Open Mathematics Courses

Cotangent complexes and deformations in ringed topoi

Homotopical algebra in a ringed topos and the cohomology of diagrams, as foundations for the cotangent complex of morphisms of ringed topoi: homotopy sheaves and the free abelian Whitehead theorem; derived functors of non-additive functors of complexes, the Deligne-Lazard theorem and the shift formulas; décalage, nerves of internal categories and the homotopy type of a category; strict simplicial objects and Deligne's contractibility theorem; derived colimits; fibred topoi, their total topoi, direct images and modules; the cohomology of total topoi and of simplicial topoi, and cohomological descent; the second total topos and local systems; left derived functors of left adjoints; a duality formula for diagrams; equivariant sheaves on simplicial topoi, the nerves of induced and coinduced modules, and the cohomology and Ext groups of equivariant modules; spectral diagrams, the stabilization of non-additive functors and stable derived functors; simplicial rings and modules, their calculus of fractions, triangles, Ext groups, derived tensor products and their spectral sequences; differential graded modules in a topos and their comparison with simplicial modules; graded monoids, the nerves of their actions, graded rings and the spectra of modules; diagonal rings and the adjunction formulas; the stable ring of a ring and the Mac Lane resolution; Ext groups of modules as cohomology of spectral nerves; quasi-coherent and perfect complexes on big fppf and fpqc sites; deformations of diagrams of schemes; equivariant cotangent complexes; equivariant extensions and deformations; flat group schemes as local complete intersections; torsors, co-Lie complexes and the Atiyah class; deformations of torsors; the co-Lie complex of a group scheme; the adjoint co-Lie complex of a group scheme; deformations of flat group schemes and of homomorphisms; Lie complexes of commutative group schemes; Lie complexes of module schemes; deformations of module schemes; graded modules and the graded cotangent complex; the Atiyah class of a module; deformations of modules and of their morphisms; filtered derived categories; filtered tensor products on a ringed topos; filtered internal Hom and the formulas of Cartan; perfect filtered complexes and duality; traces and cup products of perfect filtered complexes; filtered and graded powers of complexes; filtered Koszul complexes and the shift isomorphisms; diagonal maps of symmetric and exterior powers; L-series of endomorphisms of perfect complexes; the Atiyah class of filtered complexes; Chern classes of perfect complexes; Chern classes with supports from partial connections; weakly regular ideals; connectivity of powers of simplicial ideals; the fundamental spectral sequence; weak complete intersections; weakly regular ideals of noetherian rings; regular local rings and complete intersections; the arrow topos and relative Ext groups; deforming a morphism together with its target; de Rham complexes and Lie algebroids; formal categories with divided powers and the Čech–Alexander complex; the de Rham complex of a formal category with divided powers; décalage and the formal Poincaré lemma; cohomology of flat formal categories with divided powers; cotangent complexes of simplicial rings; graded simplicial rings and the graded transitivity triangle; localization morphisms and the cotangent complex; the derived de Rham complex; derived de Rham complexes in characteristic p; the completed derived de Rham complex of a complete intersection; the simplicial derived category of a topos; local fibrations and Kan's subdivision functor; simplicial groups and right fractions; hypercoverings and Ext groups of free simplicial modules; Giraud's characterization of topoi; deforming the target of a morphism of diagrams, equivariantly and for group and module schemes.

87 lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Marked Type II estimates for prime predecessors

OpenAI's one-sided marked Type II estimate for prime predecessors: bilinear sums of the prime indicator minus its rough-number average against arbitrary rough coefficients, at the predecessors mn - 1 and weighted by marks from fixed bands of primes, are smaller than every power of log x. A coefficient criterion; a zero-free strip at large height and long prime polynomials; Cauchy's inequality and a determinant equation; physical states, root residues and independent lines; lifespans, a memory of pending primes and contraction on clean and dirty edges; birth distinctness; the major arcs by characters, Mellin separation and sparse frequencies.

17 lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Strong Kadison–Kastler stability

Close von Neumann algebras are conjugate by a unitary near the identity, with a tolerance that depends only on the required closeness, after OpenAI's 2026 papers. Finite factors through boundary means of countable groups, measurable cocycles, double ergodicity and common tracial position; infinite factors through a fixed III_1 tensor factor, rescaled Tomita graphs, infinitesimal normalizers, analytic multipliers, a common modular action and common crossed products; all algebras through Dixmier's theorem and the central decomposition. Two counterexamples: close separable C*-algebras without unitary conjugacy, and one-sided near inclusions without small spatial embeddings.

21 lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Artin's primitive root conjecture

Every integer other than -1 that is not a square is a primitive root modulo at least c x/(log x)^2 primes in (x, 2x), after OpenAI's 2026 paper. A zero-free strip Re s > 1 - 10^-6 for the Hecke L-functions of every cyclotomic field containing the twelfth roots of unity, from cubic Gauss sums: S-integers, sextic reciprocity, Kummer characters, the cubic theta function of Kazhdan and Patterson, a Weyl reflection, a quadratic large sieve, a Poisson probe and its Euler product, zero detection; uniform complete splitting in Kummer fields; primes p = crQ + 1 from a block sieve, Bombieri-Vinogradov, marked bilinear estimates and Buchstab's densities.

19 lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Additive indecomposability of the primes

No finite modification of the set of primes is a sum A + B of two sets with at least two elements each: Ostmann's conjecture, the inverse Goldbach problem, after OpenAI's 2026 paper. Sieve bounds and collision stability; decorrelation from translated quadratic and higher characters (Bonami's inequality, Weyl sums, Heath-Brown's quadratic large sieve, Kneser's theorem, anchor codes); a supply of nonsparse Fourier transforms from coverage of every large prime; a finite-field tree comparison; a positive statistic and its contradiction.

18 lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Finite symmetric tensor categories and the Verlinde tower

Every finite symmetric tensor category in characteristic p > 0 admits a fibre functor to a higher Verlinde category, the finite case of the Benson-Etingof-Ostrik conjecture, after OpenAI's 2026 manuscript. Tilting modules for SL_2; an equivariant vector bundle on the plane and the restricted tilting theorem; higher Frobenius functors; norm maps and symmetrizers; rank varieties; descent through Frobenius-Perron dimensions; incompressible categories and reconstruction.

Nine lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Modularity of elliptic curves over imaginary quadratic fields

Every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place, after OpenAI's 2026 manuscript. Cyclic covers of a twist branched at two points; integral homology over the group ring and Picard-Lefschetz transvections; congruences between character pieces; local constancy of reduction types; Hilbert irreducibility for a curve and its conjugate; three applications of the Caraiani-Newton lifting theorem; cyclic descent; complex multiplication.

Eight lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The first moment of cubic Gauss sums

Patterson's first-moment conjecture for cubic Gauss sums, proved unconditionally after OpenAI's 2026 manuscript: the normalized cubic Gauss sums over the primary primes of Z[omega] of norm at most X add up to (6/5) c_* X^(5/6)/log X, with c_* = (2 pi)^(2/3)/(3 Gamma(2/3)). Cubic Gauss sums and their cubes; large sieves for cubic characters; the level Voronoi formula; exceptional character moments; corrected dispersion; an exact decomposition of the prime sum; the sharp cutoff; fixed angular modes and the powers G^k with k not divisible by 3.

Eight lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Grothendieck infinity-groupoids and the homotopy hypothesis

Henry's pushout conjecture for every Grothendieck coherator, after OpenAI's 2026 proof: freely adjoining to a cellular infinity-groupoid a cell parallel to a given one, and a cell joining the two, is a weak equivalence. Globular theories and their models; the homotopy calculus; coskeletal cutoffs, spines and free shapes; trees and whiskers; partial cylinders and the main theorem; the canonical semi-model structure and the Quillen equivalence with spaces.

Six lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Bounded Hochschild cohomology of von Neumann algebras

Every bounded Hochschild cocycle of degree at least two on a von Neumann algebra is the coboundary of a bounded cochain, after OpenAI's 2026 proof; with the Kadison-Sakai theorem this settles the Kadison-Ringrose cohomology problem. A rigidity theorem for first-letter identities on free Haar families; a random walk on the unitary group whose long products are asymptotically free; a Liouville theorem for its harmonic maps; bounded primitives by averaging, removal of separability and an explicit central homotopy.

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A singular surface with free tangent sheaf

A normal affine complex surface whose module of derivations is free of rank two although the surface is singular, after OpenAI's 2026 counterexample to the Lipman-Zariski conjecture. An explicit elliptic curve on the quadric with a line bundle of order four; a curve of genus 21 with fourth-order contact and a fourth root of its canonical bundle; traces, residues and Grothendieck's splitting theorem; the trace form and the residue data; two vector fields glued order by order near the curve; contraction to a singular point with free derivations; algebraization by finite determinacy with relations.

Nine lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Derivators

Grothendieck's proposal in Pursuing Stacks to keep all derived categories of diagram categories at once, with restriction functors that have adjoints on both sides. Why cones in a derived category cannot be chosen functorially; localization, prederivators, Kan's formulas and the derivator axioms; homotopy exact squares; the derivator of a ring, its strongness and its stability; recognition of cocartesian squares and shifted derivators; additivity of stable derivators, with the flip of a suspension equal to -1; the canonical triangulated structure, exact functors, the Dold-Puppe sequences, and the comparison with the classical triangles of the derived category of a ring, sign included.

Nine lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Symplectic balls and the symmetric Mahler inequality

OpenAI's theorem (2026) that for every origin-symmetric convex body K in R^n, n >= 2, every symplectic ball of capacity c < 4 embeds into int K x int K°, and its consequence, the symmetric Mahler inequality vol(K) vol(K°) >= 4^n/n!. Polar bodies, support functions and strip bodies; one- and two-forms, the Pfaffian and volume preservation, flows and Moser's method; plurisubharmonic potentials and dd^c; a ball of capacity below pi k from a holomorphic zero of order k; a planar conformal map from the arctangent series and a uniform estimate for its horizontal slice integrals; a symplectic realization inside int K x S int K° with S close to pi k/4, rescaling, and polar approximation of general bodies.

Seven lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Matrix multiplication with exponent at most 9/4

OpenAI's theorem (2026) that the exponent of matrix multiplication is at most 9/4: two n x n complex matrices can be multiplied with O(n^(9/4+eps)) arithmetic operations. Tensors, rank and Strassen's characters; a Fourier separation of blocks sharing one leg and an entropy inequality; a symmetrized profile of polynomial multiplication that is discretely concave and satisfies shifted tripling, which forces lambda(T_d) <= d^(9/4) for every character; characters that are large on matrix multiplication, from a catalytic comparison and Schauder's fixed-point theorem.

Seven lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The dyadic triangular Hilbert form

OpenAI's theorem (2026) that the dyadic triangular Hilbert form is bounded on L^3 x L^3 x L^3 uniformly in the scales and with no condition on the inputs: the absolute values of the local forms over admissible triples of dyadic intervals, summed over any finite set of scales, are at most 40 times the product of the L^3 norms. Step functions as matrices; the energy tr(P^(3/2)), its Hessian and its comparison under the matrix order; a dimension-free trace estimate for a triangle of matrices; energy increase under splitting into four children, and telescoping.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Ehrhard's inequality for convex sets

A complete proof of Ehrhard's inequality for closed convex sets: Phi^{-1} of the Gaussian measure of lambda K + (1 - lambda) L is at least lambda Phi^{-1}(gamma(K)) + (1 - lambda) Phi^{-1}(gamma(L)). Gaussian heat semigroups, the heat equation and a moving function under the semigroup; the sharp gradient bound |grad Q_t f| <= t^(-1/2) I(Q_t f) from a rearrangement inequality; the interpolation method of Neeman and Paouris with the heat semigroup: a quantity monotone along the flow of a pair of correlated Gaussian vectors; Lipschitz approximation of convex sets. Concavity along translates of a convex set follows.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The complete Crouzeix inequality

OpenAI's theorem (2026) that ||P[A]|| <= 2 max over the numerical range W(A) of ||P(z)|| for every complex square matrix A and every polynomial P with matrix coefficients of any size, with the sharp constant 2: the complete form of Crouzeix's conjecture. Convex domains with analytic polar boundaries approximate W(A) from outside; the Riemann map of the inverted exterior extends across the boundary; the Faber polynomials of the domain give a nonnegative kernel and a two-sided estimate of boundary norms by coefficients; the resolvent has positive Hermitian part on the circle, and testing it with the singular vectors of F[A] and the products FG and GF gives the constant 2.

