Open Courses — foundations and further study

These are source-declared reading routes, not a certificate that every prerequisite proof has been checked. Advanced lessons are currently in English. The foundation interface offers English and Bahasa Indonesia with actual material languages marked.

40 Foundation courses · 72 Further courses

Advanced catalogue update: OpenAI Codex — GPT-6 Astra, Ultra effort. This snapshot includes 72 public courses and 1,061 lessons; course routes are not proof certification. Local foundation drafts remain separate.

Foundation courses

Prealgebra and Quantitative Foundations

Programme home — Prealgebra and Quantitative Foundations

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No further course is declared in this snapshot.

Elementary Algebra

Programme home — Elementary Algebra

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No further course is declared in this snapshot.

Intermediate Algebra

Programme home — Intermediate Algebra

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No further course is declared in this snapshot.

Precalculus and Trigonometry

Programme home — Precalculus and Trigonometry

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No further course is declared in this snapshot.

Proof, Logic, and Discrete Structures

Programme home — Proof, Logic, and Discrete Structures

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No further course is declared in this snapshot.

Differential Calculus

Programme home — Differential Calculus

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No further course is declared in this snapshot.

Integral Calculus

Programme home — Integral Calculus

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No further course is declared in this snapshot.

Linear Algebra

Read original English — 34 sections through Complex Vector Spaces (partial book)

Read the English foundation sections — systems, vector spaces and bases (partial book)

Programme home — Linear Algebra

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Multivariable Calculus

Programme home — Multivariable Calculus

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Vector Calculus

Programme home — Vector Calculus

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No further course is declared in this snapshot.

Ordinary Differential Equations and Dynamical Systems

Programme home — Ordinary Differential Equations and Dynamical Systems

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Mathematical Computing and Reproducible Experiments

Programme home — Mathematical Computing and Reproducible Experiments

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No further course is declared in this snapshot.

Calculus-Based Probability

Programme home — Calculus-Based Probability

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No further course is declared in this snapshot.

Applied Statistics and Data Analysis

Programme home — Applied Statistics and Data Analysis

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No further course is declared in this snapshot.

Real Analysis I

Programme home — Real Analysis I

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Real Analysis II

Programme home — Real Analysis II

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Abstract Algebra I

Programme home — Abstract Algebra I

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Abstract Algebra II

Programme home — Abstract Algebra II

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Complex Analysis

Programme home — Complex Analysis

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Number Theory and Cryptology

Programme home — Number Theory and Cryptology

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Applied Combinatorics

Programme home — Applied Combinatorics

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No further course is declared in this snapshot.

Mathematical Logic, Set Theory, and Computability

Programme home — Mathematical Logic, Set Theory, and Computability

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No further course is declared in this snapshot.

Point-Set Topology

Programme home — Point-Set Topology

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Euclidean and Non-Euclidean Geometry

Programme home — Euclidean and Non-Euclidean Geometry

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No further course is declared in this snapshot.

Numerical Analysis

Programme home — Numerical Analysis

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No further course is declared in this snapshot.

Mathematical Modeling and Nonlinear Dynamics

Programme home — Mathematical Modeling and Nonlinear Dynamics

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No further course is declared in this snapshot.

Linear and Integer Optimization

Programme home — Linear and Integer Optimization

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No further course is declared in this snapshot.

Mathematical Statistics

Programme home — Mathematical Statistics

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No further course is declared in this snapshot.

Measure and Integration

Programme home — Measure and Integration

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Functional Analysis

Programme home — Functional Analysis

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Measure-Theoretic Probability and Stochastic Processes

Programme home — Measure-Theoretic Probability and Stochastic Processes

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No further course is declared in this snapshot.

Partial Differential Equations

Programme home — Partial Differential Equations

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Smooth Manifolds and Differential Geometry

Programme home — Smooth Manifolds and Differential Geometry

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Algebraic Topology

Programme home — Algebraic Topology

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Graduate Algebra

Programme home — Graduate Algebra

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Category Theory and Homological Methods

Programme home — Category Theory and Homological Methods

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Advanced Optimization and Convex Analysis

Programme home — Advanced Optimization and Convex Analysis

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No further course is declared in this snapshot.

Algebraic Geometry Bridge

Programme home — Algebraic Geometry Bridge

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Formalized Mathematics in Lean

Programme home — Formalized Mathematics in Lean

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No further course is declared in this snapshot.

