Open Mathematics Courses
A growing library of mathematics for independent study, teaching and further work: explanations, complete arguments, worked examples, exercises and solutions, with editable sources and credited references.
Available now
Subellipticity and unique continuation
Degenerate energy, finite-type subelliptic estimates, sign geometry, weighted uniqueness, simple and multiple characteristic roots, convexity and compact-contact continuation, with complete arguments and exact prerequisite readings.
English · 62 lessons · 353 solved exercises · 38 original figures · editable Markdown. Self-checked by the writing AI.
Poincaré duality on manifolds
Poincaré duality for topological manifolds with coefficients in any commutative ring: local homology and orientations, the canonical orientation of complex manifolds, fundamental classes, cap products and compactly supported cohomology, duality over fields and over self-injective rings such as Z/m, vanishing above the dimension, and the identification of the sheaf cohomology of a manifold with singular cohomology, with supports and with compact supports.
Four lessons · proofs and exercises · self-checked by Claude Opus 5.5
Ideal theory in rings
Noether’s irreducible and primary components, socle counts, isolated components, modules and elementary divisors, generic zeros, absolute primality and resultant forms.
English · five lessons · 25 solved exercises · editable sources and historical readings
The Riemann existence theorem
Finite étale covers of a scheme of finite type over the complex numbers are the same as finite covering spaces of its complex points, with no normality hypothesis: covers of punctured polydiscs, the smooth case through compactification and GAGA, descent along proper surjective morphisms, the general theorem by descent along the resolution, its analytic form, and the comparison of étale and topological fundamental groups.
Six lessons · proofs and exercises · self-checked by Claude Opus 5.5
Resolution of singularities in characteristic zero
Functorial resolution of singularities for schemes of finite type over a field of characteristic zero: smooth blow-ups, derivative ideals, maximal contact, order reduction for ideals and marked ideals, principalization, resolution with a simple normal crossings exceptional divisor, smooth compactification and elimination of indeterminacy.
Ten lessons · proofs and exercises · self-checked by Claude Opus 5.5
Complex analytic spaces and coherent sheaves
Holomorphic germs, coherence, complex spaces, Dolbeault theory, L² estimates, coherent vanishing and finiteness.
Twelve lessons · proofs and exercises · self-checked by Claude Opus 5.5
Elliptic operators and boundary problems
Elliptic operators, parametrices, boundary calculus, Fredholm operators and index theory, with complete supporting proofs, worked examples, solved exercises and reproducible figures. This edition contains 56 lesson texts and 4 editorial supplements; 56 lessons have matching PDF and TeX readers.
Start reading · Earlier proofs · Authorship and component terms
Measured relations and operator algebras
Ergodic actions, orbit relations, Krieger algebras, diagonal normalizers, groupoid cocycles, isotropy, approximately finite constructions and amenability, with full worked exercises.
English · 57 current lessons · 459 complete worked solutions · 69 editable diagrams · exact stated prerequisites
Cohomology of quasi-coherent sheaves
Sheaf and affine cohomology, Serre’s theorems, proper direct images, base change, formal geometry, duality and algebraic–analytic comparison.
English · eighteen lessons · 108 solved exercises · exact prerequisites and proof boundaries · editable Markdown and LaTeX
Representations of finite groups
Character theory and orthogonality, induction and Frobenius reciprocity, full integer Brauer induction, symmetric groups, Schur–Weyl duality and general linear groups over finite fields.
English · seventeen lessons · complete arguments and solved exercises · editable Markdown
Reductive group schemes
Six main lessons and eight supporting proof lessons on tori, centralizers, root groups, Bruhat decomposition, integral pinned classification, and forms and flag schemes, with worked examples and solved exercises. Includes integral Bruhat and Schubert cells for finite-field counting and torification.
Outer conjugacy and hyperfinite factors
Projection correction, moving tensor factors, period and obstruction, independent orbit copies, Gaussian amenable towers, relative eigenunitaries, tracial representations and general finite free actions.
English · twenty-six lessons · 155 solved exercises
Projection orthogonalization · Moving tensor factors · Period and obstruction · Independent orbit copies · Gaussian amenable towers · Relative character eigenunitaries · Topology and tracial representations · Finite free actions
Positive maps and finite-dimensional approximation
Completely positive maps, nuclearity, infinite tensor products, central sequences, MASAs and injectivity through finite-dimensional approximation.
English · fifty lessons · 504 solved exercises · complete arguments with stated prerequisites · editable Markdown
Harmonic analysis on locally compact abelian groups
Characters, inversion and Plancherel, Pontryagin duality, Poisson summation, structure, spectral representation, Wiener and Tauberian theorems, almost periodic functions, and p-adic and adelic analysis.
