodine biochemistry, and autoimmune thyroid disease | none | — |
18.08.01 | Renal physiology — homeostasis and the nephron | none | — |
18.08.02 | Nephron function: filtration, tubular reabsorption, secretion, and the countercurrent multiplier | none | — |
18.08.03 | Acid-base balance: the bicarbonate buffer system, respiratory and metabolic compensation | none | — |
18.08.04 | The renin-angiotensin-aldosterone system: RAAS physiology, hypertension, and ACE-inhibitor therapy | none | — |
18.09.01 | Reproductive biology | none | — |
18.09.02 | Gametogenesis: spermatogenesis and oogenesis, and hormonal control by FSH and LH | none | — |
18.09.03 | Fertilization and early development: implantation, HCG signaling, and placental formation | none | — |
18.09.04 | The menstrual cycle: HPO axis, follicular and luteal phases, and Knobil's discovery of pulsatile GnRH | none | — |
18.10.01 | Immunology | none | — |
18.10.02 | Inflammation: innate recognition via TLRs and NLRs, cytokine storm, and resolution mechanisms | none | — |
18.10.04 | Vaccines and immunological memory: antigen presentation, B-cell clonal selection, and the principle of herd immunity | none | — |
18.11.01 | Embryology and morphogenesis | none | — |
18.11.02 | Gastrulation and axis formation: organizer signaling and Hox gene patterning | none | — |
18.11.03 | Organogenesis: inductive interactions, morphogen gradients, and stem cell niches | none | — |
18.11.04 | Induced pluripotent stem cells: the Yamanaka factors, reprogramming, and the regenerative-medicine revolution | none | — |
18.12.01 | Plant physiology — transport, photosynthesis, hormones, and stress | none | — |
18.12.02 | Photosynthesis pathways (C3/C4/CAM) and plant water relations | none | — |
18.12.03 | Plant hormones: auxin, gibberellin, ethylene, abscisic acid, and the control of plant development | none | — |
18.13.01 | Sensory systems — vision, hearing, balance, taste, smell | none | — |
18.13.02 | Hair cell mechanotransduction and cochlear frequency tuning | none | — |
18.13.03 | Chemosensation: taste, smell, and the Buck-Axel olfactory receptor revolution | none | — |
18.13.04 | Phototransduction: rod and cone physiology, the cGMP cascade, and the molecular basis of vision | none | — |
Ecology & evolution
45 units · full: 0 · partial: 0 · none: 45 · stub: 0
| id | title | status | module |
|---|---|---|---|
19.01.01 | Mendelian genetics — segregation and dominance | none | — |
19.01.02 | Linkage, crossing over, and genetic maps: LOD scores and recombination frequencies | none | — |
19.01.03 | Sex-linked inheritance, dosage compensation, and X-inactivation | none | — |
19.02.01 | Hardy-Weinberg equilibrium | none | — |
19.02.02 | Extensions of Hardy-Weinberg: multiple alleles, X-linked loci, and inbreeding | none | — |
19.02.03 | Mutation-selection balance: the equilibrium frequency of deleterious alleles | none | — |
19.02.04 | Migration and gene flow: the island model, FST, and genetic structure | 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 | — |
19.04.02 | Neutral theory: Kimura's neutral theory and synonymous versus nonsynonymous substitution rates | none | — |
19.04.03 | Coalescent theory: the genealogy of a sample, TMRCA, and linkage disequilibrium | none | — |
19.05.01 | Quantitative genetics — heritability and the breeder's equation | none | — |
19.05.02 | Twin studies and the estimation of heritability: additive versus dominance variance | none | — |
19.05.03 | Polygenic adaptation and GWAS: quantitative trait loci and the infinitesimal model | none | — |
19.06.01 | Speciation — allopatric and sympatric | none | — |
19.06.02 | Reinforcement, reproductive isolation, and alternatives to the biological species concept | none | — |
19.06.03 | Hybrid zones and introgression: tension zones and the mosaic model | none | — |
19.06.04 | Founder-effect and peripatric speciation: Mayr, genetic drift, and the founder-flush model | none | — |
19.07.01 | Phylogenetics — tree reconstruction | none | — |
19.07.02 | Molecular clock hypothesis: calibration, rate heterogeneity, and Bayesian divergence dating | none | — |
19.07.03 | Character evolution: ancestral state reconstruction, Brownian motion, and Pagel's lambda | none | — |
19.08.01 | Macroevolution | none | — |
19.08.02 | Mass extinctions: the Big Five, recovery dynamics, and the kill curve | none | — |
19.08.03 | Trends in the fossil record: Cope's rule, the Red Queen hypothesis, and evolutionary stasis | none | — |
19.09.01 | Population ecology — Lotka-Volterra | none | — |
19.09.02 | Metapopulation dynamics: the Levins model, rescue effect, and habitat fragmentation | none | — |
19.09.03 | Age-structured populations: the Leslie matrix, stable age distribution, and reproductive value | none | — |
19.10.01 | Community ecology — interactions and food webs | none | — |
19.10.02 | Succession: primary and secondary succession and the intermediate disturbance hypothesis | none | — |
19.10.03 | Food webs, interaction strength, and trophic cascades | none | — |
19.11.01 | Ecosystem ecology | none | — |
19.11.02 | Ecosystem energy flow: trophic pyramids, primary productivity, and ecological efficiency | none | — |
19.11.03 | Ecosystem stoichiometry: the Redfield ratio and nutrient limitation | none | — |
19.12.01 | Biogeography | none | — |
19.12.02 | Island biogeography and the species-area relationship | none | — |
19.13.01 | Coevolution | none | — |
19.13.02 | Coevolutionary arms races and the Red Queen hypothesis | none | — |
19.13.02 | Conservation strategies: population viability analysis, corridor design, and adaptive management | none | — |
19.14.01 | Conservation biology | none | — |
19.15.01 | Origin of life — mechanistic scenarios | none | — |
19.15.02 | RNA world and prebiotic chemistry | none | — |
19.16.01 | Nutrient cycles: carbon, nitrogen, and phosphorus cycles and their anthropogenic disruption | none | — |
Health & medicine
39 units · full: 0 · partial: 0 · none: 39 · stub: 0
| id | title | status | module |
|---|---|---|---|
35.01.01 | The human body: organ systems and homeostasis | none | — |