Research Reading and Reproducible Mathematical Work

Programme home — Research Reading and Reproducible Mathematical Work

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No further course is declared in this snapshot.

Further courses

Harmonic analysis on locally compact abelian groups

English · draft · 17 Lessons

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Lessons
  1. Fourier analysis on finite abelian groups
  2. Characters and the dual group
  3. The dual group as the Gelfand spectrum of L^1(G)
  4. Functions of positive type
  5. Raikov's theorem and the Gelfand–Raikov theorem
  6. Bochner's theorem
  7. The Fourier inversion theorem and the dual Haar measure
  8. The Plancherel theorem
  9. The Pontryagin duality theorem
  10. Subgroups, quotients and annihilators
  11. The Poisson summation formula
  12. The structure of locally compact abelian groups
  13. Unitary representations of abelian groups: the spectral theorem
  14. Closed ideals of L^1(G) and Wiener's theorem
  15. Tauberian theorems and spectral synthesis
  16. Almost periodic functions and the Bohr compactification
  17. Totally disconnected groups, the p-adic numbers and the adèles
Sources and rights

GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Number fields

English · draft · 17 Lessons

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Lessons
  1. Algebraic integers and rings of integers
  2. Discriminants and integral bases
  3. Discrete valuation rings and Dedekind domains
  4. Norms of ideals, the ideal class group, and modules over Dedekind domains
  5. Decomposition of primes in extensions
  6. Hilbert's ramification theory in Galois extensions
  7. Lattices, Minkowski's theorem and the Minkowski embedding
  8. Finiteness of the class number
  9. Dirichlet's unit theorem
  10. Quadratic fields: ideal classes and binary quadratic forms
  11. Orders in number fields and their Picard groups
  12. Cyclotomic fields
  13. Units and class numbers of cyclotomic fields; Kummer's theorem for regular primes
  14. The different and the discriminant
  15. Counting ideals of bounded norm
  16. The Dedekind zeta function and the analytic class number formula
  17. Abelian number fields and Dirichlet L-functions at s = 1
Sources and rights

GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

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Flat, smooth and étale morphisms

English · draft · 7 Lessons

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Lessons
  1. Flat morphisms
  2. Flatness criteria, dimension and the flat locus
  3. Unramified morphisms
  4. Étale morphisms and their local structure
  5. Smooth morphisms
  6. Infinitesimal lifting and the invariance of étale morphisms under thickenings
  7. Étale neighbourhoods, henselization and quasi-finite morphisms
Sources and rights

GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Foundations of von Neumann algebras

English · draft · 20 Lessons

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Lessons
  1. Hahn–Banach, Baire and the basic theorems on Banach spaces
  2. Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian
  3. Hilbert spaces and compact operators
  4. Cauchy's theorem for cycles and its consequences
  5. Banach algebras, spectrum, holomorphic functional calculus and Gelfand theory
  6. C*-algebras: continuous functional calculus, automatic continuity, positive cones, approximate identities and quotients
  7. The spectral theorem for bounded self-adjoint operators
  8. Representations and positive functionals: the GNS construction and the Gelfand–Naimark theorem
  9. AF-algebras
  10. Compact and trace-class operators, the predual of B(H), and the operator topologies
  11. The double commutant theorem
  12. Kaplansky's density theorem and its consequences
  13. Abelian operator algebras
  14. The universal enveloping von Neumann algebra of a C*-algebra, and W*-algebras
  15. Projections and types of von Neumann algebras
  16. Polar decomposition of functionals and weak compactness in preduals
  17. Completely positive maps
  18. Integral representations of states
  19. Polish spaces and standard Borel spaces
  20. Numerical ranges, positive matrices and co-Souslin sets
Sources and rights

Claude Opus 5.5 (Anthropic); CC0-1.0

What this prepares you for

Measurable fields and direct integrals

English · draft · 5 Lessons

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Lessons
  1. Measurable fields of Hilbert spaces and their direct integrals
  2. Decomposable operators and the diagonal algebra
  3. Vector-valued functions, tensor products with L^p, and preduals
  4. The Effros Borel structure
  5. Direct integrals of von Neumann algebras
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Claude Opus 5.5 (Anthropic); CC0-1.0

What this prepares you for

Tensor products of operator algebras

English · draft · 2 Lessons

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Lessons
  1. Spatial tensor products of von Neumann algebras
  2. Tensor products of Banach and Hilbert spaces, Jordan homomorphisms and isometries of C*-algebras
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Claude Opus 5.5 (Anthropic); CC0-1.0