English · seventeen lessons · 78 solved exercises · complete arguments and worked examples · editable Markdown
Haar measure and quotient integration
Four lessons on measure and Hilbert-space tools, Haar measure on locally compact groups, Weil’s integration formula and arithmetic covolumes. Includes Haar uniqueness through shrinking neighborhoods and seven supporting prerequisite chapters.
English · 19 solved exercises · rendered mathematics · editable Markdown
Perverse sheaves and intersection cohomology
Constructible complexes and the gluing of t-structures; the perverse t-structure, intermediate extensions and intersection complexes; Artin vanishing, nearby and vanishing cycles; weights and purity over finite fields and the decomposition theorem; small resolutions and Springer theory; equivariant perverse sheaves and the dictionary with D-modules and trace functions.
12 lessons with full arguments, examples, exercises with solutions and editable sources. Written by GPT-6.1 Sol and GPT-6 Astra (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI.
Derived categories of sheaves
Sheafification, resolutions, derived operations, inverse limits, quasi-coherent sheaves and unbounded projection and base change.
English · twelve lessons and one common reading · proofs and solved exercises · editable Markdown
Foundations of von Neumann algebras
Functional and complex analysis, C*-algebras, GNS representations, tensor products, states and operator topologies.
English · twenty lessons · full proofs and solved exercises · three supporting chapters · editable Markdown
Basic Analysis I & II — Jiří Lebl
Human-authored real analysis: real numbers, sequences, continuity, differentiation, integration, metric spaces, multivariable analysis and approximation.
English · both volumes · preserved author text, proofs, exercises and figures · editable TeX · CC BY-SA 4.0
Operator algebras: representations, derivations and core traces
Integrated representations and C*-duality, logarithmic products and power strips, inner derivations, and a direct faithful-state construction of the continuous-core trace.
English · 133 lessons
Representations of compact groups
Unitarity and Peter–Weyl, maximal tori and highest weights, Weyl formulas, Laplacians, tensor products and homogeneous spaces. Includes profinite characters, Schur–Weyl duality, trace moments and positive spherical kernels.
English · thirteen lessons · full proofs and 52 solved exercises · editable Markdown
The field with one element
Weil’s proof of the Riemann hypothesis for curves and what is missing over the integers; monoids with zero and monoid schemes, schemes relative to a symmetric monoidal category, Λ-rings, Durov’s generalized rings, blueprints, characteristic one and hyperrings with the entropy formula and the Witt construction, Γ-sets over the sphere, the arithmetic site of Connes and Consani, and Smirnov’s projective line over F₁ and the ABC conjecture.
English · eleven lessons · complete proofs, examples and solved exercises · editable Markdown
Analytic geometry and sheaf realization
Weierstrass division, subanalytic preparation and compatible manifold triangulations; nearby and vanishing cycles, holomorphic tests and positive real support; microlocal composition and pure sheaves; smooth-cutoff constructibility, cotangent models and oriented-simplex duality; analytic-boundary local systems and constructible trace classes; constructible image duality and infinite locally constant twists; microsupport with locally constant coefficients; derived realization for fixed stratifications, finite coefficients and normal toric orbits.
English · twenty-four readings · complete proofs with stated prerequisites and 125 solved exercises · native mathematical readers · editable Markdown
Analytic preparation and curves · Manifold triangulations · Locally constant coefficients · Fixed stratifications · Finite coefficients · Normal toric orbits · Nearby cycles and monodromy · Microlocal composition and pure sheaves · Constructibility and oriented duality · Analytic boundaries and constructible traces · Constructible duality and infinite twists
Function algebras and approximation
Stone–Weierstrass on locally compact Hausdorff spaces, compactification, Urysohn’s lemma, lattice approximation, Bernstein polynomials, Korovkin’s theorem and the disc algebra.
English · complete arguments, worked examples and four solved exercises · editable Markdown
Operator algebras: states, supports, stability and symmetry
States and matrix blocks, open projections and ideals, tensor norms, closed operator graphs, stable index, averaging and measurable representations.
English · 33 lessons · complete proofs, worked examples and solved exercises · rendered mathematics · editable Markdown
Modular theory and weights
Weights, semicyclic representations and the Tomita–Takesaki theory: finite domains, standard forms, modular groups and KMS states, Connes cocycles, operator-valued weights, noncommutative integration and the Haagerup and Connes–Takesaki constructions.