35.01.02 | Homeostasis and feedback regulation: set points, negative feedback, allostasis | none | — |
35.01.03 | Cellular senescence and the biology of aging: telomeres, the Hayflick limit, and the mTOR/rapamycin longevity pathway | none | — |
35.02.01 | Infectious disease, immunity, and vaccines | none | — |
35.02.02 | Bacterial pathogenesis: virulence factors, toxins, antibiotic resistance mechanisms | none | — |
35.02.03 | Viral pathogenesis: replication cycle, immune evasion, pandemic dynamics (SIR model) | none | — |
35.02.04 | Epidemiology basics: R0, herd immunity, outbreak investigation | none | — |
35.02.05 | HIV/AIDS: retroviral biology, immune pathogenesis, antiretroviral therapy, and the 40-year pandemic | none | — |
35.03.01 | Chronic disease: cardiovascular disease, diabetes, and cancer | none | — |
35.03.02 | Cardiovascular disease: atherosclerosis, risk factors, myocardial infarction pathophysiology | none | — |
35.03.03 | Cancer biology: hallmarks of cancer (Hanahan-Weinberg), oncogenes, tumor suppressors | none | — |
35.03.04 | Metabolic syndrome and diabetes: insulin resistance, adipose tissue dysfunction, complications | none | — |
35.03.05 | Neurodegenerative disease: protein misfolding, Alzheimer's, Parkinson's, and the amyloid/tau/synucleinopathies | none | — |
35.04.01 | Nutrition: macronutrients, micronutrients, and diet | none | — |
35.04.02 | Macronutrient metabolism: carbohydrates, fats, proteins — energy balance and metabolic fates | none | — |
35.04.03 | Micronutrients and deficiency diseases: vitamins, minerals, and their biochemical roles | none | — |
35.04.04 | The obesity epidemic: leptin signaling, the thrifty-gene hypothesis, and the Set-Point theory of body weight | none | — |
35.05.01 | Mental health: disorders, stigma, and treatment | none | — |
35.05.02 | Mental health epidemiology: global burden, social determinants, stigma | none | — |
35.05.03 | Neurobiology of addiction: reward circuitry, dopamine dysregulation, withdrawal | none | — |
35.05.04 | Ketamine and the glutamatergic revolution in depression: NMDA antagonism, BDNF, and the rapid antidepressant effect | none | — |
35.06.01 | Public health, epidemiology, and health systems | none | — |
35.06.02 | Health disparities and social determinants: income, race, geography, and life expectancy | none | — |
35.06.03 | Vaccine science: immunization schedules, herd immunity calculation, vaccine hesitancy | none | — |
35.06.04 | The US opioid epidemic: OxyContin, the Sackler family, and the public-health response | none | — |
35.06.05 | Vaccine hesitancy: Wakefield, the MMR-autism fraud, and the public-health communication challenge | none | — |
35.07.01 | Pharmacology: how drugs work and ethics | none | — |
35.07.02 | Pharmacokinetics: ADME (absorption, distribution, metabolism, excretion), half-life, drug interactions | none | — |
35.07.03 | Drug classes and mechanisms: antibiotics (MOA), antihypertensives, psychotropics | none | — |
35.07.04 | Cytochrome P450 pharmacogenomics: CYP3A4/2D6 polymorphism, drug-drug interactions, and precision dosing | none | — |
35.08.01 | Future of medicine: genomics, AI, and global health | none | — |
35.08.02 | Genomic medicine: GWAS, polygenic risk scores, gene therapy (CRISPR-Cas9 therapeutic applications) | none | — |
35.08.03 | Precision medicine and AI in healthcare: biomarker discovery, clinical decision support | none | — |
35.08.04 | CRISPR therapeutics: Casgevy, the first FDA-approved gene-editing drug, and the clinical translation of genome editing | none | — |
35.09.01 | Medical diagnostics and imaging — clinical reasoning and seeing inside the body | none | — |
35.09.04 | Magnetic resonance imaging physics: Bloch equations, relaxation, and the Lauterbur-Mansfield Nobel revolution | none | — |
35.10.01 | Surgery and emergency medicine | none | — |
35.10.02 | Cardiopulmonary bypass and the heart-lung machine: Gibbon's decades-long quest, Bigelow's hypothermia, and the open-heart-surgery revolution | none | — |
35.10.03 | Organ transplantation: HLA matching, rejection immunology, and the Murray-Calne Nobel revolution | none | — |
Earth science
37 units · full: 0 · partial: 0 · none: 37 · stub: 0
| id | title | status | module |
|---|---|---|---|
27.01.01 | Plate tectonics and continental drift | none | — |
27.01.02 | Plate boundaries: divergent, convergent, transform; seafloor spreading and Wilson cycles | none | — |
27.01.03 | Mantle convection and driving forces: slab pull, ridge push, hot spots | none | — |
27.01.04 | Mantle plumes, hot spots, and large igneous provinces: Hawaii, Yellowstone, and the Deccan Traps | none | — |
27.02.01 | Minerals, rocks, and the rock cycle | none | — |
27.02.02 | Rock cycle and igneous processes: crystallization, Bowen's reaction series | none | — |
27.02.03 | Metamorphic and sedimentary rocks: facies, diagenesis, classification | none | — |
27.02.04 | Metamorphic petrology: Barrovian facies, index minerals, and the interpretation of pressure-temperature paths | none | — |
27.03.01 | Earthquakes, volcanoes, and geologic hazards | none | — |
27.03.02 | Seismic waves: P, S, surface waves; seismograph interpretation and Earth's interior | none | — |
27.03.03 | Volcanic hazards: eruption styles, pyroclastic flows, volcanic risk assessment | none | — |
27.03.04 | Megathrust earthquakes and the seismic cycle: subduction-zone seismicity, Cascadia 1700, and the 2011 Tohoku event | none | — |
27.04.01 | Atmosphere, weather, and climate basics | none | — |
27.04.02 | Atmospheric circulation: Hadley, Ferrel, Polar cells; jet streams and Coriolis effect | none | — |
27.04.03 | Weather systems: fronts, cyclones, severe weather formation | none | — |
27.04.04 | El Nino-Southern Oscillation (ENSO): the Bjerknes feedback, Walker circulation, and global teleconnections | none | — |
27.05.01 | Oceanography: currents, tides, and marine ecosystems | none | — |
27.05.02 | Ocean circulation: thermohaline conveyor, gyres, upwelling zones | none | — |
27.05.03 | Ocean chemistry: salinity, carbonate system, ocean acidification | none | — |