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A nonspectrahedral hyperbolicity cone

OpenAI's counterexample (2026) to the geometric generalized Lax conjecture: a polynomial of degree 20 in 23 variables, built from the Choi–Lam biquadratic form, is hyperbolic, and no linear matrix inequality of any size describes its hyperbolicity cone. Hyperbolic polynomials and Gårding's convexity theorem, the Choi–Lam form and the absence of dominated bilinear squares, a positive map and the slices of the cone, the obstruction to every pencil through a norm identity and its first variation, and OpenAI's exact lift of the same cone as the projection of a spectrahedron of size 100 with 307 auxiliary variables.

Six lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Uniform exclusion of Landau–Siegel zeros

OpenAI's theorem (2026) that (1 - beta) log q >= c for an absolute constant c > 0 and every real zero beta of every primitive real Dirichlet L-function of conductor q >= 3, the logarithmic form of the Landau–Siegel problem. The prime bias forced by a real zero near 1, an interpolation theorem for polynomials with separately bounded degrees on a projected integer box, a weighted determinant of conjugates in Q(sqrt d, sqrt 2), and its two bounds: Hadamard's inequality, and divisibility by the Frobenius congruence at the primes with chi(p) = -1. A zero-free region with no exceptional zero follows.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Quadratic stabilization of the Foulkes–Howe map

OpenAI's theorem (2026) that the canonical Hermite–Hadamard–Howe map Sym^b(Sym^a V) -> Sym^a(Sym^b V) is surjective for b >= a(a-1) and every finite-dimensional complex V. Symmetric powers and averaging, the image of the map, the diagonal criterion for surjectivity, an injectivity lemma for directional derivatives with the Fischer inner product, the induction that proves the vanishing on products, and an equivariant embedding of the kind asked for by Foulkes' conjecture.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Projection bodies and products of simplices

OpenAI's product counterexample (2026) to the simplex maximum for projection-body volume: the affine invariant |Pi K|/|K|^(d-1) of the product of two ten-dimensional simplices exceeds the simplex value, and products of simplices exceed it by an exponential factor in all large dimensions. Cauchy's formula for polytopes, affine covariance, convex bodies by circumscribed polytopes, multiplicativity on products, the simplex value from a cube plus one segment, and the first dimension, twenty, in which a product of simplices exceeds the simplex value.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A smooth affine maximal graph in dimension ten

OpenAI's smooth solution (2026) of the affine maximal equation on R^10 with positive definite Hessian that is not a quadratic polynomial. The equation as the Euler–Lagrange equation of the affine area; graphs sqrt(F^2 + t^2) and the reduction to two equations for F and an auxiliary function P; a radial solution through the origin by the contraction mapping principle; and an invariant box for three logarithmic derivatives, whose corner comes from the singular example sqrt(|x|^9 + t^2) of Trudinger and Wang, giving the solution for all radii.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A power saving for planar halving lines

OpenAI's theorem (2026) that n points in the plane with no three on a line have at most C n^(4/3 - eps) halving lines for an absolute eps > 0, the first improvement of Dey's exponent 4/3; the same bound for the switches of rank sweeps uniformly in the rank, and the bound C n (k+1)^(1/3 - eps0) for k-sets. Dey's bound through convex chains and crossings, the induction from a packing estimate to a power saving, and the packing estimate by contradiction: rank coordinates, weak limits of measures, monotone functions of two variables, three transverse foliations, a zoom with radial comparison, a spatial measure and a count of boundary activity.

Thirteen lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The classification of finite Euclidean Ramsey sets

OpenAI's classification (2026) of the finite Euclidean Ramsey sets: a finite set is Ramsey exactly when a linear system over the tensor square of its coordinate field has a solution. The colouring obstruction from rational functionals, a finite lattice identity through the flatness of completion, two-scale configurations by Gaussian smoothing, the Hales–Jewett theorem, ultrafilter semigroups and a path criterion in a free group, and the correction of endpoints through derived series. Examples on circles: at most five concyclic points are Ramsey, including kites that embed in no finite transitive set; nine generic points and three rotated squares are not.

Nine lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The Erdős–Gallai cycle decomposition theorem

OpenAI's proof (2026) of the Erdős–Gallai cycle decomposition conjecture: the edges of every graph with n vertices can be partitioned into at most Cn cycles and single edges. Lovász's path-and-cycle theorem, Haxell's hypergraph condition, routing in graphs with cut expansion, splitting at a scale, random vertex samples, closing paths through reservoirs, pair folding with lifting of cycles, and an induction over scales with a sublinear potential.

Eight lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The Gaussian propeller bound

OpenAI's proof (2026) of the Gaussian propeller conjecture of Khot and Naor: the Gaussian first moments of the cells of any finite measurable partition of a Euclidean space have squared lengths summing to at most 9/(8 pi), with equality for three planar sectors of angle 2 pi/3. Duality with Gaussian maxima, half-space and spherical-cap rearrangement, Ehrhard concavity of gap probabilities, residual covariances and a probability budget with exact margins; a sharp inequality for Gaussian maxima and the threshold for kernel clustering.

Six lessons · proofs and exercises · self-checked by Claude Opus 5.5

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An algebra of infinite little finitistic dimension

OpenAI's counterexample (2026) to the finitistic dimension conjecture: a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. A finitely presented group algebra with a selection process of unbounded extinction is realized by a single complex of bimodules over a three-vertex algebra, simulated by an ordinary bimodule, and turned into projective dimension by a trivial extension.

Eight lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Cell complexes, fundamental groups and covering spaces

Cell complexes, fundamental groups and covering spaces as used by the courses on group rings and classifying spaces: homotopy equivalences of cell complexes, trees, triangulations and simplicial approximation; fundamental groups of graphs and of complexes with attached cells, presentation complexes; universal covers with lifted cells and deck transformations, covers for normal subgroups, and cellular chains over the group ring with the Cayley complex.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Crossing numbers of complete and complete bipartite graphs

Zarankiewicz's formula for cr(K_{m,n}) and the Harary-Hill formula for cr(K_n), proved for unrestricted drawings: normal drawings, the cycle pairing of signed crossings, interpolation, a subspace intersection inequality, and the matching drawings (OpenAI, 2026).

Eight lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Van Kampen diagrams

Van Kampen diagrams without topology: plane maps and their operations, free groups, presentations and Cayley graphs, and van Kampen's lemma in three parts; the foundation for the diagram arguments of the courses on irrational traces, the Kadison-Kaplansky conjecture and reduced Baum-Connes injectivity.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A torsion-free counterexample to reduced Baum–Connes injectivity

The Baum-Connes conjecture asserts that the assembly map from the K-homology of a classifying space to the K-theory of the reduced group C*-algebra is an isomorphism. OpenAI constructed in 2026 a finitely generated torsion-free group in which the K-homology class of a torus has infinite order but reduced assembly image zero, so reduced assembly is not even rationally injective. The group contains the Heisenberg group and is defined by the closed walks of a voltage cover of an infinite labelled tower with Heisenberg voltages; an integral cocycle of the Heisenberg group extends to the whole group and detects the torus, while bounded families of spread Bott symbols, realized by finite Cayley formulas, kill the Bott class by a swindle in a norm quotient. The course is an exposition of OpenAI's proof (2026).

18 lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A torsion-free counterexample to the Kadison–Kaplansky conjecture

Kadison and Kaplansky conjectured that the reduced group C*-algebra of a torsion-free group contains no projections other than 0 and 1. OpenAI constructed in 2026 a finitely generated torsion-free group with a projection whose canonical trace lies strictly between 0 and 1/2. The group is defined by the closed walks of a voltage cover of an infinite layered expander, the tower, with voltages in a torsion-free nilpotent group; a weighted adjacency matrix over its group ring has a separated spectral band of small trace, and compression after a free product gives the scalar projection. The tower is built from streams and gated permutations and pruned so that long paths are recovered from their words; the voltages realize a model operator assembled from Schwartz idempotents of twisted group algebras, with one random draw per edge. The course is an exposition of OpenAI's proof (2026).

18 lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A projection of irrational trace and the Baum–Connes conjecture

The reduced group C*-algebra of a countable group carries a canonical trace, and the Baum-Connes conjecture predicts its K-theory from proper actions. By a theorem of Lück every predicted class has a rational trace. OpenAI constructed in 2026 a finitely generated group with a projection in its reduced group C*-algebra whose trace is the irrational number sum p^(-k_i), so the reduced Baum-Connes conjecture without coefficients fails. The group is a semidirect product of a graphical small cancellation group, whose Cayley graph contains labelled expanders isometrically, with a module over F_p defined by linear codes on the copies of the expanders. A Fourier transform turns a Laplacian-type element into random subgraph Laplacians with a uniform spectral gap, and its kernel projection counts the copies whose random pattern is the prototype. The course is an exposition of OpenAI's proof (2026).

Seven lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Nonsingular systems of equations over groups

A system of equations over a group G has coefficients in G and unknowns x_1, ..., x_n; it is nonsingular if the matrix of exponent sums of the unknowns has full row rank over Q. Howie conjectured that every nonsingular system has a solution in some group containing G; OpenAI proved this in 2026 for every group, which includes the Kervaire-Laudenbach conjecture. The proof draws a relation that kills an element of G as a planar surface of disks and bands and measures its boundary words with the spectral phase of unitary operators over the group von Neumann algebra. The new ingredient is a theorem on simultaneous fixed spaces of block unitary matrices in several unitary variables, proved with the rational cohomology of unitary groups. The course is an exposition of OpenAI's proof (2026) and builds on the course on the Kervaire conjecture.

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The Bass trace conjecture for group rings

A finitely generated projective module over a group ring has a trace with one coefficient for each conjugacy class. Bass conjectured that over the integers only the identity class carries a coefficient, and over the complex numbers that every class of elements of infinite order has coefficient zero. OpenAI proved both in 2026, for every group. For torsion-free groups this gives Kaplansky's idempotent conjecture in characteristic zero: the group ring has no idempotents other than 0 and 1. The proof turns an idempotent into simplicial cycles of every even dimension, detects its trace coefficient with a Pfaffian cocycle, and shows by a separating-set argument that such sparse cycles carry no invariant cohomology in high dimension. The course is an exposition of OpenAI's proof (2026); it also proves Linnell's restriction for integral group rings by an elementary congruence for traces of p-th powers.

Nine lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Kaplansky's direct-finiteness conjecture

Does ab = 1 imply ba = 1 in every group algebra? Kaplansky asked this, and the answer is yes in characteristic zero and for sofic groups. OpenAI showed in 2026 that it is no in general: over a finite field of characteristic two there are a finitely presented group G and a, b with ab = 1 and ba != 1. The group is not sofic and carries an injective cellular automaton that is not surjective, disproving Gottschalk's conjecture. The course proves Kaplansky's theorem in characteristic zero and the Gromov-Weiss theorem, then the construction: copies of F_l^2 glued along a system of affine lines of incidence girth 12, character packing, low-degree interpolation, and a line system built from holes, random slabs and the local lemma. The course is an exposition of OpenAI's proof (2026).

Nine lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The Kervaire conjecture

Adding one generator and one relation to a nontrivial group never gives the trivial group: the Kervaire conjecture, proved by OpenAI in 2026. For a word whose exponent sum in the new generator is +-1, a relation killing a coefficient is drawn as a planar surface of disks and bands; the boundary is measured by the trace of the spectral phase in the group von Neumann algebra, which exists for every group, and a degree computation modulo two supplies a unitary with a large fixed space. Comparing an upper bound from subadditivity with a computation along the boundary circles shows that one boundary word cannot be the only nontrivial one. The course is an exposition of OpenAI's proof (2026).

Six lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A finitely generated counterexample to the Eilenberg-Ganea conjecture

A finitely generated residually finite group with cohomological dimension 2 and geometric dimension 3: the Eilenberg-Ganea conjecture fails. The group is the kernel of the height homomorphism of the right-angled Artin group of an acyclic flag complex whose fundamental group maps nontrivially into the unit quaternions; a degree argument on finite quotients and compactness show that a two-dimensional classifying space would force small quaternion labellings that are trivial around triangles to be trivial around every closed path, and a slowly varying map of the level complex into the flag complex produces a labelling that is not. The course is an exposition of OpenAI's proof (2026).

Eleven lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A torsion-free group algebra with zero divisors

There is a finitely presented torsion-free group G, with a finite two-dimensional classifying space, whose group algebra over the field with two elements has zero divisors: Kaplansky's zero-divisor conjecture fails. Two random immersed labelled graphs with prescribed letter sets of odd intersections, built from the projective plane over the field with 128 elements, make every contribution to a product cancel in pairs; a word-counting estimate, planar separation and cone pictures show that the group of the graphs is torsion-free and that the factors are nonzero. The course is an exposition of OpenAI's proof (2026).