English · 136 lessons
Entropy of automorphisms
Operator convexity, Jensen’s inequality and the sharp continuity bound for entropy: one complete lesson with proofs, equality cases and three solved exercises.
English · editable sources · selected lessons from an expanding course
Hermite functions and the Schwartz kernel theorem
One complete lesson on Hermite functions, tempered distributions and the Schwartz kernel theorem, with three solved exercises and a supporting Bott-operator proof.
English · editable sources · selected lessons from an expanding course
Class field theory
Twenty-four lessons on local and global reciprocity, Lubin–Tate fields, idèles, explicit class fields, Artin L-functions, Chebotarev, Weil and Brauer groups, and class field towers, with ninety-six solved exercises.
English · twenty-four lessons · editable Markdown and LaTeX
Start reading · Free references and prerequisite proofs
Two higher-dimensional local prerequisite proofs remain incomplete; their exact statements and providers are recorded in the course.
Crossed products and groupoid C*-algebras
Full and reduced crossed products, induction, duality and the Thom isomorphism; groupoid algebras, diagonals, ideals, Morita equivalence, foliations and geometric index constructions.
English · seventeen lessons · 77 solved exercises · supporting proof readings · editable Markdown
Measure theory: lifting and differentiation
The lifting theorem and Besicovitch covering and differentiation, with full lesson proofs and five solved exercises.
Two English lessons · editable sources · linked measure-theory prerequisites in Bahasa Indonesia
Cyclic cohomology and noncommutative differential geometry
Connections and curvature, frequency calculus for actions, cyclic cohomology, norm-controlled cyclic forms, transverse classes, geometric cycles and the Godbillon–Vey invariant.
Seven working lessons · 110 solved exercises · English · editable sources and offline reader. Written and self-checked by GPT-6.1 Sol (OpenAI), Ultra. Further lessons are in progress.
Subfactors, index and finite-depth classification
A course from the Jones construction to standard invariants and classification, with complete finite-trace, corner, cup-tail, residual-completion, hypertrace ideal/entropy, state-averaging, conditional-variance and controlled-tunnel arguments at their stated hypotheses. The course is not yet complete, and some of its prerequisites are not yet proved in these courses.
Read the course · Sources and proof status
99 readings · 96 teaching chapters · 3 supporting proof readings · incomplete
Sites, topoi and étale cohomology
Sites and sheaves, topoi and their points, cohomology on sites and hypercoverings; topologies on schemes, the étale site and its points, direct and inverse images, Galois cohomology of fields, Brauer groups and Tsen’s theorem, the cohomology of curves, proper and smooth base change, compact support, cohomological dimension and the Künneth formula, Poincaré duality for curves and smooth varieties, and the comparison with singular cohomology.
English · nineteen lessons · complete proofs and solved exercises · editable Markdown
ℓ-adic cohomology and trace formulas
Adic cohomology and trace methods, with full arguments, examples and solved exercises.
English · eight lessons · linked supporting readings · editable Markdown
Spectra, resolvents and scattering
Spectral measurements, polynomial and rough differential operators, radiation, modified waves, wave traces and spectral counting, connected by exact models and explicit proof routes.
English · 51 lessons · 255 exercises with full solutions · 18 original figures · editable sources
Group schemes
Seven lessons on Hopf algebras, Lie algebras and smoothness, group schemes over fields, quotients and torsors, diagonalizable groups and tori, abelian varieties and Néron models. Complete arguments, examples, exercises and solutions accompany the editable sources.
K-theory of C*-algebras
Idempotents, vector bundles, stabilization, exact sequences, Bott periodicity, trace determinants and crossed-product computations.
English · 23 lessons · 115 solved exercises · editable Markdown · continuing proof expansions with explicit prerequisite notes
Algebraic geometry: flat, smooth and étale morphisms
Five selected lessons with 28 commutative-algebra and scheme-morphism prerequisites, an integral-descent reading, and supporting Stacks proofs. Two supplementary lessons identify their remaining proof inputs.
Distributions, kernels and analytic singularities
Seventy-two lessons on distributions, support geometry, Fourier growth and analytic singularities, with prerequisite proofs, complete worked exercises and an offline reader with editable sources.
Self-checked by the writing AI. Later course topics are in preparation.
Foliations and their operator algebras
Transverse measures and currents, holonomy C*-algebras, Hilbert modules on the leaf space, measured index formulas and geometric K-theory.
English · five lessons · proofs, examples and solved exercises · original illustrations · editable Markdown. The course states its remaining mathematical questions and proof gaps.
Constant-coefficient equations and solvability
Polynomial symbols, support geometry, regularity, Cauchy and boundary problems, supported solvability, and smooth nonuniqueness.