27.05.04 | The biological pump: marine snow, the ocean carbon cycle, and climate regulation | none | — |
27.06.01 | Hydrology: the water cycle and groundwater | none | — |
27.06.02 | Groundwater systems: aquifers, Darcy's law, recharge and depletion | none | — |
27.06.03 | Surface hydrology: watershed dynamics, flood frequency analysis | none | — |
27.07.01 | Climate change: evidence, impacts, and mitigation | none | — |
27.07.02 | Radiative forcing and feedback loops: greenhouse gases, ice-albedo, water vapor | none | — |
27.07.03 | Climate proxies and paleo-climate: ice cores, tree rings, ocean sediment records | none | — |
27.07.04 | Impacts and adaptation: sea-level rise, extreme events, mitigation scenarios | none | — |
27.07.05 | Stratospheric ozone depletion: CFC chemistry, the Antarctic ozone hole, and the Montreal Protocol | none | — |
27.08.01 | Earth history and the geologic time scale | none | — |
27.08.02 | Geologic time scale: radiometric dating, stratigraphy, and mass extinction boundaries | none | — |
27.08.03 | Evolution of the atmosphere and biosphere: Great Oxidation Event, Snowball Earth | none | — |
27.08.04 | The Permian-Triassic mass extinction: the Siberian Traps, the 'Great Dying,' and the collapse of the Paleozoic biosphere | none | — |
27.09.01 | Structural geology — stress, strain, folds, and faults | none | — |
27.09.02 | Folding, faulting, and rock deformation mechanics | none | — |
27.10.01 | Soil science — formation, profiles, and the critical zone | none | — |
27.10.02 | Soil classification, horizons, and the critical zone | none | — |
27.10.03 | Soil carbon cycling and the rhizosphere microbiome: mycorrhizae, biochar, and the largest terrestrial carbon sink | none | — |
Astronomy
33 units · full: 0 · partial: 0 · none: 33 · stub: 0
| id | title | status | module |
|---|---|---|---|
28.01.01 | The solar system: planets, moons, and small bodies | none | — |
28.01.02 | Formation of the solar system: nebular hypothesis, accretion, late heavy bombardment | none | — |
28.01.03 | Planetary interiors and surfaces: comparative planetology, magnetic fields | none | — |
28.01.04 | The Nice model: solar-system dynamical instability, giant-planet migration, and the Late Heavy Bombardment | none | — |
28.02.01 | Stars and stellar evolution | none | — |
28.02.02 | Stellar structure: hydrostatic equilibrium, nuclear burning, the PP chain and CNO cycle | none | — |
28.02.03 | Stellar evolution: main sequence to red giant, HR diagram tracks | none | — |
28.02.04 | Stellar endpoints: white dwarfs (Chandrasekhar limit), neutron stars, pulsars, black holes | none | — |
28.02.05 | Stellar nucleosynthesis: the B²FH process network, nuclear burning stages, and the origin of the elements | none | — |
28.03.01 | Galaxies and the Milky Way | none | — |
28.03.02 | Galaxy classification and structure: Hubble sequence, bulge/disk/halo, dark matter halos | none | — |
28.03.03 | Active galactic nuclei and quasars: accretion onto supermassive black holes | none | — |
28.03.04 | Galaxy formation and evolution: the cosmic star-formation history, hierarchical merging, and the Hubble sequence | none | — |
28.04.01 | Cosmology: the Big Bang, expansion, and fate of the universe | none | — |
28.04.02 | Big Bang nucleosynthesis: primordial abundances of H, He, Li | none | — |
28.04.03 | Cosmic microwave background: temperature fluctuations, acoustic peaks, LCDM parameters | none | — |
28.04.04 | Dark matter and dark energy: evidence, candidates, and the accelerating expansion | none | — |
28.04.05 | Large-scale structure formation: the Press-Schechter formalism, dark-matter halos, and the cosmic web | none | — |
28.05.01 | Exoplanets: detection methods and habitability | none | — |
28.05.02 | Exoplanet detection methods: radial velocity, transit photometry, direct imaging | none | — |
28.05.03 | Exoplanet demographics: occurrence rates, habitability zone, atmospheric characterization | none | — |
28.05.04 | Exoplanet atmospheres: transmission spectroscopy, the habitable zone, and the search for biosignatures | none | — |
28.06.01 | Space exploration: history and future | none | — |
28.06.02 | Space telescope science: multiwavelength astronomy, gravitational-wave observatories | none | — |
28.06.04 | The Apollo program: the Cold War Moon race, the Saturn V, and the twelve Moonwalkers | none | — |
28.07.01 | The interstellar medium and star formation | none | — |
28.07.02 | Molecular clouds and protostellar evolution: Jeans collapse, the Hayashi track, and the H-R diagram pre-main-sequence | none | — |
28.08.01 | High-energy astrophysics — compact objects, accretion, and cosmic explosions | none | — |
28.08.02 | Pulsars and neutron stars: the Lindblad-Bell discovery, the Tolman-Oppenheimer-Volkoff limit, and magnetars | none | — |
28.09.01 | Observational astronomy — telescopes, detectors, and spectroscopy | none | — |
28.09.02 | Adaptive optics, interferometry, and high-angular-resolution astronomy | none | — |
28.10.01 | Astrobiology — life in the universe | none | — |
28.10.02 | Extremophiles, habitable zones, and biosignatures | none | — |
Computer science
13 units · full: 0 · partial: 0 · none: 13 · stub: 0
| id | title | status | module |
|---|---|---|---|
50.01.01 | Computational thinking and algorithms | none | — |
50.02.01 | Data structures: arrays, trees, and graphs | none | — |
50.03.01 | Algorithmic complexity and Big-O notation | none | — |
50.04.01 | Programming paradigms: functional, OOP, and beyond | none | — |
50.05.01 | Operating systems: processes and memory | none | — |
50.06.01 | Computer networks and internet architecture | none | — |
50.07.01 | Databases: relational, NoSQL, and data modeling | none | — |
50.08.01 | Cybersecurity, encryption, and privacy | none | — |
50.09.01 | Artificial intelligence and machine learning | none | — |
50.10.01 | Software engineering and design patterns | none | — |
50.11.01 | Distributed systems and consensus | none | — |
50.12.01 | Computing ethics and societal impact | none | — |
50.13.01 | Computer architecture — from logic gates to modern CPUs | none | — |
Philosophy
56 units · full: 0 · partial: 0 · none: 56 · stub: 0