Ten lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A translational tile with no periodic tiling in dimension three

A finite set T in Z^3 tiles Z^3 by translations, but no tiling of Z^3 by T is invariant under a subgroup of finite index, and the union of the unit cubes at the points of T has the same property in R^3, even with arbitrary real translations. An arithmetic rule for arrays along lines of integer slope has a solution with no constant column but none with a vertical period; tiling equations in Z^2 x V with V cyclic encode the rule through the dependence of outputs on input digits; one explicit common solution, a stacking with a fresh prime and a rigid passage to three dimensions complete the proof. The course is an exposition of OpenAI's proof (2026) that the periodic tiling conjecture fails in dimension three.

Seven lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The Erdős similarity conjecture for geometric progressions

For every q in (0, 1), a compact subset of [0, 1] of measure as close to 1 as we like contains no affine copy x + s{q^n : n ≥ 1}, s ≠ 0, so geometric progressions are not measure universal. Random tables on nested dyadic grids route every point through a finite ordered tree; the first default choice on the route of a center offers many independent chances to hit a copy, windows of indices with gaps keep earlier decisions unchanged, and finitely many dilations represent all of them. The course is an exposition of OpenAI's proof (2026) of the geometric-progression case of the Erdős similarity conjecture.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Global uniqueness for isotropic elasticity

The displacement-to-traction map of an isotropic elastic body determines both Lamé moduli: on a bounded connected domain in R^3 with smooth boundary, smooth moduli with μ > 0 and 3λ + 2μ > 0 are recovered from the map from boundary displacements to boundary tractions. Adjoining the divergence as an unknown gives a system with principal part the Laplacian; exact solutions of complex geometrical optics type have leading terms governed by a transport equation; a comparison of the transport equations of the two media depends holomorphically on the direction, which ranges over a conic in the complex projective plane, and a bundle over this conic forces it to be the identity. The course is an exposition of OpenAI's proof (2026). Boundary determination, interior elliptic regularity, a Carleman estimate for the Laplacian and holomorphic frames over a disk are used as named results from the literature.

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Exact quantum factoring

A polynomial-time uniform family of quantum circuits over one fixed finite gate set (NOT, CNOT, Toffoli, the Hadamard gate and the phase gate S, with their inverses and singly controlled versions) outputs the complete prime factorization of every integer N ≥ 2 with probability one. A transform with a piecewise-linear phase replaces the Fourier transform of order finding; exact power sums and classical corrections make the probability of the true order explicit; a factoring process produces, with the factorization, verifiable factorizations of p − 1 from which the probabilities of all its steps are computed after the run; filters computed afterwards make the success probability exactly one quarter, and one step of amplitude amplification gives certainty. The course is an exposition of OpenAI's construction (2026), with a deterministic primality test of Agrawal–Kayal–Saxena type.

Six lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Semidefinite lifts of the perfect matching polytope

The shifted matching matrices, with entries |M ∩ δ(U)| − 1 + ρ for odd sets U and perfect matchings M of the complete graph K_n, have positive semidefinite rank 2^{Ω(n)} for every fixed 0 < ρ < 1; consequently every exact semidefinite lift of the perfect matching polytope has exponential size. Averaging row factors over restrictions of cuts to cells, an orthogonal splitting inequality and a Fourier smoothing theorem turn a small factorization into an approximation of a parity function on an expanding graph by squares of low degree, which a functional of Grigoriev type rules out. The course is an exposition of OpenAI's proof (2026), with a further smoothing theorem for averages over perfect matchings.

Seven lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A counterexample to Naimark's problem

A unital, infinite-dimensional, simple C*-algebra with a faithful tracial state whose nonzero irreducible representations are all unitarily equivalent, constructed in ZFC: a negative answer to Naimark's problem without additional set-theoretic axioms. Excision of pure states, expectations onto algebras of peaking projections, reduced amalgamated free products and reduced HNN adjunctions give a bridge that joins two chosen classes of pure states while every pure state keeps a unique extension; transfinite iteration with closed subiterations and countable determination treats every pure state. The course is an exposition of OpenAI's construction (2026); it uses the homogeneity theorem of Kishimoto, Ozawa and Sakai without proof.

Seven lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Open homomorphisms of solvably closed Galois groups

Uchida's conjecture: for number fields F_1, F_2 and Galois extensions E_i/F_i with E_i solvably closed, every continuous homomorphism Gal(E_1/F_1) -> Gal(E_2/F_2) with open image is induced by a unique field embedding E_2 -> E_1. An l-adic logarithm rank theorem shows that Frobenius values at infinitely many primes of degree one determine an additive character; regular Kummer quotients and Frobenius generation make the homomorphism surjective and compatible with cyclotomic logarithms; a uniform bound on missed primes and power-residue separation match residue characteristics, and the homomorphism is identified with conjugated restriction. The course is an exposition of OpenAI's proof (2026); it uses two theorems of Uchida and a criterion of Hoshi without proof.

Eight lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A three-manifold without conjugate points

A closed three-manifold with a Riemannian metric without conjugate points that admits no metric of nonpositive sectional curvature. Two holed-torus-times-circle pieces are glued so that the boundary classes satisfy h_2 = h_1 + f_1 and f_2 = h_1; shifts of axes and a Pythagorean length formula in spaces with the CN inequality show that no nonpositively curved metric exists, and a concave cometric on the neck, with an index-form analysis of every visit of a geodesic to the neck, excludes conjugate points. The course is an exposition of OpenAI's proof (2026).

Seven lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Threshold repetition for entangled games

For every finite two-player one-round game with entangled value v < 1, the probability of winning at least a fraction v + delta of k parallel repetitions decays exponentially in k, uniformly over all finite-dimensional entangled strategies and correlated question distributions. The proof: sampling tested coordinates turns a conditioning estimate into a threshold bound; resolvent purifications put conditioned states in common spaces with exact answer probabilities; an entropy telescope controls how they depend on the other player's question; one reference vector per support component yields legal one-game strategies and a delta^3 rate. A universal delta^5 rate rests on Song's conditioning estimate, a named hypothesis. The course is an exposition of OpenAI's proof (2026).

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Strict hot spots on simply connected planar domains

On a bounded simply connected planar domain with smooth boundary, no eigenfunction of the first positive Neumann eigenvalue has an interior critical point, so its maximum and minimum are attained only on the boundary: Burdzy's simply connected hot-spots conjecture, in strict form and for every eigenfunction. The proof: Filonov's gap between the Neumann and Dirichlet eigenvalues; a conformal map smooth up to the boundary, boundary signs and the multiplicity bound; Rohleder's variational principle for tangent vector fields; boundary kernels of the disc with a subcritical potential and a multiplier identity; conditional negativity of a reciprocal kernel through matrices with one positive eigenvalue; a Hilbert-space embedding of the circle. Standard facts on Sobolev spaces and elliptic regularity are named hypotheses. The course is an exposition of OpenAI's proof (2026).

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Cotype, K-convexity and the approximation property

A nonzero real Banach space with the approximation property is K-convex if and only if both the space and its dual have finite Rademacher cotype; without the approximation property this fails. The proof: Walsh norms of operators, Kahane's inequality and moment lifts; a Fourier coefficient of binary rank two for functions of sign matrices; growth of Walsh norms from a slow decrease, through frames over F_2; uniform decay for finite-rank operators; a compact domain that carries the decay to the identity through the approximation property. The decay theorem is proved in full; two classical theorems of Pisier are named hypotheses. The course is an exposition of OpenAI's proof (2026).

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Characters of central sequences and Jiang–Su stability

Kirchberg and Rørdam asked whether a unital separable C*-algebra A absorbs the Jiang–Su algebra Z whenever its central sequence algebra F_ω(A) has no characters. The answer is yes. The core is a finite theorem: every nonzero unital C*-algebra without characters has a finite maximal tensor power that receives a unital homomorphism from the dimension-drop algebra I(2,3). Its proof: the Rørdam–Winter criterion in an arbitrary unital C*-algebra; labels that stay linearly independent in every irreducible representation of a tensor power; a full square-zero element and a homomorphism from the matrix cone; a cube model of tensor powers of the cone. Proved in full from two named hypotheses: the theorem of Kirchberg and Rørdam relating Z-stability to F_ω(A), and (for infinite tensor powers) the theorem of Dadarlat and Toms. The course is an exposition of OpenAI's proof (2026).

Six lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Unitary synthesis from a Boolean oracle

One quantum oracle circuit, generated in polynomial time from n alone, with polynomially many qubits, gates H, T, T*, CNOT and oracle calls, approximates every n-qubit unitary within diamond-norm distance 1/2 once a suitable Boolean oracle is chosen. The proof: programmable slots; a diagonal block of small norm from random signs and Walsh-Hadamard transforms; the transfer function of the block matrix and two nearly isometric field maps with few nonzero Fourier coefficients per row; amplitude amplification and a recursion that calls the smaller circuit once per level; compilation of every slot into words in H, T, T* read from one oracle, with the phases kept. The course is an exposition of OpenAI's proof (2026).

Six lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A hyperbolic group that is not residually finite

A torsion-free word-hyperbolic group that is not residually finite: the fundamental group of a finite complex of Euclidean triangles with three vertices, built from punctured lines over a finite field, in which one fixed nonidentity element dies in every finite quotient and in every finite-dimensional linear representation. The proof: a criterion for complexes whose vertex links have no short closed walks, through reduced planar fillings, angular Gauss-Bonnet, thin diagrams and a stability lemma, giving torsion-freeness, hyperbolicity and nontrivial closed paths; then a comparison of ranks over the reals and over a fixed prime field, through arrays of permutation matrices, local factorizations and a polynomial certificate over the lines. The course is an exposition of OpenAI's proof (2026).

Seven lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Classical capacity of generalized amplitude damping

Every qubit generalized amplitude-damping channel, for all damping strengths and stationary populations, has Holevo information of n uses equal to n times that of one use, an explicit maximum over one variable attained by two pure states with opposite phases; the channel is also additive with every finite-dimensional partner channel for the Holevo information, the minimum output entropy and the regularized Holevo information. The proof: entropy differences as integrals of log-determinants; a jointly convex Schur-complement log-determinant and a concave scalar deficit; an entropy inequality for block matrices with a zero block, extended to the thermal block structure; a site-by-site bound for the output entropy of n uses in terms of the input populations. One named hypothesis, the classical coding theorem, turns the results on Holevo information into results on the classical capacity. The course is an exposition of OpenAI's proof (2026).

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Stability of Brenier maps

For the uniform measure on a convex body, quadratic optimal transport maps depend on the target in a one-third Hölder way for the Wasserstein distance, with one constant for all targets in a fixed compact set. The exponent one third is optimal, already on a cube with three-atom targets, which disproves a conjectured square-root estimate. The proof: derivatives of Laguerre cells; a uniform estimate of moment changes by mass changes, from a midpoint coupling by Knothe's rearrangement; moving the sites of finite targets with fixed masses keeps the potentials within O(W_2); a slice interpolation inequality turns this into the exponent one third. Proved in full in the programme, with no named hypothesis. The course is an exposition of OpenAI's proof (2026).

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Taming implies compatibility

Donaldson asked whether, on a closed four-manifold, an almost complex structure tamed by a symplectic form admits a compatible symplectic form. The answer is yes. If not, separation gives a positive current that annihilates closed invariant forms, and a closed lift adds an anti-invariant correction so that the sum annihilates all closed forms. Resolvent regularization shows that the correction is square integrable and that nearby complex lines of the current align; at points of positive density the current is planar, and its restriction there is closed. A final pairing removes the correction, and the current then annihilates the taming form, which is impossible. Proved in the programme from one named hypothesis, the short-time asymptotics of the heat kernel of the Hodge Laplacian. The course is an exposition of OpenAI's proof (2026).

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Griffiths positivity on the quadric

Griffiths asked whether every ample vector bundle carries a Hermitian metric with positive curvature in the sense of Griffiths. The answer is no, already in rank two on P¹ × P¹: an explicit bundle glued along the graphs of three projective transformations gives a family of ample bundles of which all but finitely many carry no such metric. The proof: symmetric-power filtrations for ampleness; for the metrics, a plurisubharmonic quadratic limit of dual norms along power maps, and an invariant measure that two transformations of order three with a hyperbolic product cannot share. Proved in full in the programme, with no named hypothesis. The course is an exposition of OpenAI's proof (2026).