English · 148 lessons · proofs, worked examples and solved exercises · supporting proof readings · editable Markdown. A growing course; prerequisites that are not yet proved in these courses are named where they are used.
Elliptic boundary reduction
Arbitrary boundary measurements, the complete compact-cylinder model, regularity and smooth-error solvability, with supporting operator proofs.
English · one selected lesson with supporting proofs · 17-page PDF · editable Markdown and TeX
Categorical and derived tools for analytic sheaves
Categorical foundations, formal ind and pro objects, representability, module methods, tensor structures and coherent gluing, with complete arguments, worked examples and solved exercises. This growing course currently contains 86 lessons.
English · 86 lessons · 344 solved exercises · editable Markdown
Kasparov’s KK-theory
Extensions and lifting obstructions, analytic K-homology and the index pairing, graded correspondences, Kasparov products, Thom classes, equivariance and descent.
English · fourteen available lessons · complete arguments, worked examples and solved exercises · PDF and editable mathematical sources
Matrix extension
A smooth change of frame extends invertible matrix data to an exterior domain. The complete proof includes ordered transport, smooth parameter families and the exact relation between any two factor choices, with finite-matrix and calculus foundations.
Core mathematics and languages
The core mathematics programme provides the existing foundation courses. Its Indonesian interface and editions are also available. Each course identifies its actual languages.
Authorship and use
Lessons credit their human sources and identify AI contributions by Claude Opus 5.5 (Anthropic), GPT-6 Astra (OpenAI), Ultra, and GPT-6.1 Sol (OpenAI), Ultra. Each lesson states how it was checked.
See the source and authorship records and component terms.
The collection is growing. Available lessons are listed above; this edition does not claim to contain the entire mathematics programme.
Semisimple Lie algebras
Brackets, complete reducibility, roots and finite-type Serre presentations, highest weights, characters, compact forms and category O.
English · eighteen written lessons · 73 solved exercises · eight figures · editable Markdown. Supplementary unproved statements and prerequisite scopes remain explicit.
Characteristic classes
Vector bundles, integral characteristic classes, curvature forms, manifold pairings, bordism, signature, PL comparison and exotic spheres.
English · twenty-five chapters · 155 solved exercises · editable Markdown
Bott suspension and Weyl traces
The index-one Bott oscillator, vector-bundle suspension and reduction to Euclidean space, together with exact Weyl kernel, trace and trace-class proofs. Complete prerequisite readings, source, TeX, PDFs and reproducible figures accompany both lessons.
Adèles, idèles and Tate’s thesis
Restricted products, adèles and idèles, ray class groups, Fourier analysis and Poisson summation, Tate zeta integrals and Hecke L-functions, and the local tree and spherical Hecke algebra of GL₂.
Twelve self-checked lessons with full arguments, examples, 48 solved exercises and editable sources. Exact prerequisite readings and source credits accompany the course.
Local fields
Valuations and completions, p-adic lifting, extensions, locally compact fields, ramification, multiplicative groups and explicit local extensions.
12 lessons with full arguments, worked examples, complete exercise solutions and editable sources. Written and self-checked by GPT-6.1 Sol (OpenAI) in Codex.
The Riemann zeta function
A graduate course on the analytic theory of the Riemann zeta function: Dirichlet series, prime-counting estimates, the Gamma function and the functional equation, entire-function products, growth and zeros, the prime number theorem and explicit formulas, mean values, critical-line zeros and short intervals.
19 lessons with full arguments, examples, exercises with solutions and editable sources. Written and self-checked by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI.
Dirichlet L-functions and primes in progressions
Dirichlet characters and L-functions, from Dirichlet's theorem to average distribution and the least prime in a progression. The course develops conductors, Gauss sums, character sums, functional equations, special values, zero-free regions, exceptional zeros, explicit formulas, the prime number theorem for progressions, Siegel and Siegel–Walfisz estimates, the large sieve, Vaughan's identity, the Bombieri–Vinogradov and Barban–Davenport–Halberstam theorems, and Linnik's theorem. The opening lessons prove the character constructions, quadratic reciprocity and elementary analytic prerequisites used in their arguments.
English · 17 lessons
Algebraic spaces and stacks
Algebraic spaces from étale quotients, stacks and gerbes from descent and torsors, Artin's algebraicity criteria and moduli stacks, the stack of curves and stable reduction, and simplicial models for higher stacks.
12 lessons with full arguments, examples, exercises with solutions and editable sources. Written and adapted by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI.