| id | title | status | module |
|---|---|---|---|
20.01.01 | Epistemology: knowledge, justification, and truth | none | — |
20.01.02 | Theories of justification: foundationalism, coherentism, reliabilism | none | — |
20.01.03 | Epistemological skepticism and responses: Cartesian doubt, contextualism, externalism | none | — |
20.01.04 | Social epistemology: testimony, trust, virtue, and the social structure of knowledge | none | — |
20.02.01 | Theories of justice: Rawls, Nozick, and fairness | 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 | — |
20.02.06 | Ethics of artificial intelligence | none | — |
20.02.07 | Consequentialism in depth: act vs. rule utilitarianism, preference satisfaction, effective altruism | none | — |
20.02.08 | Deontology: Kant's categorical imperative, rights theory, contractualism (Scanlon) | none | — |
20.02.09 | Virtue ethics: Aristotle's eudaimonia, contemporary neo-Aristotelianism, care ethics | none | — |
20.03.01 | The measurement problem in quantum mechanics | none | — |
20.03.02 | Philosophy of quantum mechanics: interpretations (Copenhagen, Many-Worlds, Bohmian, QBism) | none | — |
20.03.03 | Space, time, and relativity: substantivalism vs. relationalism, the direction of time | none | — |
20.03.04 | The anthropic principle and cosmic fine-tuning: Carter, Barrow-Tipler, and the multiverse response | none | — |
20.04.01 | Aesthetics: beauty, art, and judgment | none | — |
20.04.02 | Kant's Critique of Judgment: the beautiful, the sublime, and disinterested pleasure | none | — |
20.04.03 | The sublime and the beautiful: Burke, Kant, and the eighteenth-century origins of aesthetics | none | — |
20.05.02 | The unit of selection | none | — |
20.05.03 | Natural selection and teleology: function, design, and Darwinian explanation | none | — |
20.05.04 | Reductionism in biology: levels of organization, multiple realizability, emergence | none | — |
20.05.05 | Evo-devo: evolutionary developmental biology, deep homology, and the genetic tool-kit | none | — |
20.06.01 | Consciousness: the hard problem, qualia, and the mind-body debate | none | — |
20.06.02 | The hard problem of consciousness: qualia, the explanatory gap, and physicalist responses | none | — |
20.06.03 | Theories of consciousness: Global Workspace, Integrated Information Theory (IIT), higher-order theories | none | — |
20.06.04 | The neuroscience of consciousness: neural correlates, anesthesia, and disorders of consciousness | none | — |
20.07.01 | Democratic theory: participation, deliberation, and representation | none | — |
20.07.02 | Deliberative democracy and the public sphere | none | — |
20.08.01 | Philosophy of science: demarcation, falsification, and paradigms | none | — |
20.08.02 | Scientific realism vs. anti-realism: structural realism, constructive empiricism (van Fraassen) | none | — |
20.08.03 | Causation and explanation: counterfactual analysis, mechanistic explanation, causal inference | none | — |
20.08.04 | Kuhn, Lakatos, and Laudan: paradigms, research programmes, and the theory-ladenness of scientific change | none | — |
20.09.01 | Philosophy of mathematics: Platonism, constructivism, and the nature of numbers | none | — |
20.09.02 | Foundations of mathematics: logicism (Russell), formalism (Hilbert), intuitionism (Brouwer) | none | — |
20.09.03 | Mathematical ontology: Platonism, nominalism, structuralism, fictionalism | none | — |
20.09.04 | Gödel's incompleteness theorems: the demise of Hilbert's program and the limits of formal reasoning | none | — |
20.10.01 | Confucianism: ethics, society, and the exemplary person | none | — |
20.11.01 | Buddhism: the Four Noble Truths, the Eightfold Path, and the question of suffering | none | — |
20.11.01 | Metaphysics — existence, identity, causation, modality | none | — |
20.11.02 | Metaphysics — depth: grounding, persistence, and possible worlds | none | — |
20.11.03 | Personal identity: Parfit, fission, teleportation, and the reductionist view of the self | none | — |
20.12.01 | Advaita Vedanta and Hindu philosophy: Brahman, Atman, and the question of reality | none | — |
20.12.01 | Philosophy of language — reference, meaning, and use | none | — |
20.12.02 | Frege, Russell, and theories of reference | none | — |
20.12.03 | Speech act theory: Austin, Searle, Grice, and how words do things | none | — |
20.13.01 | Daoism: wu wei, the Dao, and natural harmony | none | — |
20.13.01 | Philosophy of mind — consciousness, physicalism, and the mental | none | — |
20.13.02 | Embodied, embedded, enacted, extended cognition | none | — |
20.14.01 | Political philosophy — justice, authority, and the state | none | — |
20.14.02 | Social contract theory: Hobbes, Locke, and Rousseau on sovereignty and legitimacy | none | — |
20.14.03 | The capabilities approach: Sen, Nussbaum, and justice as the freedom to live a valued life | none | — |
20.15.01 | History of philosophy — ancient, medieval, modern, and continental | none | — |
20.15.02 | Kant's Critique of Pure Reason: transcendental idealism, the synthetic a priori, and the categories | none | — |
20.15.03 | Husserl's phenomenology: intentionality, the reduction, and transcendental consciousness | none | — |
Psychology
59 units · full: 0 · partial: 0 · none: 59 · stub: 0
| id | title | status | module |
|---|---|---|---|
29.01.01 | Introduction to psychology and research methods | none | — |
29.01.02 | The replication crisis and Open Science: reproducibility, preregistration, and the reform of psychological research | none | — |
29.01.02 | Research designs: experiments, quasi-experiments, correlational, and meta-analysis | none | — |
29.01.03 | Statistical reasoning in psychology: effect size, power, replication crisis | none | — |
29.02.01 | Neuroscience: brain and behaviour | none | — |
29.02.02 | Brain regions and function: lobes, limbic system, basal ganglia, cerebellum | none | — |
29.02.03 | Neurotransmitter systems: dopamine, serotonin, norepinephrine, GABA, glutamate | none | — |
29.02.04 | Neuroplasticity and neurogenesis: LTP, critical periods, recovery from brain injury | none | — |