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Modular theory and weights

English · draft · 96 Lessons

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Lessons
  1. Curriculum and full result contracts
  2. Finite domains and representations of general weights
  3. Weights and semicyclic representations: the exact opening package
  4. Comparing weights through finite-energy vectors
  5. A bounded-operator kernel for weight arguments
  6. Detecting normal weights by finite observations
  7. Finite domains, null directions, and support corners
  8. Concrete preduals from Hilbert tensors
  9. The predual-valued map of a normal weight
  10. Countable amplification and ultraweak bicommutant closure
  11. Normal positive maps and their preadjoints
  12. The vector retained by a comparison map
  13. A faithful weight on the GNS commutant
  14. Spectral calculus with its domains retained
  15. Recovering operators from energy forms
  16. Closing energy domains and comparing resolvents
  17. Compatible pairs and complex interpolation
  18. Closing an involution and recovering its modular data
  19. Building the two multiplication actions of a Hilbert algebra
  20. Haar convolution as left and right Hilbert algebras
  21. Polynomial cutoffs recover an operator's range
  22. Right Hilbert algebra density and commutants
  23. The full left Hilbert algebra obtained by dualizing twice
  24. Weights and the Hilbert spaces of multiplication
  25. Changing algebra coordinates and scaling a weight
  26. Analytic kernels for unbounded modular operators
  27. Approximating multiplication without losing its bound
  28. The modular group and its analytic algebra
  29. The positive cone of a standard representation
  30. Recovering a representation from its positive cone
  31. Contractive retractions and conditional expectations
  32. Spatial comparison on an arbitrary representation
  33. Constructing spatial energy from finite observations
  34. Adding spatial energies and changing the reference weight
  35. Spatial energy and modular time
  36. The KMS boundary condition determines the modular group
  37. Operators recovered from small trace defects
  38. Reconstructing a weight from a modular cocycle
  39. Recovering a weight from spatial energy
  40. Fixed elements and changes of density
  41. Trace densities and noncommutative integration
  42. Tracial multiplication actions and the preserving expectation
  43. Spectral layers in a semifinite trace algebra
  44. Changing the reference of a weight cocycle
  45. Forced C*-unitization
  46. Finite weight domains and GNS
  47. Bounded functionals and cyclic vectors
  48. States and the positive spanning family
  49. Weak compactness and second adjoints
  50. Banach holomorphy, generators, and resolvent limits
  51. Finite corners, faithful traces, and affine group fixed points
  52. The universal C*-bidual
  53. Lower semicontinuous C*-weights
  54. From a C*-modular condition to the GNS von Neumann algebra
  55. Supported GNS representations and the balanced cocycle
  56. Countable modular sums and infinite-weight boundaries
  57. Joint spectral measures and commutator estimates in standard form
  58. Measurable Hilbert fields and their diagonal commutant
  59. Central decomposition from countable operator equations
  60. Regular representations, Fourier algebras, and Borel group measures
  61. Compatible traces and the coupling operator
  62. Canonical L2 and standard implementations
  63. Relative tensor products and fusion
  64. Tensor operators and tensor weights
  65. Conditional expectations from modular invariance
  66. Corner weights tested on all positive energies
  67. Recognizing invariant weights by their densities
  68. Inner modular flow and rigidity of finite weight data
  69. Weight domination and exact half-strip endpoints
  70. Analytic generators and exact finite weight domains
  71. Commuting modular flows and relative half-strip multipliers
  72. Central cocycles and common finite domains
  73. Normal-functional polar decomposition and invariant pairs
  74. Ergodic factors and trace normalization
  75. Semifinite corners and bounded modular operators
  76. Modular partitions and the full finite domain
  77. Compact-time continuity along automorphism orbits
  78. Norm continuity of relative modular time
  79. A supported inverse weight cocycle
  80. Partial cocycles and their exact mixed KMS domains
  81. Comparing supported weights and transporting their cuts
  82. Uniform geometry of faithful weights
  83. When countable sums detect extended positive energy
  84. Finite calculus and composition of operator-valued weights
  85. Detecting and determining operator-valued weights
  86. Modular invariants of operator-valued weights
  87. Compatible modular weights and operator-valued reconstruction
  88. Homogeneous operators in the continuous core
  89. Commutant duality for operator-valued weights
  90. Six laboratories on expectations and operator-valued averages
  91. Compatible measurable GNS fields and both representation transports
  92. Measurable weight fields and their integrals
  93. Separable GNS spaces and disintegration of C*-weights
  94. The realization boundary in measurable weight fields
  95. Measurable algebra cores and exact selection
  96. Direct integrals of Hilbert and Tomita algebras
Sources and rights