Commutative algebra for geometry
Rings and modules for algebraic geometry, from spectra and flatness to dimension, regularity, infinitesimal lifting, complete local rings and henselian neighborhoods.
English · 19 lessons
Descent and the étale fundamental group
Coherent pullbacks and gluing, faithfully flat descent, Galois categories, finite etale covers, proper families, specialization and tame ramification, followed by comparison over the complex numbers.
English · 7 lessons
Flat, smooth and étale morphisms
Flatness, local criteria and dimension; unramified, étale and smooth morphisms; infinitesimal lifting; henselization and étale localization.
English · 7 lessons
Hilbert and Picard schemes, deformations and representability
Parameter spaces for families of subschemes, sheaves and line bundles, with their deformation theory and representability proofs.
English · 11 lessons
Local cohomology, Lefschetz theorems and purity
Local cohomology and duality, connectedness, formal geometry, Lefschetz comparison, Picard groups, purity and specialization. All eight lessons include proofs, examples and solved exercises.
English · 8 lessons
Morphisms of schemes
The geometry of maps between schemes: separation, finiteness, limits, valuative criteria, properness, projectivity, line bundles, fibre dimensions, Zariski’s Main Theorem, normalization, rational maps, blow-ups and divisors.
English · 17 lessons
Sheaves and schemes
The language of schemes: sheaves on topological spaces, ringed spaces and their modules, affine schemes and the equivalence between modules and quasi-coherent sheaves, schemes glued from affine pieces, the functor of points and representability, fibre products, local properties, quasi-coherent sheaves and closed subschemes, and Proj with projective space and projective bundles. The course cites the site's lesson on sheaves of modules for the homological facts it proves and supplies the sheaf-theoretic background that lesson assumes. Lessons prove what they teach; the Stacks project (read in its AI Integrated Stacks Project edition) is cited by tag.
English · 12 lessons
Fourier-integral operators
Oscillatory integrals, homogeneous phase equivalence and stabilization, Maslov index and signed crossings, prescribed canonical coordinates, real and complex function normal forms, quadratic Hamilton maps and positive complex Lagrangian planes, conic isotropic geometry and clean Lagrangian pairs, Lagrangian distributions, invariant symbols, Fourier-integral composition, graph and corank continuity, fold geometry and densities, Airy representations, and invariant fold continuity on every real Sobolev scale. Positive Lagrangian ideals and their damped distribution representations include uniform complex division, exact Gaussian branches and complete two-way symbol constructions. Complex-phase Fourier-integral operators include transverse ideal and operator composition and the universal order-zero L2 criterion, with full endpoint damping estimates. First-order scalar Cauchy theory includes weighted energy, existence for integrable forcing and exact spatial wavefront transport by the real principal Hamiltonian. Higher-order Cauchy theory develops simple characteristic roots, weighted jets, Green duality, supported and restricted mixed regularity, semiglobal existence, branch factorization, forced propagation and intrinsic boundary wavefront equality. Supported Cauchy solvability forces real normal roots and noncharacteristic finite cone support; the principal-type boundary alternative has positive double contact, with full finite transport tests and exact support hypotheses. Principal-type boundary double roots are treated through forward and backward singular-weight energy estimates, quotient reconstruction, supported Cauchy existence, an explicit Fourier trace lift and microlocal half-step propagation. Smooth complex degree-one division and nonnegative-symbol gradient estimates have complete supporting proofs. The higher-order Cauchy lesson supplies the all-real supported/restricted mixed half-space duality, common minimal extension, exact normal-step inequality and full differential normal recovery, with worked exponential and boundary-delta examples. Boundary chart transport, invariant transversal order and the full bundle-valued boundary differential trace theorem include exact real-order coordinate bounds, finite patching and a solved comparison of two order-three traces. After canonical straightening, an all-order ordinary-symbol construction removes arbitrary finite matrix lower terms under a scalar principal direction, retaining ordered inverse flows, all differentiated estimates and both parametrix identities. Smooth real-principal propagation follows from exact signed model kernels, every diagonal and flow wavefront direction, a common microlocal transport tube and full canonical conjugation, including radial and zero Hamilton fields. Exact Sobolev propagation uses trace-free distributional averaging, an L2 time primitive, full compact-microsupport localization and the graph orders that retain the precise gain m minus one, including endpoint membership. One homogeneous canonical tube follows an injective compact cosphere segment. Compatible Maslov data, proper graph quantization, both parametrices and full lower-term transport give conjugation on the entire tube. Prescribed finite characteristic singularities are realized with any exact Sobolev threshold. A positive-frequency Gaussian profile and endpoint regularization series give the entire singular ray and precisely two forcing rays, transferred