29.02.05 | The neuroscience of sleep: NREM and REM, the Saper wake-sleep flip-flop, and the functions of sleep | none | — |
29.03.01 | Sensation and perception: how the brain constructs reality from sensory data | none | — |
29.03.02 | Visual perception: depth cues, object recognition, top-down vs. bottom-up processing | none | — |
29.03.03 | Auditory and somatosensory systems; pain and the gate-control theory | none | — |
29.03.04 | Hubel and Wiesel's visual cortex architecture: orientation columns, ocular dominance, and hypercolumns | none | — |
29.04.01 | Learning and memory: conditioning, cognitive maps, encoding, storage, retrieval, and forgetting | none | — |
29.04.02 | Classical and operant conditioning: schedules of reinforcement, extinction, latent learning | none | — |
29.04.03 | Memory systems: episodic, semantic, procedural; encoding, consolidation, retrieval | none | — |
29.04.04 | Forgetting and false memory: interference, decay, misinformation effect | none | — |
29.04.05 | Motor learning and the cerebellum: Marr-Albus theory, long-term depression at Purkinje cells, and adaptive error correction | none | — |
29.05.01 | Cognition and intelligence: thinking, reasoning, and the measurement of mind | none | — |
29.05.02 | Decision-making and judgment: heuristics, biases (Kahneman-Tversky), dual-process theory | none | — |
29.05.03 | Language acquisition: critical period, Chomsky's nativism vs. connectionist accounts | none | — |
29.05.04 | Working memory: the Baddeley-Hitch model, the phonological loop, and the visuospatial sketchpad | none | — |
29.06.01 | Developmental Psychology Across the Lifespan | none | — |
29.06.02 | Cognitive development: Piaget's stages, Vygotsky's ZPD, theory of mind | none | — |
29.06.03 | Moral development: Kohlberg's stages, Gilligan's critique, moral foundations theory | none | — |
29.06.04 | Adult development and aging: identity formation, midlife, and cognitive change in aging | none | — |
29.06.05 | Adolescent brain development: prefrontal cortex maturation, synaptic pruning, and the dual-systems model of risk-taking | none | — |
29.07.01 | Social psychology: social influence, group dynamics, prejudice, and relationships | none | — |
29.07.02 | Attitudes and persuasion: cognitive dissonance, elaboration likelihood model | none | — |
29.07.03 | Group dynamics: conformity (Asch), obedience (Milgram), groupthink, social loafing | none | — |
29.07.04 | Prejudice and discrimination: implicit bias, stereotype threat, contact hypothesis | none | — |
29.07.05 | Interpersonal attraction and close relationships: adult attachment, social-exchange theories, and the Sternberg triarchic model | none | — |
29.08.01 | Personality theories and assessment | none | — |
29.08.02 | Trait theories: Big Five (OCEAN), psychometric approaches, cross-cultural validity | none | — |
29.08.03 | Psychodynamic and humanistic theories: Freud, Erikson, Rogers, Maslow | none | — |
29.08.04 | The person-situation debate: Mischel 1968, the challenge to trait theory, and the density-distribution resolution | none | — |
29.09.01 | Psychological disorders: diagnosis, controversy, and the limits of classification | none | — |
29.09.02 | Mood disorders: depression, bipolar; neurobiology and cognitive models | none | — |
29.09.03 | Anxiety and trauma disorders: GAD, PTSD, OCD; fear conditioning models | none | — |
29.09.04 | Psychotic and neurodevelopmental disorders: schizophrenia, ADHD, autism spectrum | none | — |
29.09.05 | Eating disorders: anorexia nervosa, bulimia nervosa, binge-eating disorder, and the biology of starvation | none | — |
29.10.01 | Therapy and treatment approaches | none | — |
29.10.02 | Evidence-based therapies: CBT, DBT, exposure therapy; efficacy meta-analyses | none | — |
29.10.03 | Biological treatments: pharmacotherapy mechanisms, ECT, TMS, psychedelic-assisted therapy | none | — |
29.10.04 | Psychodynamic psychotherapy: Freud, Jung, and the object-relations tradition; transference, interpretation, and the therapeutic alliance | none | — |
29.11.01 | Motivation and emotion: drives, needs, feelings, and the forces that shape behaviour | none | — |
29.11.02 | Emotion theories: James-Lange, Cannon-Bard, Schachter-Singer, appraisal theory | none | — |
29.11.03 | Stress and health psychology: allostatic load, coping strategies, psychoneuroimmunology | none | — |
29.11.04 | Self-determination theory: Maslow, Deci-Ryan, and the intrinsic motivation of autonomy, competence, and relatedness | none | — |
29.12.01 | Cross-Cultural and Indigenous Psychology | none | — |
29.12.02 | Cross-cultural cognition: Nisbett's geography of thought, East Asian holistic vs Western analytic reasoning | none | — |
29.13.01 | Psychometrics — test construction, reliability, and validity | none | — |
29.13.02 | Classical test theory, reliability, and factor analysis | none | — |
29.13.03 | Item response theory and the Rasch model: latent-trait measurement | none | — |
29.13.04 | Intelligence testing history: Binet-Simon 1905, Spearman's g, the Flynn effect, and the controversy over IQ | none | — |
29.14.01 | History of psychology — from philosophy of mind to cognitive neuroscience | none | — |
29.14.02 | The cognitive revolution and the computational theory of mind | none | — |
29.15.01 | Industrial and organizational psychology — work, selection, and motivation | none | — |
29.15.02 | Personnel selection and validity generalization: Schmidt-Hunter meta-analysis and the science of hiring | none | — |
Sociology
28 units · full: 0 · partial: 0 · none: 28 · stub: 0
| id | title | status | module |
|---|---|---|---|
30.01.01 | The sociological imagination and research methods | none | — |
30.01.02 | Sociological research methods: surveys, ethnography, experiments, mixed methods | none | — |
30.01.03 | Classical theory: Marx, Weber, Durkheim; contemporary theoretical perspectives | none | — |
30.02.01 | Culture and society: a global perspective | none | — |
30.02.02 | Cultural diversity and ethnocentrism; cultural relativism and hybridization | none | — |
30.02.03 | Media and culture industry: Adorno-Horkheimer, moral panic, media framing | none | — |