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Small Cohen–Macaulay modules

Does every complete Noetherian local domain have a finitely generated module of maximal depth? The answer is no: the completed cone over a suitable surface has none. The proof: a small Cohen-Macaulay module is free over a power series ring and extends to a sheaf on a resolution of the cone; balancing its restriction to the exceptional surface and comparing Riemann-Roch on the surface and on the plane gives K² - 2c₂ ≤ 6H²; a cover of the plane branched along the complete quadrangle violates this. Proved in full from five named hypotheses on smooth projective surfaces, among them Bogomolov's inequality and Noether's formula. The course is an exposition of OpenAI's proof (2026).

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Kaplansky's quasitrace problem

Kaplansky's quasitrace problem: is every normalized 2-quasitrace on a unital C*-algebra a tracial state? The answer is no: a separable unital C*-algebra has quasitraces, and every one of them fails additivity by at least 1/144 on one fixed pair of positive elements. The proof: a defect lemma from two relations between finitely many elements, bundles over products of projective spaces with a Chern class obstruction, generic expanding maps normalized by a determinant potential, and an algebra of weighted shifts. Proved in full from two named hypotheses: the existence theorem for quasitraces of Cuntz, Blackadar and Handelman, and standard facts on smooth manifolds. The course is an exposition of OpenAI's proof (2026), with a sharper constant.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Kirchberg's embedding problem

Kirchberg's embedding problem: does every separable C*-algebra embed into the norm ultrapower of the Cuntz algebra O₂? The answer is no: the full C*-algebra of an explicit group has no unital embedding into the norm ultrapower of any nuclear C*-algebra. The proof: a spectral test for representations of a group with property (T), a finiteness statement in nuclear algebras, and an endomorphism of finite index that forces chains of representations. Proved in full from two named hypotheses: property (T) of the group ℤ³ ⋊ SL₃(ℤ), and the nuclearity of O₂. The course is an exposition of OpenAI's proof (2026).

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Lipschitz-free spaces and bounded approximation

The Lipschitz-free space of a metric space, and the approximation property and the bounded approximation property of Banach spaces. The theorem: there is a countable metric space with all distances at least 1 whose free space has the approximation property and not the bounded one; and the space ℓ₁ has equivalent norms without the metric approximation property. The proof: bounded blocks glued at a point; graphs of large degree whose small balls look like a tree with independent colors; signed measures of an operator; walks that can be followed by finitely many points; test functions on the edges and an average. The main theorems need no hypothesis; one consequence rests on a named theorem of Kalton. The course is an exposition of OpenAI's construction (2026).

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Metric Markov cotype two of ℓ₁

Metric Markov cotype two: moving points on the states of a reversible Markov chain so that neighbouring states carry nearby points, at a cost controlled by the variation of the old points over a time interval. The theorem: the sequence space ℓ₁ has this property, with an explicit constant. Its proof: cuts of finitely many points, a cubic map of a cube that is flat at the vertices, a fourth-power potential, and a walk that stops at a geometric random time; the new points are means of medians of three. Also: Hilbert space, the spaces L₁(μ), complex scalars, and the extension of Lipschitz maps from subsets of Hilbert space into ℓ₁ (from one named hypothesis, the extension theorem of Mendel and Naor). The course is an exposition of OpenAI's proof (2026).

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Markov type and superreflexivity

Markov type of normed spaces: how far a stationary reversible Markov chain can travel in t steps. Uniformly convex norms, superreflexivity, finite representability. The theorem: a real Banach space with Markov type p > 1 is superreflexive. Its proof: a norm on sequences that remembers the order of the entries, random increasing maps and a reversible chain of them, four homeomorphisms of the line and words in a free group, and a reversible walk that moves at linear speed. One named hypothesis, the theorem of James and Enflo; everything else is proved. The course is an exposition of OpenAI's proof (2026).

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A counterexample to Wall's D(2) problem

Wall's finite D(2) problem: is a finite complex of dimension three that looks two-dimensional to homology and cohomology with all coefficients homotopy equivalent to a finite complex of dimension two? Chain complexes twisted by a character of the fundamental group; Fox derivatives; an obstruction for complexes of dimension two proved by a tower of infinite cyclic covers; the finite plus construction; an explicit group that gives a counterexample. The new arguments are proved in full; of six standard theorems of algebraic topology, five are proved in the programme and the homotopy invariance of twisted homology is a named hypothesis. The course is an exposition of OpenAI's construction (2026).

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Randomized versus quantum query complexity

How many input bits must an algorithm read to compute a Boolean function: deterministic, randomized and quantum query complexity. The quantum query model with a proof that measurements can be deferred; amplitude amplification; the adversary bound; Huang's theorem on the sensitivity graph; the inequality D(f) <= C Q(f)^4 for total functions, with complete proofs. Functions with quantum complexity about sqrt(m) and randomized complexity about m^(2-1/k), which show that the exponent 4 is optimal; the randomized lower bound uses one named hypothesis, the Forrelation lower bound of Bansal and Sinha. The course is an exposition of OpenAI's proof (2026).

Six lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Depth-five circuits for iterated matrix multiplication

The (1,1) entry of a product of n matrices of size n x n, and arithmetic circuits with five alternating layers of sums and products in which every gate computes a homogeneous polynomial. The theorem: over a field of characteristic zero such a circuit needs at least n^(sqrt(n)/400) gates for large n, and n^(sqrt(n)+4) gates suffice; with a complete proof. The tools: a rank measure built from partial derivatives and multiplications, and two traces computed by sums over paths. The course is an exposition of OpenAI's proof (2026).

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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One-tape time in two-fifths-power space

Machines with one writable tape: blocks, visits, controllers and the local replay of a list of controllers. Finite fields of characteristic two, polynomial extensions of Boolean functions, and an evaluator that computes words defined by rules with three shared registers. The theorem: T steps of a machine with one writable tape are simulated in about T^(2/5) bits of storage, with a complete proof. Several writable tapes in about sqrt(T) bits; a lower bound for time from the space hierarchy theorem, which is a named hypothesis. The course is an exposition of OpenAI's proof (2026).

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Hypercontractivity on a torus and juntas for Lipschitz functions

Trigonometric polynomials and Fejer means on a torus; the heat operators and three inequalities for them. A logarithmic Sobolev inequality on the circle, with Rothaus's lemma. Hypercontractivity of the heat operators on the circle and on a torus of any dimension. Austin's theorem: a function on a cube that is Lipschitz for the sup distance is close to a function of a bounded number of coordinates, with an explicit bound that does not depend on the dimension. All proofs are complete; no theorem of the literature is assumed.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Large independent sets in three-colorable graphs

Label Cover instances; a graph built from families of phases on layers of tuples of questions, with a proper coloring by three colors when all tests can be satisfied. Lipschitz functions for the sup distance on tori: odd extension, means over coordinates, influences. Lists of answers from an independent set, and the alignment theorem for families of lists, with proofs of the minimax theorem and of Ramsey's theorem. The theorem: for every delta > 0 it is NP-hard to tell graphs with a proper coloring by three colors from graphs whose independent sets have fewer than delta |V| vertices, from two named hypotheses (the PCP theorem and parallel repetition); Austin's theorem on Lipschitz functions of many variables, which the proof uses, is proved in the course Hypercontractivity on a torus and juntas for Lipschitz functions. The course is an exposition of OpenAI's proof (2026).

Six lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The rigidity of the Turing degrees

Computability relative to a set, defined by computably enumerable sets of triples; joins and Turing degrees; real numbers as cuts. Meager sets, Baire's theorem, continuity of Borel functions on comeager sets, and the avoidance lemma. Recovery of an irrational number from four values of a Borel function with countable fibres that is compatible with sums. The rigidity theorem for the Turing degrees, from the theorem of Slaman and Woodin on Borel representations of automorphisms, which is a named hypothesis. The course is an exposition of OpenAI's proof (2026) of the rigidity of the Turing degrees.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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2-to-1 games with perfect completeness

Projection games, their value and their repetition; formulas with a gap; the clause-variable game and the base game with perfect completeness and small soundness, from two named hypotheses. Slices of matrices over the field with two elements and the inverse theorem for partial tables, from a third named hypothesis. The tree of function spaces, keys of functions, the constraints of the new game, perfect completeness. Joint folding, the functional of a response, buckets, and the law of the tests. Comparisons of the law of the tests with uniform laws: one hidden child, sparse projection, the posterior bound, bias. Two accepted levels, the response table, good advice and the prediction of the upper response. The hidden part of the upper matrix, the list of heavy characters, the two decoders. Admissible forms, two draws of the lower array, a fixed form on a slice, agreement with forms of bounded rank. Odd lists, clean children, the bound by parallel repetition. The choice of the parameters, soundness, maps that are exactly 2-to-1, equal weights, the theorem. The course is an exposition of OpenAI's proof (2026) of the 2-to-1 games conjecture with perfect completeness, from three theorems of the literature that are stated as named hypotheses.

Ten lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The logarithmic Brunn-Minkowski inequality

Support functions and Wulff bodies; the logarithmic Brunn-Minkowski inequality for origin-symmetric convex bodies and its reduction to a variance bound for soft polytopes; the additive inequality for powers between 0 and 1. Moment pairs; adjoint identities; the operator A and its test functions. The tensor estimate, with the difference written as a sum of squares. Density of the even test functions. The variance bound for soft polytopes and the logarithmic Brunn-Minkowski inequality, from two named hypotheses; the Gaussian case without hypotheses. The course is an exposition of OpenAI's proof (2026) of the logarithmic Brunn-Minkowski inequality; it proves the inequality from two statements of the literature that it names as hypotheses.

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The dimension of self-similar measures

Entropy of random variables with finitely many values: the chain rule, conditioning, mutual information, mixtures, and estimates for the binary entropy. Entropy of a law on the line at a scale, averaged over the positions of the grid: comparison of scales, concavity, convolution, uniform steps. Self-similar measures on the line; entropy at small scales and dimension; uniform nonsaturation. The two-point gain; pair capacities; the gain of a convolution from fair pairs. The pair bound for a finite law, by random nested partitions. The entropy rate of the walk of maps; the upper bound for the dimension. Block types; entropy of translations given the types; pair capacity at fine scales. The dimension formula d = min(1, h/chi); the formula without exact overlaps. The course is an exposition, with a complete proof, of OpenAI's theorem (2026) on the dimension of self-similar measures on the line; the proof assumes that the measure is exact dimensional.

Eight lessons · proofs and exercises · self-checked by Claude Opus 5.5

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A metric with no local isometric immersion

Seven lessons with a complete proof of OpenAI's theorem (2026) that there is a smooth metric on a square such that no neighbourhood of the centre has a smooth isometric immersion into three-dimensional space. The height equation of an immersed surface; the construction of the metric from small squares that accumulate at the boundaries of disks of negative curvature; the patch obstruction, proved in five lessons: regularity of solutions of the height equation across the curve where the curvature changes its sign (an estimate at a free edge for an equation of mixed type), energy estimates where the curvature is negative, and a rapidly oscillating perturbation of the metric whose effect is measured by a moment and excluded by the theorem of Baire.

Seven lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The free uniform spanning forest

Random spanning trees of finite weighted graphs: the matrix-tree theorem, inclusion probabilities and covariances of edges through the reduced Laplacian, negative correlation, and the response identity that bounds the dependence of the inclusion probabilities on the weights uniformly in the graph. The free uniform spanning forest of an infinite connected graph with finite degrees: determinant formula for the uniform spanning tree, convergence along exhaustions, the limit law and its properties. Tilts of the forest by numbers on finitely many edges, their meaning as conditional laws, the response bound, and the drift. The integral equation of the drift for arbitrary families of continuous paths: existence, uniqueness, causality, measurability. The course is the start of an exposition of OpenAI's theorem (2026) that the free uniform spanning forest is a factor of independent labels; that theorem is not yet in the course.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Coboundary expanders of bounded degree

Coboundary expansion of simplicial complexes over the field with two elements, and the complexes of OpenAI's theorem (2026) that bounded-degree coboundary expanders exist in every dimension. Definitions, weights, the relation to cohomology, examples, the passage to skeletons; coset complexes of groups; the groups of polynomial matrices and their coset complexes, with bounded degree and growth. The theorem is derived from two named statements, an expansion inequality relative to the cocycles and the vanishing of cohomology, which this course does not prove.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Universal optimality of the triangular lattice

A complete proof that the triangular lattice is universally optimal in the plane, following OpenAI's proof (2026): among all point configurations of density one it has the least energy for every completely monotone interaction of the squared distance. The linear programming bound for configurations of density one; completely monotone interactions as positive mixtures of exponentials; the reduction to one normalized statement about functions of one variable; the construction of these functions from a sine product, cardinal interpolation, Gaussian damping and the Fourier transform; the solution of the interpolation system; and the proof of the two sign conditions. Three steps are finite sets of inequalities between explicitly defined numbers, established by computations with rigorous error bounds; the programs are printed in the lessons. A last lesson derives that the triangular lattice has the least periodic energy for the interactions |x|^-s, 0 < s < 2, and -log|x| among all periodic configurations.