through both full graph inverses. Characteristic escape over a compact set gives a finite smooth adjoint obstruction and existence with the exact Sobolev gain m minus one. A complete flat-jet, seminorm-duality and summable-correction argument supplies one smooth solution for smooth data satisfying that obstruction. Global solvability modulo smooth functions is equivalent to compact adjoint singular-support bounds and compact bounds for characteristic excursions with both endpoints over a fixed compact set. The complete test-space estimate, locally finite seminorm and Hahn–Banach receiving argument are given, with an exact slit-plane obstruction. Compact characteristic returns give a proper closed trajectory relation, a smooth Hausdorff trajectory space and one global time section. The conic construction supplies an invariant positive radius, scalar complex transport with all symbol derivatives, and a characteristic relation that is closed and Lagrangian even in the ambient product cotangent space. Directional characteristic symbol transport includes complete Maslov and density frame construction, every parameter and radial estimate, closed reflexive forward and backward support hulls, and a conormal kernel example proving why the forcing boundary must be retained. Directional kernel corrections include actual symbol realization, ordinary-symbol asymptotic summation with closed wavefront control, arbitrary ordinary lower-term transport, and every residual through a smooth error. Global parametrix comparison constructs the compact-middle distribution product, proves its exact wavefront relation and same-sign uniqueness, and gives the conditional adjoint conversion to both inverse identities. Local signed parametrices preserve the full diagonal identity and place their ordinary Lagrangian errors at a strictly positive travel time, with an exact bound on every real Sobolev scale and an elliptic local difference. The complete scalar global signed construction assembles an exact conic operator partition, corrects a closed strictly directional residual and proves both inverse identities, uniqueness, exact full wavefront and a globally elliptic ordinary difference, including arbitrary real operator orders, complex lower terms and clopen mixed signs. First-order Hilbert and finite systems are treated through a uniform Hermitian lower bound, all-real and all-p energy, complete weak existence and traces, ordered evolution and Bochner forcing. The reverse criterion, fixed positive symmetrizers and three solved examples distinguish contraction, reversibility and coupled singular fronts. Oscillatory Cauchy kernels are constructed for a scalar homogeneous principal direction with arbitrary complex finite-matrix ordinary lower terms. The full phase, ordered amplitude and smoothing correction give exact local and finite-time canonical graphs, Maslov composition and every compact-data wavefront direction. Three solved examples retain initial defects, noncommuting parameter derivatives and variable-speed delta weights.
English · 10 lessons
D-modules
Algebraic differential operators and flat connections; characteristic varieties and holonomicity; inverse and direct images, duality and minimal extensions; regular singularities, Riemann–Hilbert, localization, stacks, crystals and Fourier transformation.
English · 10 lessons
The geometric Langlands conjecture
Bundles, Hecke correspondences, geometric class field theory and the rank-one categorical correspondence, together with all-rank averaging, opers, critical-level calculations and spectral categories, leading to the proof of the unramified conjecture.
9 lessons of the planned 15, with full arguments, examples, exercises with solutions and editable sources. Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI.
The affine Grassmannian and geometric Satake
From lattice counts and spherical Hecke algebras to geometric and derived Satake: loop groups, Schubert varieties, weight functors, convolution, fusion and the Langlands dual group.
English · 1 lesson
Galois representations
A fifteen-lesson course from profinite Galois groups and elliptic-curve Tate modules to Weil–Deligne representations, modular forms and Serre modularity. All fifteen lessons are written.
English · 15 lessons
Global Langlands conjectures and functoriality
The global side of the classical Langlands programme. The course sets up reductive groups and their L-groups, the Satake isomorphism for unramified groups and unramified L-factors, automorphic representations of general groups and their automorphic L-functions, and the analytic theory for GL_n (Godement–Jacquet, Rankin–Selberg, strong multiplicity one) as statements with their GL_1 and GL_2 cases recalled. It states the principle of functoriality and its known cases — symmetric powers and tensor products, base change for GL(2) (Langlands) and the Artin conjecture for two-dimensional representations (Langlands–Tunnell) — and the global reciprocity conjectures with the known cases. It states the modularity of elliptic curves over Q in its equivalent forms, outlines the Taylor–Wiles method, and proves the deductions that lead from modularity to Fermat's Last Theorem (given Ribet's theorem and Mazur's torsion theorem), to the analytic properties of L(E, s) and, with Serre's argument, to Sato–Tate from symmetric-power automorphy. It ends with the function-field case (statements) and an outlook on endoscopy and Arthur's classification. Deep theorems are stated exactly with references; the lessons prove the reductions, computations and deductions around them.