30.03.01 | Socialization and Identity Formation | none | — |
30.03.02 | Agents of socialization: family, peers, schools, media; resocialization | none | — |
30.03.03 | Identity and the self: symbolic interactionism (Mead, Goffman), stigma | none | — |
30.04.01 | Social stratification: class, race, gender, and caste | none | — |
30.04.02 | Class structure and mobility: measurement, intergenerational mobility, the American Dream myth | none | — |
30.04.03 | Race and ethnicity: systemic racism, intersectionality (Crenshaw), racial formation theory | none | — |
30.04.04 | Gender inequality: gender gap in wages, occupational segregation, feminist theory | none | — |
30.05.01 | Social institutions: family, education, religion, and media | none | — |
30.05.02 | Family structure and change: marriage forms, divorce trends, kinship systems | none | — |
30.05.03 | Education and religion as institutions: reproduction of inequality, secularization | none | — |
30.06.01 | Deviance and social control: who defines normal, who punishes difference, and why it matters | none | — |
30.06.02 | Crime and the criminal justice system: incarceration rates, mass incarceration, racial disparities | none | — |
30.06.03 | Social control theories: labeling theory (Becker), strain theory (Merton), control theory | none | — |
30.07.01 | Globalization and social movements: how people organize across borders to challenge power | none | — |
30.07.02 | Social movements: resource mobilization, framing theory, new social movements | none | — |
30.07.03 | Global inequality: world-systems theory (Wallerstein), dependency theory, development debates | none | — |
30.08.01 | Urbanization and demography: the global movement of people and the cities that shape their lives | none | — |
30.08.02 | Demographic transition: birth/death rate dynamics, population projections | none | — |
30.08.03 | Urban sociology: segregation, gentrification, urban ecology | none | — |
30.09.01 | Race and ethnicity — the social construction of difference | none | — |
30.10.01 | Gender and sexuality — sociology of | none | — |
30.10.02 | Feminist theory — intersectionality, performativity, and the gender structure | none | — |
Anthropology
18 units · full: 0 · partial: 0 · none: 18 · stub: 0
| id | title | status | module |
|---|---|---|---|
31.01.01 | Anthropology: the four fields and holism | none | — |
31.01.02 | Anthropological theory: evolutionism, diffusionism, functionalism, structuralism, postmodernism | none | — |
31.02.01 | Cultural anthropology: ethnography and fieldwork | none | — |
31.02.02 | Ethnographic methods: participant observation, fieldwork ethics, thick description (Geertz) | none | — |
31.02.03 | Kinship, marriage, and family: descent systems, alliance theory, cross-cultural variation | none | — |
31.02.04 | Religion, ritual, and symbolism: rites of passage (Van Gennep), shamanism, magic and science | none | — |
31.03.01 | Archaeology: material culture and excavation | none | — |
31.03.02 | Archaeological methods: stratigraphy, typology, dating techniques, remote sensing | none | — |
31.03.03 | Prehistoric cultures: Paleolithic tool traditions, the Neolithic revolution, state formation | none | — |
31.04.01 | Biological anthropology: evolution and hominins | none | — |
31.04.02 | Human evolution: the hominin fossil record, Out of Africa, Neanderthal admixture | none | — |
31.04.03 | Human variation and race: the non-existence of biological races, clines, and adaptation | none | — |
31.05.01 | Linguistic anthropology: language, culture, and society | none | — |
31.05.02 | Language and culture: Sapir-Whorf hypothesis, language endangerment, code-switching | none | — |
31.05.03 | Historical linguistics: proto-languages, sound change, language families | none | — |
31.06.01 | Applied anthropology: globalization, ethics, and decolonization | none | — |
31.06.02 | Medical anthropology: culture and illness, ethnomedicine, global health disparities | none | — |
31.06.03 | Development and humanitarian anthropology: critiques of development, participatory approaches | 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
19 units · full: 0 · partial: 0 · none: 19 · 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 | — |
22.04.01 | Reading guide: The Catcher in the Rye (Salinger) | none | — |
22.04.02 | Reading guide: The Great Gatsby (Fitzgerald) | none | — |
22.04.03 | Reading guide: The Old Man and the Sea (Hemingway) | none | — |
22.04.04 | Reading guide: Nineteen Eighty-Four (Orwell) | none | — |
Linguistics (science of language)
10 units · full: 0 · partial: 0 · none: 10 · stub: 0
| id | title | status | module |
|---|---|---|---|
51.01.01 | Phonetics — the physical science of speech sounds | none | — |
51.02.01 | Phonology — phonemes, features, and sound systems | none | — |
51.02.02 | Autosegmental phonology, feature geometry, and optimality theory | none | — |
51.03.01 | Morphology — morphemes, word structure, and formation | none | — |
51.04.01 | Syntax — constituency, dependency, and generative grammar | none | — |
51.05.01 | Semantics and pragmatics — meaning, truth, and use | none | — |
51.06.01 | Historical and comparative linguistics — change, families, and reconstruction | none | — |
51.06.02 | The Indo-European comparative method: Grimm, Verner, and the laryngeal theory | none | — |
51.07.01 | Sociolinguistics — variation, identity, and language in society | none | — |
51.07.02 | Variationist sociolinguistics: the Labovian method and the apparent-time construct | none | — |
Music & art
30 units · full: 0 · partial: 0 · none: 30 · stub: 0
| id | title | status | module |
|---|---|---|---|
34.01.01 | Music fundamentals: rhythm, melody, and harmony | none | — |
34.01.02 | Harmony and counterpoint: voice leading, functional harmony, modulation | none | — |
34.01.03 | Musical form and analysis: sonata form, rondo, theme and variations; set theory in post-tonal music | none | — |
34.01.04 | Rhythm, meter, and cross-cultural time: Western notation, West African polyrhythm, and additive meters | none | — |
34.02.01 | Music history: Western and world traditions | none | — |