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The Cartier–Miller evaluation of coefficient sums

Truncated powers of a polynomial of degree three or four over a prime field and the curve v^2 = f(w): the coefficient that gives the trace of the curve, the truncated sum as a sum over the points of the curve and as a character sum, the Cartier operator on the function field of the curve with its values on the two differentials that carry the sums, the points of the curve with orders and residues, functions and differentials without poles, the regular identity that evaluates the sums through a function on the curve, and the statement of D. Jordan's theorem (2026) that evaluates these sums with about log p operations through the arithmetic of the curve.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Pretrees

Sets with a betweenness relation that behaves as in a tree: the axioms, intervals, the theorem that every finite pretree is a set of vertices of a finite tree, directions on linear sets, uniqueness of medians, and dense, complete and median pretrees, with an example of a complete dense pretree that is not median, separable pretrees, whose intervals are ordered like the real interval [0,1], a theorem that recognizes dendrites by a pretree structure, flows, directed arcs with their suprema, the completion of a pretree by its flows, the three kinds of flows, the adjacent points of the completion, discrete pretrees as sets of vertices of trees, quotients of pretrees, the dense quotient, the pretree of cut points of a connected space, the map from such a space to the completion of its set of cut points, and the dendrite quotient of a continuum.

Fifteen lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The finite-energy boundary of a group

Finite-energy functions on the Cayley complex of a finitely presented group, and the boundary they give a one-ended group: a function with connected level sides, a compact metric space containing the group with square summable edge lengths, a boundary that is a continuum, paths near boundary points, the action of the group on the boundary as a convergence group with dense orbits, the dimension and first Čech cohomology of the boundary, a continuum without cut points inside the boundary, and the proof that this continuum is locally connected. The result: under named hypotheses on the group, every finite-energy gradient is a limit of gradients of finitely supported functions. This is the part of OpenAI's proof (2026) of the Singer conjecture in dimension four outside the middle degree that takes place on the group.

Ten lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Convergence groups

Convergence groups on compact metric spaces, from the definition by sequences: every element is elliptic, parabolic or loxodromic, loxodromic elements have attracting and repelling fixed points and a compact fundamental domain, and the stabilizer of a loxodromic fixed point is virtually infinite cyclic; convergence actions and minimal actions pass to equivariant quotients, a convergence group on a continuum acts as a convergence group on its dendrite quotient, the points of the completion of the cut points have supports in the continuum that control which of them a group element fixes, under conditions on the group the dendrite quotient is not a point, and when the group fixes a point of the dendrite quotient the cut points form the vertices of a tree on which the group acts. A group with a finite generating set and a one-ended Cayley graph has the fixed-point property for trees that these results assume.

Eight lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Continua, arcs and cut points

Locally connected continua with complete proofs: connected open sets are path connected, every path contains an arc, first Čech cohomology of a compact metric space by finite covers, and the theorem that a Peano continuum of dimension at most one with vanishing first Čech cohomology is a dendrite and has cut points, which is the last step of OpenAI's proof (2026) of the Singer conjecture in dimension four outside the middle degree.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Depth-three circuits beyond the square root

OpenAI's theorem (2026) that one explicit polynomial-time language needs depth-three OR-AND-OR circuits with more than 2^(A sqrt n) gates for every constant A: a restriction bound for CNFs that does not depend on the number of clauses, a disjoint sparse partition by sunflowers, slices of a hash-and-polynomial language with limited independence, and a count of all gates.

Seven lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Scheduling unit jobs on three machines

OpenAI's theorem (2026) that unit jobs with arbitrary precedence constraints can be scheduled on three identical machines with minimum makespan in polynomial time: a dynamic program over Boolean descriptions of bounded size, made complete by a hierarchy of schedule intervals cut at separators of shared cutoffs, with short lists of upsets and boundary invariants that keep every description bounded at every depth.

Eight lessons · proofs and exercises · self-checked by Claude Opus 5.5

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One sample suffices for matroid prophet inequalities

OpenAI's theorem (2026) that one independent sample per element suffices for a constant-competitive prophet inequality on every finite matroid, even when the arrival order is chosen by an adversary who sees all samples, values and the rule's random seed; through density expansions, guarded paths of flats, reverse exposure and a marked-pivot count.

Nine lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Heilbronn's triangle problem

OpenAI's theorem (2026) that n points in the unit square can have every triangle of area at least a constant times n^(-2+eta) for a fixed eta > 0; through field norms written in a large base, residue orbits, a cap at an auxiliary prime, lattice counts and a deletion argument.

Eight lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Configuration gaps in bin packing

OpenAI's theorems (2026) that the configuration linear program of bin packing can fall short of the optimum by any additive constant, refuting the modified integer round-up conjecture, and that additive-constant approximation is NP-hard; through competing trees, interval coverage encoded in five-item bins, and a vertex-cover reduction.

Six lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The entropy conjecture for C¹ maps

OpenAI's theorem (2026) that Shub's entropy conjecture fails for C^1 maps: a noninvertible C^1 self-map of a compact manifold with zero topological entropy and the eigenvalue 2 on second homology; through entropy localization, a clock with prediction registers, an attraction theorem and the degree of squaring.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Packing and covering in infinite matroids

OpenAI's theorem (2026) that the infinite matroid packing/covering conjecture is false in ZFC; through density ideals, an ultrafilter with an ordinal rank, local bases defined by probes, and a self-dual uniform matroid placed along a double ray.

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Borsuk's conjecture fails in dimension nine

OpenAI's theorem (2026) that the rank-one projections of R^4, a copy of projective three-space in R^9, cannot be covered by ten sets of smaller diameter; through admissible maps, an odd map on symmetric matrices, degree modulo two and the combinatorics of six-label triangle systems.

Six lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The square-root degree bound

OpenAI's theorem (2026) that the linear Fourier coefficients of a Boolean function can sum to more than any constant multiple of the square root of its degree; through observations of independent signs, a tagged report on Gaussian intervals, and amplification by independent copies.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Superexponential van der Waerden numbers

OpenAI's theorem (2026) that van der Waerden numbers grow super-exponentially for every number of colours, W_r(k) > k^(ck floor(log2 r)); through circle coordinates and adaptive meshes, a balanced colouring chosen by the local lemma, and a sparse perturbation by squared-norm bands.

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Barnette's conjecture

OpenAI's proof (2026) of Barnette's conjecture that cubic bipartite 3-connected planar graphs are Hamiltonian; through sphere maps, pairs of face-vertex matchings, a cancelling exponential sum and two induced trees in the dual triangulation.

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Polynomial removal fails for ordered binary matrices

OpenAI's counterexample (2026) to polynomial removal for ordered binary matrices: a fixed 66 x 66 pattern and tree-indexed matrices that are far from avoiding it but contain very few copies of it.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Seymour's second neighborhood conjecture

OpenAI's proof (2026) of Seymour's second neighborhood conjecture, that every finite oriented graph has a vertex with at least as many second as first out-neighbors; through minimal counterexamples, transitive tournament blow-ups, a pruning theorem proved with generic ranks, and an extremal argument; with weighted forms and consequences for directed triangles.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Dimension reduction in L_p

OpenAI's theorem (2026) that n points of L_p, 1 < p < infinity, p != 2, embed into l_p with any fixed distortion above 1 using n^o(1) coordinates, while exact embeddings need about n^2/16; through weighted moments, the localization of electrical flows, and Poisson sums.

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The Lane-Emden conjecture

OpenAI's proof (2026) of the Lane-Emden conjecture and its weighted Henon-Lane-Emden form: below the critical hyperbola the elliptic system has no positive solution on R^n, without any condition at infinity, by a localized virial identity for Newton potentials and a comparison along parallel lines.

Five lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Three-marginal Coulomb transport

OpenAI's counterexample (2026) to the Monge ansatz: for the Coulomb cost of three particles in space, and every Riesz cost in dimension at least two, a smooth density for which the optimal coupling exists but is given by no pair of maps, although maps come arbitrarily close. A fourth lesson restates the result with infinitesimals: an internal pair of maps at an infinitely fine resolution exceeds the minimum by a positive infinitesimal, and its standard part is the optimal coupling.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Sampling graphs with given degrees

OpenAI's proof (2026) that the switch chain on simple graphs with given degrees mixes in polynomial time for every graphical degree vector, through the operator inequality H^2 >= H for pair resamplings, and an exact uniform sampler with expected polynomial running time.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Shortest common superstrings

OpenAI's polynomial-time algorithm (2026) that finds a common superstring at most twice as long as the shortest one, for every finite family of strings. Forced occurrence counts give a balanced graph of cost at most the optimum; periodic layers and a rule for periodic words show how to connect it at no more than the same cost again.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Thresholds and expectation thresholds

OpenAI's proofs (2026) of four conjectures on thresholds: the second Kahn–Kalai conjecture (containment thresholds of graphs within a logarithm of the expectation threshold), Talagrand's conjecture that integral and fractional expectation thresholds agree up to a constant, Talagrand's discrete convexity conjecture, and the Ascoli–He–Park–Talagrand decomposition of graphs into pieces that appear at a constant multiple of the integral expectation threshold. Planted resampling, tree covering, multiscale selectors, biased Fourier weights and embedding arguments, all proved in the course.

Eight lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Sharp thresholds for graph properties

OpenAI's proofs (2026) of the Friedgut–Kalai conjectures: increasing graph properties of G(n,p) have threshold width O(log(1/ε)/(log n)²), and r-uniform hypergraph properties width O((log n)^(−r/(r−1))). The course develops biased Fourier analysis and hypercontractivity, restricts symmetric functions to the edges meeting a random block of vertices, and captures each Fourier support at one vertex.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Circulant Hadamard matrices

OpenAI's proof (2026) of the circulant Hadamard conjecture, usually attributed to Ryser: real circulant Hadamard matrices exist only in orders 1 and 4. The course works in the group ring of the cyclic group: it proves the local structure of cyclotomic integers near a prime and Turyn's restriction to orders 4u² with u odd, shows that alternating products of character values are roots of unity of odd order, and derives a contradiction modulo a prime above 2; even-length Barker sequences follow.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Metric Markov cotype of ℓ₁

OpenAI's answer (2026) to a question of Mendel and Naor: the space ℓ₁ has metric Markov cotype two. The course compares geometric and uniform averages of time along stationary Markov chains, writes finite subsets of ℓ₁ through weighted cuts, proves a flatness inequality for the cubic 3r²−2r³, and builds the required points from a killed random walk; they turn out to be expected medians of three stopped-walk endpoints.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Average sensitivity of threshold functions

OpenAI's proof (2026) of the asymptotic Gotsman–Linial conjecture: a polynomial threshold function of degree d on the n-dimensional Boolean cube has average sensitivity at most 8d√n. The course proves a one-sided coupling inequality, builds a grading of the functions on the cube weighted by the polynomial, counts sensitive edges through an anticommutator, and turns the degree bound into a binomial coupling; it also derives a bound on noise sensitivity.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The Lang–Plaut problem

OpenAI's negative answer (2026) to the Lang–Plaut problem: a doubling subset of Hilbert space that admits no bi-Lipschitz embedding into any finite-dimensional space. The course builds the set from coloured strips at rapidly decreasing scales and proves its doubling bound, differentiates the Lipschitz sheets of a supposed embedding near a lexicographically maximal derivative, proves two energy estimates, and turns crossings of rows and columns into a packing contradiction in Euclidean space.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Sensitivity and block sensitivity

OpenAI's superquadratic separation (2026): Boolean functions whose block sensitivity exceeds any constant multiple of the square of their sensitivity, answering negatively the quadratic form of the Sensitivity Conjecture. The course proves the composition and OR lemmas, a classical quadratic example, the construction of nested predicates on a randomly labelled tournament, and the sensitivity recurrences that give the separation and a fixed power larger than 2.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The duality conjecture for metric entropy