English · 5 lessons
The local Langlands correspondence for GL_n
A thirteen-lesson course connecting representations of general linear groups over local fields with Weil–Deligne parameters. It covers the real correspondence, representation classification, local factors and the converse theorem, monodromy blocks, explicit rank-two factor matching, dihedral supercuspidals and rectifiers, modular forms and CM elliptic curves, inner forms and Jacquet–Langlands, existence strategies, base change, induction and formal degrees, and parameters and packets for reductive groups. Proofs and complete exercise solutions accompany the explicit calculations; deep existence and classification theorems are stated with precise references.
English · 12 lessons
Lie's theory of transformation groups
Analytic local transformation groups, parameters, flows, complete systems, the three fundamental theorems, structure and homogeneous actions, jets and contact transformations, classification, and the Riemann–Helmholtz problem. Twelve lessons with proofs and solved exercises. The full Lie–Tresse and classification derivations are not yet included.
English · 12 lessons
Operator-algebraic statistical mechanics and the Bost–Connes system
Arithmetic equilibrium states, Hecke algebras, thermal time and current observables.
English · 1 lesson
Spectral triples and the local index formula
Logarithmic operator traces, regular spectral triples, residue index cocycles, and hypoelliptic transverse geometry.
English · 3 lessons
The abstract theory of ideals
Noether’s axiomatic ideal theory: global ideal invertibility and its converse, composition series, arbitrary orders and discriminants, three differents, tame normal integral bases, conductors and adjoints.
English · 6 lessons
Hypercomplex systems and representations
Operator groups, semisimple rings and representations lead to central simple algebras, crossed products, Galois descent and the principal genus theorem. The final arithmetic lesson separates its proved reductions from its stated class-field-theory inputs.
English · 3 lessons
Invariants and finiteness
Classical covariants and symbolic notation; Hilbert finite generation; finite groups, the Noether degree bound and Molien series; polarization and vector invariants; finiteness over every characteristic and Noetherian bases.
English · 5 lessons
Rational function fields and equations with prescribed group
Subfields and polynomial subrings of rational function fields; rational invariants and prescribed Galois groups; coefficient tests for absolute irreducibility and reduction modulo primes.
English · 2 lessons
Symmetries and conservation laws
The first-variation identity, both Noether theorems and their converses, mechanics and gauge fields, the stationary black-hole first law, null Lagrangians, Euler covariance, and the reduction of Riemannian differential invariants to curvature.
English · 3 lessons
Number fields
Algebraic integers, ideal factorization, ramification, geometry of numbers, units and class groups, quadratic and cyclotomic fields, and zeta and L-functions.
English · 17 lessons
Explicit formulas and positivity
The explicit formulas of prime number theory and the reformulations of the Riemann hypothesis they lead to. Weil's explicit formula for ζ with general test functions (including functions with jumps) and for Dirichlet L-functions, Guinand's summation formula, Weil's positivity criterion in the compact-support form (each test function sees finitely many primes), the archimedean distribution and positivity for small support, Li's criterion, the heat flow of ξ and the de Bruijn–Newman constant, Montgomery's pair correlation theorem, and the random-matrix predictions with their evidence. It continues the F1 course's lesson on Weil's proof, which proves smoothed explicit formulas and Weil's criterion for Gaussian-type test functions, and it rests on NT-ZETA and NT-DIRL.
English · 8 lessons
Microlocal sheaves
A research course developing conic sheaves and kernel operations into specialization, microlocal morphisms and microsupport geometry.
English · 5 lessons
Constructible and perverse sheaves
Microlocal kernels and contact transformations, pure and simple sheaves, constructibility and stratifications, integral characteristic cycles, Euler indices, Morse inequalities, conormal orientations, Lefschetz traces of constructible correspondences, local cutoffs, tangent specialization, expanding and shrinking localization, complex fixed-point traces, constructible functions, Euler integration, homogeneous Fourier calculus, constructible specialization, the integral characteristic-cycle correspondence, Euler convolution, compact convex inverses, odd-dimensional Euler vanishing, vector-field indices, complex stalk-costalk Euler equality, complex constructible-function duality, compact Morse models with full cotangent projections, truncation triangles with abelian hearts, t-exact functors with adjoints between hearts, real perverse support, costalks and truncation triangles, perverse descent with fibre dimension bounds, complex middle perversity with exterior products, holomorphic Morse exhaustions on Stein manifolds, and both global Stein cohomology vanishing statements.