34.02.02 | Western art music survey: Baroque to Romanticism; landmark works and social contexts | none | — |
34.02.03 | 20th-century and world music: modernism, jazz, non-Western traditions, popular music | none | — |
34.02.04 | From Gregorian chant to Notre Dame polyphony: Western music, c. 600-1300 | none | — |
34.03.01 | Visual art: elements, principles, and composition | none | — |
34.03.02 | Color theory and composition principles: Itten, Albers; figure-ground, gestalt | none | — |
34.03.03 | Gestalt psychology and visual composition: proximity, similarity, closure, and the laws of perceptual organization | none | — |
34.04.01 | Art history: cave paintings to contemporary | none | — |
34.04.02 | Modern and contemporary art movements: Impressionism through Abstract Expressionism to Conceptual Art | none | — |
34.04.03 | Italian Renaissance art: from Giotto to Michelangelo, c. 1300-1520 | none | — |
34.05.01 | Film and photography as visual storytelling | none | — |
34.05.02 | Cinéma vérité and direct cinema: the 1960s observational documentary revolution | none | — |
34.06.01 | Architecture and design of the built environment | none | — |
34.06.02 | Architecture and urban design: form follows function, modernism, sustainability | none | — |
34.06.03 | The Bauhaus: Gropius, Dessau, and the modernist synthesis of art, craft, and industry, 1919-1933 | none | — |
34.07.01 | Aesthetics theory: taste, judgment, and culture | none | — |
34.07.02 | Theories of aesthetic experience: Kant's sublime, Dewey's pragmatist aesthetics, institutional theory | none | — |
34.07.03 | The end of art and aesthetic universalism: Danto, Dickie, and the institutional theory | none | — |
34.08.01 | Digital media, art, and technology | none | — |
34.08.02 | Generative art and creative coding: algorithms as aesthetic medium | none | — |
34.09.01 | Theater and drama — structure, history, and craft | none | — |
34.09.02 | Dramatic structure — from Aristotle to Brecht and Beckett | none | — |
34.09.03 | Greek tragedy and comedy: Aeschylus, Sophocles, Euripides, and Aristophanes at the Dionysia | none | — |
34.10.01 | Dance — technique, history, and choreography | none | — |
34.10.02 | Modern dance technique — Graham, Cunningham, and Balanchine | none | — |
34.10.03 | Petipa and the Imperial Russian ballet: classical ballet, Sleeping Beauty, and the 19th-century synthesis | none | — |
World history
52 units · full: 0 · partial: 0 · none: 52 · stub: 0
| id | title | status | module |
|---|---|---|---|
32.01.01 | Prehistory and human migration out of Africa | none | — |
32.01.02 | Human dispersal: anatomically modern humans, migration routes, genetic evidence | none | — |
32.02.01 | Mesopotamia and the Fertile Crescent | none | — |
32.02.02 | Mesopotamia — Sumer, Akkad, Babylon, and the birth of the city | none | — |
32.03.01 | Ancient Egypt and Nubia | none | — |
32.03.02 | Ancient Egypt and Nubia — kingdoms of the Nile | none | — |
32.04.01 | Indus Valley Civilization and Vedic India | none | — |
32.04.02 | The Indus Valley civilization — Harappa, Mohenjo-daro, and an undeciphered script | none | — |
32.05.01 | Ancient China: Shang through Han dynasties | none | — |
32.05.02 | Ancient China — from Shang to Han | none | — |
32.06.01 | Classical Greece and the Hellenistic world | none | — |
32.06.02 | Greek philosophy and democracy: Socrates, Plato, Aristotle; the polis and direct democracy | none | — |
32.07.01 | Roman Republic and Empire: from founding myths to the fall and beyond | none | — |
32.07.02 | Roman law and governance: the Twelve Tables, Roman Republic vs. Empire, legacy in legal systems | none | — |
32.07.03 | The Byzantine Empire: Constantinople, Justinian, and the thousand-year continuation of Rome | none | — |
32.08.01 | Classical India: Mauryan Empire, Gupta Golden Age, and South Indian Kingdoms | none | — |
32.09.01 | Pre-Columbian Americas: Olmec, Maya, Aztec, Inca, and North American civilizations | none | — |
32.10.01 | Islamic Golden Age and the Caliphates | none | — |
32.10.02 | Science and philosophy in the Islamic Golden Age: translation movement, algebra, optics | none | — |
32.11.01 | Medieval Europe and the Crusades | none | — |
32.11.02 | Medieval Europe — feudalism, faith, plague, and the 12th-century renaissance | none | — |
32.12.01 | Sub-Saharan African kingdoms | none | — |
32.12.02 | Sub-Saharan Africa — empires of the Sahel and the Swahili coast | none | — |
32.13.01 | The Mongol Empire and its legacy | none | — |
32.13.02 | The Mongol Empire — the largest contiguous land empire | none | — |
32.14.01 | Age of Exploration: Multiple Perspectives | none | — |
32.14.02 | The Columbian Exchange: biological and demographic consequences of contact | none | — |
32.15.01 | Colonialism and imperialism: colonizer and colonized | none | — |
32.15.02 | Colonialism and imperialism — from conquest to formal empire | none | — |
32.15.03 | Postcolonial theory: Said's Orientalism, Spivak's subaltern, and Bhabha's hybridity | none | — |
32.16.01 | The Atlantic Slave Trade: African, European, and American Perspectives | none | — |
32.16.02 | The Atlantic slave trade — scale, economics, and abolition | none | — |
32.17.01 | Enlightenment and Revolutions: American, French, Haitian, and Latin American | none | — |
32.17.02 | Atlantic revolutions compared: American, French, Haitian, Latin American independence | none | — |
32.18.01 | Industrial Revolution and its global consequences | none | — |
32.18.02 | Industrialization and global inequality: divergence, living standards debate, labor movements | none | — |
32.19.01 | Meiji Japan, Qing collapse, and the Scramble for Africa | none | — |
32.20.01 | World War I: Global Perspectives | none | — |
32.20.02 | Causes and consequences of WWI: alliance systems, total war, the peace settlement | none | — |
32.20.03 | Verdun and the Somme, 1916: industrialized attrition on the Western Front | none | — |
32.21.01 | Interwar Period and the Rise of Totalitarianism: Fascism, Communism, and Liberal Democracy in Crisis | none | — |