OpenAI's counterexample (2026) to the duality conjecture for metric entropy, posed by Pietsch in 1972: covering numbers of convex bodies are not preserved by polarity up to universal constants, already when one body is a cube. The course turns a real matrix into a pair of bodies, proves the duality theorem for convexified packing, approximates all column combinations of a matrix built from partitions by a short list, and constructs the partitions from random directions for symmetric multilinear forms over a finite field.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Fixed points of nonexpansive maps

OpenAI's solution (2026) of Kirk's problem: every nonexpansive self-map of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point, for the given norm. The course reduces to a minimal invariant set where approximate fixed points are diametral (Goebel–Karlovitz), finds an anchor point through resolvents and an idempotent in a compact semigroup of maps (Ellis), builds an infinite tree of maps whose functionals are chosen against compact families of predicted outputs, and switches its leaves to a common approximate fixed point to obtain a contradiction.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Tingley's problem

OpenAI's solution (2026) of Tingley's problem: every surjective isometry between the unit spheres of real normed spaces extends to a surjective real-linear isometry of the whole spaces. The course proves the Mazur–Ulam theorem, makes a hypothetical positive defect attained in an ultrapower, finds aligned extremal chords as a fixed point of a return map with Darbo's theorem (proved from Schauder's and Brouwer's), and rules them out by support, secant and convexity estimates.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Amenability and unitarizability

OpenAI's answer (2026) to Dixmier's question for discrete groups: a discrete group is amenable exactly when every uniformly bounded representation on a Hilbert space is similar to a unitary one. The course proves the Day–Dixmier theorem and, for every nonamenable group, constructs representations with all norms at most 1 + ε that cannot be made unitary: nonamenability gives a contracting average of translations, Hall's theorem assigns the edges of a Cayley-type graph to their endpoints with small loads, random sign frames turn the assignment into operators with bounded conjugation differences, and a Hilbert–Schmidt energy bound shows that their cocycle has no bounded implementer. Corollaries for Thompson's group F and for group C*-algebras.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Thompson's group F and amenability

OpenAI's proof (2026) that Thompson's group F is not amenable, answering Geoghegan's 1979 question. The course develops F through dyadic partitions of the unit interval: normalized restrictions, images of fine uniform partitions, elements moving pairs of separated dyadic intervals, and covariance. It constructs an explicit Lipschitz map of the closed unit ball of a Hilbert space into its unit sphere that moves every point by at least 1/2, through a curve with a bounded, uniformly separated tail. It then colors dyadic partitions recursively with this map and proves that one finite set of elements of F has a uniformly large relative boundary on every finite subset, so that F has no Følner sets.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The hyperinvariant subspace problem

OpenAI's negative answer (2026) to the hyperinvariant subspace problem: on every infinite-dimensional separable complex Hilbert space there is a nonzero operator S with ‖S^n‖^{1/n} → 0 whose commutant is a proper, weakly closed, unital, transitive algebra, so S has no hyperinvariant subspace other than 0 and the whole space. The course proves Domar's theorem that a bilateral weighted shift with regular, quadratically decaying logarithmic weights has only the tails as invariant subspaces; builds a measurable field of such shifts over the 2-adic integers with the odometer and proves the estimate ‖S^n‖ ≤ exp(−10⌊(n+1)²/4⌋); shows that operators commuting with the scalar multipliers are fields of operators; and constructs a contraction commuting with S that moves every tail one step backward, so that a hyperinvariant subspace would carry a strictly decreasing label along the odometer, which a probability space does not allow.

Three lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Isomorphisms of free group factors

OpenAI's theorem (2026) that the free group factors L(F_n), 2 ≤ n < ∞, are all isomorphic. The first part proves OpenAI's rank step L(F_n) ≅ L(F_{n+1}) for n ≥ 3: free independence and freely generating Haar tuples, a bounded logarithm of the extra generator and a norm bound for free sums that does not depend on the number of summands; trace-preserving polynomial flows with displacement bounds; a small cocycle with a prescribed word value, through free prefixes, Catalan moments and cutoff inversion; and a limit with adaptive error budgets whose tuple generates the whole factor. The second part proves Voiculescu's corner formula, that the corner of L(F_n) cut down by a projection of trace ½ is L(F_{4n−3}), and deduces L(F_2) ≅ L(F_3): the semicircle law, a freeness criterion on the full Fock space, free semicircular families, 2×2 matrices with semicircular and circular entries that are free from matrices over the other variables, and the polar decomposition of a circular element. The third part extends the corner formula to every trace 1/k, with k×k semicircular matrices, and shows that every positive rational number lies in the fundamental group of L(F_2).

Nine lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Relative bicentralizers and core commutants

Relative bicentralizers of inclusions with expectation and the commutant of the bottom algebra in the continuous core. The relative bicentralizer, its description in the Ocneanu ultrapower, its expectation as a limit of averages over almost centralizing unitaries; Connes' transition isomorphisms, the relative bicentralizer flow of Ando, Haagerup, Houdayer and Marrakchi, and the state-independent algebra generated by the bottom algebra and the relative bicentralizer; Marrakchi's binormal identity, proved through a tracial corner of the continuous core; approximate eigenoperators and Marrakchi's spectral shift maps; and OpenAI's proof (2026) that the two relative products of the bottom algebra and the relative bicentralizer differ exactly by the flow, through an iterated completely positive kernel, an estimate uniform in total modular energy and repeated random tensor tests. Consequences: the core fixed point theorem and a criterion for subalgebras with the same relative commutant in the core. Finally, finite-dimensional subfactors, state changes and the lifting of matrix units from the asymptotic centralizer, and the main theorem of OpenAI for bottom factors of type III_1: an approximately finite-dimensional subalgebra, invariant under the modular group of a suitable state, with the same relative commutant in the continuous core, built by an inductive construction after Ando, Haagerup, Houdayer and Marrakchi; and, for a single factor of type III_1, Haagerup's theorem that the faithful normal states with irreducible centralizer are norm dense. The last lesson treats semifinite bottom factors: Popa's hyperfinite subfactor with the same relative commutant for factors of type II_1, a matrix amplification with an almost periodic state for type II_infinity, and the resulting theorem for every semifinite bottom factor.

Eleven lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Kadison's similarity problem

Kadison's similarity problem, solved: every bounded unital homomorphism of a unital C*-algebra into the bounded operators on a Hilbert space is similar to a *-homomorphism (OpenAI, 2026). Completely bounded homomorphisms and Paulsen's similarity theorem; implementation of derivations with half the completely bounded norm and Arveson's distance formula; the noncommutative little Grothendieck inequality, row and column estimates, finitely generated homomorphisms and cyclic domains; Kirchberg's passage from inner derivations to similarity with a polynomial bound; words in free subalgebras; Schur-Weyl invariants, propagation from small corners and a Hankel test, which give a free-product corner estimate independent of the corner; transfer to II_1 factors through Popa's free independence in ultrapowers; the uniform commutator estimate for all von Neumann algebras; inner derivations, the similarity theorem with the bound ||pi||_cb <= ||pi||^(2C), and universal hyperreflexivity of von Neumann algebras.

Ten lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Bicentralizers of type III₁ factors

Bicentralizers of type III₁ factors and Connes' bicentralizer problem. Connes–Størmer transitivity makes the asymptotic centralizer of every faithful normal state a II₁ factor; the bicentralizer, its description in the Ocneanu ultrapower, its state-preserving expectation and the Houdayer–Isono self-bicentralizing property; Connes' isomorphism and the bicentralizer flow, built in a single ultrapower; modular averaging, bounded recovery and spectral intertwining rigidity (OpenAI, 2026), which make the flow trivial; Haagerup's criterion for a trivial bicentralizer; Marrakchi's strongly invariant states; and the theorem that every type III₁ factor with separable predual has trivial bicentralizer.

Eight lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Projective modules and locally polynomial algebras

Projective modules over polynomial rings and algebras that are locally polynomial. Quillen's patching theorem for extended modules and the method of Quillen induction; Horrocks' theorem, with an elementary proof over local rings; the Quillen–Suslin theorem that finitely generated projective modules over a polynomial ring over a field are free; automorphism groups of symmetric algebras, axiom Q and the factorization of an automorphism over two localizations; and the theorem of Bass, Connell and Wright that a finitely presented algebra which is locally a symmetric algebra is the symmetric algebra of a finitely presented module, so that locally polynomial algebras over k[y₁,…,y_r] are polynomial rings.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Generators of II₁ factors

Every II₁ factor with separable predual is generated by one operator, equivalently by two self-adjoint operators (OpenAI, 2026). Haar unitaries and freeness of a unitary from a C*-algebra in tracial von Neumann algebras; tracial ultraproducts; Popa's free independence theorem for irreducible inclusions, proved by patching partial isometries with local quantization in corners; Haagerup's inequality for words of length one and a perturbation of a unitary whose correlation with a fixed vector stays uniform along long words; relative generation W*(P,u) = M for a dense G_δ set of unitaries u; irreducible hyperfinite subfactors, two self-adjoint generators of the hyperfinite factor and a 2×2 matrix encoding. Finally the generator problem in full: every von Neumann algebra with separable predual is singly generated, through properly infinite algebras (Wogen), finite factors and direct integrals with a Borel choice of fibre generators (Willig); and the vanishing of the generator invariant of Dykema, Sinclair, Smith and White.

Six lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The irrationality exponent of π

The theorem that the irrationality exponent of π is 2: for every ν > 2, |π − p/q| > q^(−ν) for all large q. Irrationality exponents, Dirichlet's and Liouville's theorems; a weighted multiplicity estimate along commuting vector fields, proved with formal flow coordinates and Bézout's inequality; a curve inequality for logarithmic curves with separated weights, by a zero estimate and logarithmic residues; interpolation of jets on logarithmic curves through a blow-up, Kleiman's theorem and Serre vanishing; an interpolation determinant at Gaussian rational centres with an arithmetic lower bound and an analytic upper bound from Taylor collisions; and the convergence of the Flint–Hills series, with the exact convergence region of the sums of n^(−a)|sin n|^(−b).

Six lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Intersection numbers and positivity

Intersection numbers of line bundles on proper schemes, defined through Snapper's polynomial of Euler characteristics, with multilinearity, reduction to components and divisors, degrees on curves, pullback and positivity for ample bundles; the Nakai-Moishezon criterion for ampleness; nef line bundles and Kleiman's theorem that they have nonnegative intersection numbers, so that nef plus ample is ample; and Bézout's inequality for the isolated zeros of n polynomials in n variables, with a weighted version.

Four lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Large gaps between primes

Large gaps between consecutive primes. The average gap, and the theorem that for every C > 0 a positive proportion of the gaps p_(n+1) - p_n exceed C log p_n, which answers a question of Erdős and Prachar on the increases of p_n/n. Smooth divisor sums and their averages, with a prime factor through the Bombieri-Vinogradov theorem; the singular series and Gallagher's theorem on its average; correlations of smooth divisor sums by Euler products near the pole of the zeta function; a square of signed divisor sums whose prime mass in an adjacent interval is controlled by neighbouring dimensions; and an alternating choice of test functions that makes this mass arbitrarily small.

Six lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Irrationality of special values

Irrationality proofs by small linear forms and by determinants. The irrationality of ζ(2) and Apéry's theorem that ζ(3) is irrational, by Beukers' integrals and an involution of the unit interval; then the complete proof that Catalan's constant G = 1 - 1/9 + 1/25 - ... is irrational: determinants of mixed moments with Chebyshev rows whose Taylor contact cancels ζ(2), lower bounds for their denominators at the prime 2 and at the odd primes by base-p digit reductions, nonvanishing at prime scales by Frobenius and a finite certificate, and an upper bound at the real place by an integral formula, bounded interpolation, Chebyshev expansions of logarithmic kernels and an exact root certificate.

Seven lessons · proofs and exercises · self-checked by Claude Opus 5.5

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The Gaussian moat problem

Walks through Gaussian primes with bounded steps. The arithmetic of split primes and the reduction of the uniform component bound to a finite periodic sieve; Shannon entropy, Pinsker's inequality, short lists of candidates and averages over random coordinates; Hoeffding's inequality and the isoperimetry of the discrete cube; many differences of a walk stretch and their separation by randomly signed products of Gaussian primes; entropy enrichment by sampling differences; coverage transferred backwards with shared continuations; a common sampling schedule; and the information cost of avoiding zero classes. Together they prove that for every step bound the connected components of the Gaussian-prime graph have bounded size, so that one cannot walk to infinity on Gaussian primes with bounded steps.