English · 4 lessons
Transcendental numbers
From Liouville's explicit constructions and algebraic heights to Hermite and Lindemann, auxiliary functions, elliptic transcendence, Roth's theorem, E-functions and algebraic independence.
English · 13 lessons
Analytic elements, strips and KMS
Holomorphy in Banach spaces, one-parameter groups and their generators (Stone's theorem, cores, analytic vectors, convergence in the strong resolvent sense), holomorphic extension of one-parameter groups, analytic and entire elements, conjugate-linear closed operators, and the strip arguments behind the KMS condition.
English · 2 lessons
Automorphic forms and representations of GL(2)
A course on how congruence conditions, Fourier coefficients and local representations fit together for GL(2). It develops the passage from classical modular forms to adelic representations, local and global Whittaker models and L-functions, newvectors, the cuspidal and Eisenstein spectra, and quaternionic forms and the trace formula.
English · 14 lessons
Hilbert algebras
Left and right Hilbert algebras: commutative examples, the group algebra of a locally compact group, finite-rank examples, equivalence of Hilbert algebras, and fully solved exercises.
English · 1 lesson
Hilbert C*-modules and Morita equivalence
Hilbert C*-modules, adjointable and compact operators, finite projective modules, tensor products, stabilization, Fredholm index, regular operators and continuous fields; completely positive maps, imprimitivity bimodules, induced representations, stable isomorphism and Morita equivalence in K-theory and noncommutative tori.
English · 14 lessons
Injective factors
This course classifies the injective factors with separable predual that are not of type III₁: the hyperfinite factor is the only injective factor of type II₁, there is only one of type II∞ and one of each type IIIλ with 0 < λ < 1, and the injective factors of type III₀ are the Krieger factors, classified by their flow of weights. It also shows that injectivity, approximate finite dimensionality, semidiscreteness and property P coincide for factors with separable predual (for type III₁ with the help of a later uniqueness theorem, used without proof), building on trace inequalities, property Γ and approximately inner and centrally trivial automorphisms. Basic references are [Connes 1976], [Connes 1973], [Takesaki III], [Anantharaman–Popa] and [Connes 1994].
English · 6 lessons
Noncommutative integration and spatial theory
This course develops integration on spaces that ordinary measure theory cannot see, such as the space of orbits of an ergodic group action or the leaf space of a foliation. It starts with the spatial derivative, which compares a weight on a von Neumann algebra with a weight on its commutant, and then builds transverse measures on measured groupoids, the von Neumann algebra of random operators, and the weights, formal dimension and index that come with them. Basic references are [Connes 1979], [Connes 1980a], [Connes 1982] and [Connes 1994].
English · 5 lessons
Tensor products of operator algebras
Spatial tensor products of von Neumann algebras on Hilbert spaces of any dimension: normal functionals, slice maps, the commutation theorem and coproducts. Tensor products of Banach and Hilbert spaces with the injective and projective norms, trace duality, and Jordan homomorphisms and isometries of C*-algebras. Tensor products of C*-algebras: the minimal norm, Takesaki's minimality and simplicity theorems, and the product map of a factor and its commutant.
English · 3 lessons
Traces and noncommutative integration
Traces on von Neumann algebras, from finite and semifinite traces to the center-valued trace; integration for a trace, the commutation theorem and the extended center-valued trace; operators measurable with respect to a trace: examples, convergence in measure and nearly everywhere, and the commutant of left multiplication; and the multiplicity of a representation, measured by the coupling function.
English · 3 lessons
Type III factors: central sequences, full factors and almost periodic states
This course studies factors of type III through bounded sequences that asymptotically commute with the whole algebra and its normal states. It shows that a factor with separable predual is full, meaning that its inner automorphisms form a closed subgroup, exactly when all such sequences are trivial; it builds the invariants Sd and τ from almost periodic weights and the modular group; and it constructs full factors of type III₁ that have no almost periodic weight and are not crossed products of a semifinite algebra by a discrete abelian group. Basic references are [Connes 1974], [Connes 1973], [Connes 1976] and [Takesaki II].
English · 4 lessons
Modular forms
Modular forms from the geometry of the upper half-plane: modular curves, holomorphic forms, Hecke operators and newforms, L-functions, theta series, cohomology and modular symbols, complex elliptic curves, non-holomorphic Eisenstein series and the Rankin–Selberg method.
19 lessons with full arguments, examples, exercises with solutions and editable sources. Written and self-checked by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI.
Linear forms in logarithms and their applications
From elementary height and approximation bounds to Baker's theorem, effective logarithmic estimates and their Diophantine applications.
8 lessons with full arguments, examples, exercises with solutions and editable sources. Written and self-checked by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026.