32.21.02 | The interwar period — Versailles, depression, and the rise of totalitarianism | none | — |
32.22.01 | World War II: Global Theaters and Multi-Perspective Histories | none | — |
32.22.02 | World War II — causes, course, Holocaust, and the atomic age | none | — |
32.23.01 | Decolonization: India, Algeria, Vietnam, Congo, and the end of empire | none | — |
32.23.02 | Decolonization — the end of European empires and the Third World | none | — |
32.24.01 | The Cold War: US, Soviet, Chinese, and Non-Aligned Perspectives | none | — |
32.24.02 | The Cold War — superpower rivalry, proxy conflicts, and the nuclear revolution | none | — |
32.25.01 | Globalization, Neoliberalism, and the Post-Colonial World | none | — |
32.25.02 | Globalization — trade, finance, networks, and backlash | none | — |
32.26.01 | The 21st Century: Digital Revolution, Climate Crisis, and Shifting Power | none | — |
32.26.02 | Globalization Backlash, the Anthropocene, and the Shifting Global Order | none | — |
History of science
17 units · full: 0 · partial: 0 · none: 17 · stub: 0
| id | title | status | module |
|---|---|---|---|
33.01.01 | Ancient science: Mesopotamia, Greece, China, and India | none | — |
33.01.02 | Greek and Hellenistic science: Aristotle's natural philosophy, Euclidean geometry, Archimedean mechanics | none | — |
33.01.03 | Ancient science — Presocratics, Aristotle, Euclid, Archimedes, Ptolemy, Galen | none | — |
33.02.01 | Islamic Golden Age and medieval European science | none | — |
33.02.02 | Medieval European science: universities, scholasticism, and the recovery of Aristotle | none | — |
33.03.01 | The Scientific Revolution: Copernicus to Newton | none | — |
33.03.02 | Copernicus to Newton: heliocentric model, Kepler's laws, Galileo's telescopic observations | none | — |
33.04.01 | Industrial Revolution, chemistry, and electromagnetism | none | — |
33.04.02 | Chemistry's revolution: phlogiston to Lavoisier, atomic theory (Dalton), periodic table (Mendeleev) | none | — |
33.05.01 | The relativity and quantum revolutions | none | — |
33.05.02 | Quantum revolution: Planck, Einstein, Bohr, Heisenberg, Schrödinger and the Copenhagen consensus | none | — |
33.06.01 | Genetics, DNA, and the molecular biology revolution | none | — |
33.06.02 | The double helix and molecular biology: Franklin, Watson-Crick, and the central dogma | none | — |
33.07.01 | The digital revolution: computing and the internet | none | — |
33.07.02 | Computing history: Turing, von Neumann architecture, transistor, and the internet | none | — |
33.08.01 | Contemporary science: challenges, open science, and the future | none | — |
33.08.02 | Big science and global collaboration: CERN, space race, Human Genome Project | none | — |
Economics
41 units · full: 0 · partial: 0 · none: 41 · 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 | — |
52.01.01 | Microeconomics — scarcity, choice, and equilibrium | none | — |
52.01.02 | Revealed preference and Afriat's theorem: cyclical consistency and the rationalizability of demand | none | — |
52.01.03 | The Edgeworth box, Pareto efficiency, and the two welfare theorems | none | — |
52.02.01 | Macroeconomics — aggregates, growth, and cycles | none | — |
52.02.02 | Solow growth model, convergence, and endogenous growth | none | — |
52.03.01 | Econometrics — identification, estimation, and inference | none | — |
52.04.01 | Game theory — Nash equilibrium and strategic interaction | none | — |
52.04.02 | Signaling games and Bayesian Nash equilibrium: Spence, Akerlof, and the intuitive criterion | none | — |
52.05.01 | International trade and finance — comparative advantage and open-economy macro | none | — |
52.06.01 | Behavioral economics — bounded rationality, biases, and nudges | none | — |
52.06.02 | Neuroeconomics and the dopamine reward prediction error | none | — |
Civics
15 units · full: 0 · partial: 0 · none: 15 · 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 | none | — |
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 | — |
Media literacy
12 units · full: 0 · partial: 0 · none: 12 · stub: 0
| id | title | status | module |
|---|---|---|---|
36.01.01 | Media foundations: history and theory | none | — |
36.01.02 | Media business models: advertising, subscriptions, attention economy, platform economics | none | — |
36.02.01 | News and journalism: verification and source evaluation | none | — |
36.02.02 | Investigative journalism: source verification, FOIA, data journalism methods | none | — |
36.03.01 | Propaganda and persuasion: rhetorical analysis | none | — |
36.03.02 | Disinformation ecosystems: deepfakes, bot networks, coordinated inauthentic behavior | none | — |
36.04.01 | Digital literacy: algorithms, filter bubbles, and echo chambers | none | — |
36.04.02 | Algorithmic curation: filter bubbles, recommendation systems, and epistemic consequences | none | — |
36.05.01 | Visual literacy: images, data visualization, and manipulation | none | — |
36.05.02 | Data visualization and infographics: misleading charts, chartjunk, and honest design | none | — |
36.06.01 | Media ethics: responsible consumption and production | none | — |
36.06.02 | Platform accountability and regulation: Section 230, GDPR, content moderation tradeoffs | none | — |
rhetoric-composition
1 units · full: 0 · partial: 0 · none: 1 · stub: 0
| id | title | status | module |
|---|---|---|---|
22.05.01 | Rhetoric and composition theory — from Aristotle to modern argument | none | — |
language
2 units · full: 0 · partial: 0 · none: 2 · stub: 0
| id | title | status | module |
|---|---|---|---|
22.05.02 | Aristotle's rhetorical proofs and Toulmin's model of argument | none | — |
22.06.02 | Prose rhythm, sentence acoustics, and the architecture of parataxis | none | — |
creative-writing
1 units · full: 0 · partial: 0 · none: 1 · stub: 0
| id | title | status | module |
|---|---|---|---|
22.06.01 | Creative writing — craft across fiction, poetry, and nonfiction | none | — |
world-literatures
1 units · full: 0 · partial: 0 · none: 1 · stub: 0
| id | title | status | module |
|---|---|---|---|
22.07.01 | World literatures and literary theory | none | — |