Eight lessons · proofs and exercises · self-checked by Claude Opus 5.5

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Incompressible flow: foundations and forced equations

Forces, energy, vorticity, exact scaling, pressure recovery, whole-space and periodic projections, constant modes, the viscous integral equation, and periodic and whole-space weak and H1 strong solutions. Fifty-four lessons supply complete existence and compactness proofs, spatial-tail estimates, original forces and viscosity, energy endpoints, pressure recovery, stability, maximal continuation, nonendpoint velocity criteria, the small L3 endpoint and two hundred and seventy solved exercises. Critical mild solutions include full pressure, homogeneous data and small-data global existence with arbitrary periodic means. Vorticity and strain continuation, finite-sum criteria, a small endpoint tail and exact source corrections are proved. Exact boundary maps retain wall work and surface forces. Euler local construction, top-norm continuity and logarithmic vorticity/strain continuation retain the original force and mean. Actual velocity-tail continuation, concentration examples and two finite weighted heat estimates retain the original viscosity, residual and time endpoints. Moving-annulus energy retains full sphere, harmonic, stretching and force terms; the weighted Sobolev and local curl arguments are complete. Global nonlinear energy and total speed retain the original force, viscosity, full heat/Leray kernels and all frequency contributions. Actual annulus selection, full local heat propagation and a closed energy budget give local L4-time velocity with both force classes. The complete localized vorticity equation retains every boundary term. Complete positive-time heat iteration, full Newton maps and an explicit smallness construction give actual arbitrarily small annular velocity, vorticity and first derivatives for the unforced or compact-force receiver. Original-viscosity frequency backpropagation retains the same amplitude and every exterior and frequency contribution. The exact source hierarchy and finite backward chains are proved. Quantitative total-speed iteration and the exact original source time separation are proved. Full persistence intervals strengthen the constants, and selected earlier frequencies retain a quantitative reciprocal-square sum. Original vorticity mass and an actual quantitative regularity interval are proved, including full heat and Newton boundary terms and every finite Gaussian coefficient bound. The same mass survives on the actual shorter interval. The complete outward Gaussian and final-time annular transfer now gives original final-time vorticity and velocity mass in both weighted-transfer cases. Smaller-shell and optimized covering-radius consequences are proved. The complete uniform original scale family now gives positive scale-independent velocity mass and the frequency-event bound. Exact overlap, the full Lp curl receiver, stronger mass coefficients and joint radius-and-shell optimization are proved. The complete unforced whole-space bounded-L3 energy endpoint is proved, retaining the full source sixth term and original heat field. Exact frequency absorption, all-order smoothing and common restart preserve the original viscosity and pressure. The improved envelopes, two actual projection bands and joint heat-weight/time-window optimization have complete receiving bounds.

The twenty-first constructs the full periodic residual stress and pressure maps, a sharp inverse-divergence norm and explicit rational Beltrami families. It proves every finite correction term with original viscosity and force, and supplies five solved exercises on pressure freedom, the sharp trace-free positive domain, mixed families, resonant mean correction and minimum finite force cost.

The twenty-second proves the original intermittent covering and norms, complete derivative bounds and a stronger periodic product estimate with the missing amplitude retained. Five solved exercises give exact fourth moments, a sharp boundary resonance, rectangular periods and the full temporal pair correction with pressure and constant modes.

The twenty-third proves the full intermittent velocity, pressure and residual stress with original viscosity and every quadratic term. Explicit periodic kernels give complete oscillatory inverse bounds. Five solved exercises give exact pair matrices, square norms, the temporal correction cost, an optimal frequency split and the entire energy increment.

The twenty-fourth constructs the original mollified equation, complete stress cutoffs and smooth energy amplitudes, including their zeros and every derivative scale. It corrects the source energy-volume factor by an exact comparison. Five solved exercises retain the actual shear pressure, cutoff geometry, arbitrary volume, smoothing loss and every term in the full physical energy error.

The twenty-fifth proves the complete original velocity step, including both small corrections, the full increment and every derivative through order three. It constructs the periodic projection kernel and proves all finite-exponent bounds and the required smooth-input estimate. Five solved exercises give exact projections, optimized kernel bounds, the full projected fourth moment and the original amplitude tail.

The twenty-sixth proves the complete original stress step: every viscous, transport, correction, oscillation and mollification term, the exact pressure projection and all first derivatives. It corrects the printed parameter example and proves a larger admissible range. Five solved exercises give exact stress and curl maps, the complete parameter region, a changing-mean force example and sharper base thresholds.

The twenty-seventh completes the prescribed-energy weak solution: every energy contribution, the original time endpoints, convergence, pressure means, continuous integrable vorticity and exact time locality. Five solved exercises cover sharper base thresholds, Lipschitz energy profiles, all pressure modes, the original heat cancellation and a continuum sharing the same initial velocity.

The twenty-eighth constructs the original vanishing-viscosity limit, retaining the full starting tensor, energy index, coefficient dependence and time intervals. One fixed base gives strong positive-Sobolev and pressure convergence. Five solved exercises test the physical equation and prove explicit rates in the original viscosity.

The twenty-ninth constructs the original radial vortex maps, sharp Green bounds, a complete smooth background and its arbitrarily deep spectral well. It proves the exact eigenfunction dictionary, the integer threshold window and full slow-tail cutoff estimates. Five solved exercises retain the physical pressure, energy, circulation and sign costs. The following lesson supplies the nonreal Rayleigh eigenvalue.

The thirtieth constructs an actual unstable Euler vortex with an integer angular mode. It proves the full oriented boundary integral, localized operator bounds, uniform complex root construction, smooth physical mode and original pressure. Five solved exercises include exact origin coefficients and sharp weighted moment thresholds.

The thirty-first constructs the compact unstable vortex and its full weighted Euler generator. It proves spectral persistence with a definite cutoff radius, the complete Biot-Savart kernel, the closed transport domain and finite algebraic multiplicity. Five solved exercises retain the zero mode, strengthen the weight range and give an explicit ring correction with its full physical volume.

The thirty-second constructs the complete original three-dimensional ring kernel and transfers the actual unstable spectral space. Exact axis and transport maps, every curvature term, the physical pressure and complete Euler and viscous background forces are proved. Five solved exercises strengthen the stretching bound and preserve the whole enclosed algebraic multiplicity.

The thirty-third proves the original physical ring generator and the entire heat-and-dilation domain. Full cylindrical volume, small stretching, compact coupling, precise transport domains and smooth-input bounds receive the actual Euler mode. The exact viscous semigroup retains all coefficients, energy and smoothing terms. Five solved exercises include explicit radius thresholds, a velocity-integrability counterexample and the corrected amplitude and pressure in the curl map.

The thirty-fourth proves the complete original viscous instability and full velocity receiver. Exact domains, semigroups, resolvents and compact spectral transfer retain every viscosity, amplitude and dilation term. Endpoint vorticity integrability, an explicit counterexample family and the entire velocity and pressure maps are proved. Five solved exercises construct the full velocity generator, retain the nonlinear residual and calculate the exact physical tail and heat-semigroup norm.

The thirty-fifth proves the actual velocity spectral growth and complete smoothing at every real Sobolev order, retaining the original viscosity, amplitude and drift. The full solenoidal space and the axisymmetric no-swirl space have their own proved maximal rates. Five solved exercises calculate exact Jordan and real-mode norms, construct the original compact force with its energy identity, and prove the complete nonlinear integral map.

The thirty-sixth constructs the complete original nonlinear correction and distinct suitable Leray flows with one force and zero initial velocity. Every physical coefficient, pressure, time endpoint and energy identity is retained. Five solved exercises give a sharper contraction interval, an explicit second-order correction, a continuum of flows with the same force, both pressure orders and the exact initial concentration.

The thirty-seventh constructs the exact averaged operator, original-viscosity local evolution, energy defect and full force and pressure maps. An explicit Gaussian example and a complete dilation correction test the source formulas. Five solved exercises retain the full frequency relation, optimize the Gaussian rotation, construct the Sobolev cancellation correction and weighted evolution, and prove the original dyadic local flow with both boundary energy fluxes.

The thirty-eighth constructs the original local cascade with its sharp Sobolev maps, complete local evolution and an exact energy-transfer pump. Five solved exercises prove real and complex averaging composition, solve every initial state on the pump plane, compute the full diffusion defect and construct its infinite-dimensional invariant space with a global viscous evolution and complete physical force.

The thirty-ninth proves the complete original local-cascade averaging theorem, including all tensor coefficients, the entire rotation fiber, full Haar and surface measures, a constructed smooth extension, every Fourier multiplier and the complete scale sum. Five solved exercises give exact closing moments, the minimum tensor kernel, the full retraction derivative, a sharp bounded-weight classification for the original pump and quantitative stability of its actual viscous flows.

The fortieth constructs the actual original-viscosity cascade on its maximal interval, with complete band equations, every boundary energy term, the full orthogonal diffusion defect and an exact positive memory kernel. Five solved exercises give the entire heat response and resolvent, a stronger annular-width estimate, the signed scale-energy current and the derivative of the actual flow with respect to viscosity.

The forty-first constructs every original quadratic gate and the complete scheduled dissipative transfer, including its sharp exponent endpoint. Five solved exercises prove the exact gate correspondence, a stronger transfer estimate, the unique damped-output peak, global energy-class continuation and exact terminal regularity. Critical weighted square-sum divergence holds throughout the final stages, with its full physical-time rate.

The forty-second proves the complete original autonomous delayed transfer, with visible product-sign, energy-sign, interval and coefficient-table repairs. Five solved exercises classify every quadratic polynomial integral, retain all amplitude and clock factors and the full inter-scale pump, construct a neighborhood of persistent finite transfer, and give a much sharper trigger-time interval.

The forty-third proves strict positive viscous activation at every scale, exact physical history maps and complete receiver bounds. Five solved exercises retain the full activation-coefficient product, quantify the positive memory series, construct the exact moment maps, prove the strictly positive orthogonal heat remainder and strengthen all receiver bounds with their original viscous decay. All six viscous supporting notes are included; their complete uniform induction proves the original averaged-equation finite endpoint.

The forty-fourth proves the full original viscous rotation corrector and actual first transfer, including every intermediate forcing interval, all upper bands and the corrected receiving-drive range. Five solved exercises determine exact energy positivity, strengthen the complete velocity transfer, retain all physical Sobolev factors, solve the weighted-tail comparison and compute the exact scalar forcing asymptotic.

The forty-fifth proves the full original-viscosity uniform cascade and its finite maximal endpoint, retaining every physical history, incoming pump, weighted tail and corrected checkpoint. Five solved exercises strengthen amplitude and lifetime bounds, prove Sobolev growth throughout every interval, retain the complete band sums, and establish matching critical growth and its exact time-integrability threshold.

The forty-sixth derives the complete original exterior and interior cylindrical profile maps, including dynamic clocks, signed swirl, curvature, the regular axis, moving period, source elliptic operator and periodic pressure. Five solved exercises retain the original time endpoint, characterize the axial pressure obstruction, prove the exact flux-energy decomposition, construct a full smooth swirl example and strengthen the weighted stream estimate using both boundary values.

The forty-seventh reconstructs the original fields from all six profile residuals, proves the complete sinh-coordinate and sampling-measure maps, and calculates explicit finite-network derivative bounds. Five solved exercises retain the physical pressure and time, test incompatible predictions, compare exact measures, specify a six-layer network and strengthen grid control using all pure second residual derivatives and both quadratic cell-width contributions.

The forty-eighth proves the original anchored Burgers branch, its complete regularity and physical characteristic map. Explicit smooth approximants show the exact scope of residual-derivative losses, with full parameter bounds for the finite network. Five solved exercises give the cubic inverse, original characteristics, all derivative contributions, matching residual-decay coefficients and the losses of a completely specified network.

The forty-ninth proves the original CCF mode defects, signed symmetries and full Hilbert-transform error bounds. Five solved exercises calculate the anchor maps, a specified velocity and both mode defects, the sharp joint error coefficient, every original nonuniform-grid weight and a complete transform budget with its global cutoff terms.

The fiftieth proves the original physical clocks, full weighted transport and exact analytic corrections in the Chen–Hou construction. The complete frozen model identifies the precise derivative threshold and the different behavior of the full Sobolev norm. Five solved exercises calculate a logarithmic clock, a complete annular weight, a nonzero correction tail, every Gaussian derivative norm and the full three-component correction with its gradient defect.

The fifty-first proves the full original two-point transport and reflected-domain correction, the singular-cell repair budget and a rigorous enclosure of the horizontal constant below2.549. Five solved exercises retain both coupled components, an actual mesh region, the rectangle edge in a transport partner, the cell-to-global comparison and four exact cutoff-array lookups with their receiving costs.

The fifty-second certifies all3896 original horizontal upper entries, proves the sharp anisotropy map and reconstructs the full physical K2 transport formulas. It evaluates both singular constants and the horizontal residual exactly, restores the edge-kernel factor and strengthens the height-dependent edge bound. Five solved exercises carry these results through their original physical receiving coefficients.

The fifty-third proves the complete outer and residual-edge K2 bounds, exact horizontal partner maps and the full vertical velocity increments. Every cutoff band, local sign and residual mass remains. A stronger middle bound recovers the source estimate after explicit corrections to intermediate formulas. Five solved exercises quantify measure concentration, test exact physical velocity differences and strengthen whole parameter cells.

The fifty-fourth certifies both complete saved vertical derivative bounds, retaining every original cost and singular endpoint. It identifies and repairs a local extra-area error, reconstructs the weighted velocity integral and proves the actual-exponent selector despite its false all-exponent generalization. Five solved exercises integrate the full residual, quantify midpoint improvement, store an exact correction and enlarge the actual-exponent cell.

Fifty-four complete lesson drafts; the series is in preparation. Self-checked by GPT-6 Astra at Ultra.

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The forced Navier–Stokes blowup argument

The complete forced blowup construction, from profiles, stress and moment maps through the actual correction fields, compact smooth forcing, bounded energy, velocity growth and periodization at every positive viscosity. Three computation lessons prove eventually stationary forcing, rapidly decreasing forcing and direct velocity detection, with exact instruction maps, scalar estimates, limiting flows and fixed detectors. The cumulative reader contains twenty-three lessons and 115 solved exercises, with mathematical proof notes, editable sources, figures and an offline reader.

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The Jacobian conjecture and affine spaces

Polynomial maps and affine spaces: Keller maps and the reduction of the Jacobian conjecture to injectivity, with a complete hand proof that an explicit map of three-space with Jacobian determinant -2 identifies three points; coordinates and a degree-17 polynomial in four variables whose zero set is affine three-space but which is not a coordinate; locally nilpotent derivations, additive actions and their lifts to line bundles; a fourfold whose cylinder is affine five-space but which is not affine four-space, so that Zariski cancellation fails in dimension four in characteristic zero; a polynomial in five variables that becomes a coordinate after adjoining one variable, with explicit complementary coordinates, but is not a coordinate; a degree-five stable coordinate in four variables that is not a coordinate; and smooth affine-space fibrations over the line, the plane and a discrete valuation ring, all of whose fibres are affine three-space, which are not locally trivial.

Ten lessons · proofs and exercises · self-checked by Claude Opus 5.5

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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.

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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

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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

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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

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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

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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

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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.

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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

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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

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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

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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.

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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

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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

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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

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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.

Twelve lessons with examples, sixty worked exercise solutions and editable sources. Several general theorems remain conditional on the prerequisite proofs stated in the lessons. AI-generated exposition by GPT-6.1 Sol and GPT-6 Astra (OpenAI), October 2026.

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Connections, curvature and holonomy

No lessons are currently available for this course.

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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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

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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

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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

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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.

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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

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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.

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Distributions, kernels and analytic singularities

One hundred and forty-four lessons on distributions, support geometry, Fourier growth, derivative-growth classes, wavefront localization, analytic substitution, holomorphic representations, conic decompositions, prescribed wavefront sets, holomorphic wedges, homogeneous Fourier correspondence, analytic pullback, general kernel operations and analytic support boundaries, characteristic normals, analytic differential regularity, support uniqueness, exact characteristic-half-space solutions, convex-domain determination, support criteria in half-spaces and proper cones, analytic parametrices with entire errors and characteristic propagation at every derivative-growth scale, prescribed global singular rays with smooth forcing, analytic convolution regularity for compact inputs and proper support sums, microhyperbolic cones with boundary inverses and exact convex wavefront bounds, exact homogeneous waves on prescribed light rays, analytic functionals with real-carrier approximation, local holomorphic tests, Fourier–Laplace transforms, entire harmonic lifts, canonical Poisson fields and their full converse, decay normalization, smallest real carriers, finite carrier splitting with exact overlap ambiguity, completeness modulo smaller carriers, hyperfunctions on arbitrary open sets and analytic manifolds, arbitrary-cover gluing and closure-supported extension, growth-free holomorphic boundary values, analytic wavefronts of hyperfunctions, the wedge converse, joining opposite holomorphic sides, finite closed-cone splitting and exact holomorphic compatibility through sum cones, noncharacteristic hyperfunction substitution with coordinate and distribution agreement, full tensor wavefront bounds, admissible products, chain and Leibniz rules and finite products under joint noncancellation, proper hyperfunction integration, its analytic wavefront bound, fiber coordinate changes and analytic-input kernels, singular-input kernel action, contraction through zero intermediate covectors and proper convolution with full associativity, analytic-weight fiber detection, both hyperfunction support-normal signs, rough and convex boundary contact and growth-free open-set uniqueness, noncharacteristic analytic substitution through proper integrals, singular projection formulas and finite kernel associativity, quantitative analytic Cauchy construction on polydiscs and arbitrary analytic normal data, analytic hypersurface Cauchy germs, noncharacteristic boundary continuation, global convex-domain existence and complex-cone solvability, full noncharacteristic hyperfunction regularity, microlocal existence, directional quotient inverses, two-sided noncharacteristic traces and Holmgren uniqueness, supported hyperfunction division with exact support, canonical one-sided Cauchy data, exact boundary-source matrices, full distributional endpoint traces and analytic coordinate compatibility, Gaussian carrier bounds, recovery of arbitrary compact hyperfunctions and the exact Gaussian criterion for analytic regularity, with prerequisite proofs, worked exercises and an offline reader with editable sources.

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Gaussian rescaling, both tangent directions of fiber normals, closure after projection, convex fiber propagation and analytic boundary regularity at all normal frequencies are accompanied by explicit smooth-data counterexamples.

The Fourier–Laplace range for compact convex real carriers is proved using rotated Laplace rays, moment recovery and smallest real carriers, with explicit nonconvex obstructions and infinite point derivatives.

Global derivative-growth representatives modulo analytic functions are constructed by Gaussian center localization, yielding the global wavefront intersection formula and simultaneous representation of finitely many regular germs.

Smooth functions are extended across a boundary without loss of finite differentiability, then expressed through square coordinates and exchange invariants; positive majorants retain every flat boundary derivative.

Sharp distributional extension orders include the strict oscillatory endpoint, and an explicit real phase gives an order-one extension for any continuously differentiable amplitude.

Concentration and oscillatory limits include periodic point masses, a radial circle and center contribution, two-parameter boundary poles, and locally uniform convergence under gradient bounds.

Products and singular limits include coordinate principal values, quartic finite parts, logarithmic axis crossings, and exact overlap laws for moving Cauchy kernels.

Endpoint extension includes interval primitive estimates, finite-measure representations, weighted Taylor subtractions and the sharp order threshold for exponential phases.

Reciprocal compression, finite-difference moment cancellation and smooth coefficients determine the full principal-value and point-supported terms of boundary poles.

Reflected endpoint formulas, smooth division on intervals, point-jet kernels and the obstruction at a flat zero determine the solutions and their distributional order.

Degenerate differential equations admit explicit flat branches, finite point-jet inverses and locally finite families of component constants and concentrated terms.

Coordinate-power kernels, simultaneous polynomial constraints and directional uniqueness describe supported distributions; parabolic boundary poles have exact order and singular support.

Sublevel geometry determines concentration masses, integrable surface logarithms define cut boundary values, and moving logarithmic sources converge to circular arclength.

Boundary variation determines derivative order, quadratic division includes accumulating zeros, and exact moving-interface and logarithmic branch formulas describe distributional jumps.

Finite-order complex-phase estimates combine cancellation and damping with exact weights, including integrable degeneracy at critical points.

Finite stationary expansions retain precise phase and amplitude regularity, explicit differential coefficients and complex Gaussian branches; nilpotent quadratic exchange has an exact finite remainder and a sharp loss.

Paired oscillatory variables have an explicit critical point and a mixed-derivative expansion with a precise amplitude budget, factorial coefficients and sharp rounded decay orders.

Boundary interval bounds propagate by a positive harmonic envelope on connected upper domains, including domains with holes; the full asymptotic order follows from deeper truncation.

Smooth complex critical graphs determine explicit stationary coefficients with the oriented Jacobian, a fixed Gaussian branch and an absolute remainder; the proofs allow null imaginary-Hessian directions.

Curved surfaces have uniform Fourier asymptotics; the two normal phases of a convex body and an exact smoothing sandwich yield a quantitative lattice-count bound.

Boundary stationary phase has exact normal endpoint factors, complete half-integer critical expansions and a Fresnel transition across the boundary.

Finite-order phases have exact gamma coefficients, missing powers and nonlinear coordinate densities, together with two uniform canonical Airy profiles.

Cubic collisions have smooth normal coordinates, two complete amplitude sequences and uniform Airy expansions, including nondegenerate transverse directions.

Complex phases give oscillatory distributions with full symbol-range order bounds, smoothness off stationary directions and exactly normalized causal wave kernels.

Fractional difference norms give exact energy constants, limiting identities and sharp weighted Fourier embeddings.

Annular Sobolev multiplier bounds yield finite-exponent and Hölder elliptic estimates, with an explicit obstruction at the continuous endpoint.

Complex homogeneous oscillatory phases have a precise stationary covector bound and a quadratic damping example with one singular direction.

Compact frequency projections and exact conormal directions of smooth weighted densities include flat zeros and both normal signs.

Transverse smooth densities have exact restriction and product formulas, including the intersection angle factor and the support of restricted weights.

Quadratic boundary powers and logarithms have exact point-source constants and complete wavefront sets, including signature-dependent vertex directions and all forward wave powers.

Holomorphic boundary values persist along arbitrary smooth complex approaches to real submersions, with uniform finite-test estimates and explicit regular and singular critical-level examples.

Every closed conic smooth wavefront set has a realization with any prescribed finite differentiability; sign symmetry characterizes real-valued realizations, and compact singular base sets permit compact support.

Local holomorphic wedges with polynomial bounds on each smaller approach cone have a common boundary and the exact dual-cone restriction; smooth and analytic remainders have distinct effects.

Real principal type polynomials admit a uniform estimate for arbitrary additional imaginary displacements, including the affine case and explicit nonlinear examples.

Summable smoothing gives quantitative cutoffs in arbitrary norms and fixed Fourier tests for nonquasianalytic classes, with explicit Gevrey examples.

The full Denjoy–Carleman criterion relates derivative budgets to unique infinite jets and compact cutoffs, with irregular weights and sharp numerical examples.

Quadratic boundary distributions and causal wave powers have exact analytic and derivative-growth wavefronts, including every signature and the full point-source vertex fiber.

Tangential coefficients and transverse deltas have exact wavefront formulas. Analytic densities and transverse intersections include both normal signs at zeros within their coefficient support.

Regular simple arcs are minimal complex carriers. Polynomial approximation, exact endpoint tests and holomorphic weights distinguish minimal carriers from regular path images containing loops.

Complete offline reader and editable sources.

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.

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Constant-coefficient equations and solvability

Polynomial symbols, support geometry, regularity, Cauchy and boundary problems, supported solvability, convolution equations and Banach analytic methods.

English · 214 chapters · proofs, worked examples and solved exercises · supporting mathematical readings · editable Markdown and reproducible illustrations. Each chapter states its hypotheses and prerequisites.

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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

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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

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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

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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.

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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.

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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

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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.

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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 lessons with full arguments, examples, 48 solved exercises and editable sources. Exact prerequisite readings and source credits accompany the course.

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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.

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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.

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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

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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.

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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

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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

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Flat, smooth and étale morphisms

Flatness, local criteria and dimension; unramified, étale and smooth morphisms; infinitesimal lifting; henselization and étale localization.

English · 7 lessons

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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 · 12 lessons and a supporting reading

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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

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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

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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

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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

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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.

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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

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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

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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

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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

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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

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Operator-algebraic statistical mechanics and the Bost–Connes system

Arithmetic equilibrium states, Hecke algebras, thermal time and current observables.

English · 1 lesson

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Spectral triples and the local index formula

Logarithmic operator traces, regular spectral triples, residue index cocycles, and hypoelliptic transverse geometry.

English · 3 lessons

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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

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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

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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

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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

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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

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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

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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

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Microlocal sheaves

A research course developing conic sheaves and kernel operations into specialization, microlocal morphisms and microsupport geometry.

English · 5 lessons

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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.

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