GPT-6 Astra and GPT-6 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Hilbert algebras

English · draft · 1 Lessons

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Lessons
  1. Left and right Hilbert algebras
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Claude Opus 5.5 (Anthropic); CC0-1.0

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Analytic elements, strips and KMS

English · draft · 2 Lessons

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Lessons
  1. Holomorphy in Banach spaces, Stone's theorem and resolvent convergence
  2. Analytic elements and strip arguments
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Claude Opus 5.5 (Anthropic); CC0-1.0

What this prepares you for

Traces and noncommutative integration

English · draft · 4 Lessons

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Lessons
  1. Traces on von Neumann algebras
  2. Integration for a trace, the commutation theorem, and applications
  3. Measurable operators for a trace: examples, convergence and the commutant
  4. Multiplicity of a von Neumann algebra on a Hilbert space
Sources and rights

Claude Opus 5.5 (Anthropic); CC0-1.0

What this prepares you for

Crossed products and the flow of weights

English · draft · 35 Lessons

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Lessons
  1. Entry contracts and mathematical conventions
  2. Transporting crossed products through action cocycles
  3. Two crossed products and the surviving action
  4. Building an intrinsic flow from modular coordinates
  5. Central cocycles as obstructions to fixed implementers
  6. Kernels, local fixed parts, and freeness
  7. Closing a domain after spectral localization
  8. Integrating covariance with left Haar measure
  9. A coefficient domain for the dual GNS construction
  10. Moving an action and a modular parameter together
  11. Continuity of actions: scalar tests, preduals, and bounded nets
  12. Modular time on a crossed-product coefficient algebra
  13. Extending the dual weight beyond the common involution domain
  14. How the dual weight moves crossed-product generators
  15. Changing the Hilbert space of a regular crossed product
  16. Recovering multiplicity from a Weyl system
  17. Locating frequencies with Fourier filters
  18. Detecting semifinite corners through central translation
  19. Recovering a spectral coordinate from a scaled trace
  20. Comparing dual weights through matrix corners
  21. The coefficient algebra as the value space of a weight
  22. Averaging a dual action and recovering its coefficients
  23. Commuting with the original algebra inside its core
  24. Recovering a group action from its integrated operators
  25. Recovering covariance with nonunital coefficients
  26. Realizing central cocycles by spectral averaging
  27. Recovering a weight from dual invariance
  28. Removing noncentral cocycles from a trace-scaling action
  29. Recovering a crossed product from its eigenunitaries
  30. Recognizing the continuous core of a fixed algebra
  31. Haar coordinates and integration after recognition
  32. Dual weights and their supports
  33. The weight after crossing twice
  34. Derivatives of actions: domains, perturbations, and implementation
  35. Innerness and spectral tails
Sources and rights

GPT-6 Astra and GPT-6 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Type III factors: central sequences, full factors and almost periodic states

English · draft · 4 Lessons

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Lessons
  1. Ultraproducts and the asymptotic centralizer
  2. Full factors
  3. Almost periodic weights and the invariant Sd
  4. Full factors without almost periodic weights
Sources and rights

Claude Opus 5.5 (Anthropic); CC0-1.0

What this prepares you for

Injective factors

English · draft · 6 Lessons

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Lessons
  1. Trace inequalities for finite von Neumann algebras
  2. Property Γ and the algebra generated by a factor and its commutant
  3. Approximately inner and centrally trivial automorphisms
  4. Injective von Neumann algebras
  5. Uniqueness of the injective II₁ factor
  6. Classification of injective factors
Sources and rights

Claude Opus 5.5 (Anthropic); CC0-1.0

What this prepares you for

Entropy of automorphisms

English · draft · 5 Lessons

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Lessons
  1. Entropy of finite-dimensional subalgebras
  2. The entropy of a trace-preserving automorphism
  3. Entropy defect and abelian models
  4. Dynamical entropy of C*-algebras and von Neumann algebras
  5. Entropy of lattice translations in quantum spin chains
Sources and rights

Claude Opus 5.5 (Anthropic); CC0-1.0

What this prepares you for

Noncommutative integration and spatial theory

English · draft · 4 Lessons

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Lessons
  1. The spatial derivative
  2. Measured groupoids and transverse measures
  3. Square-integrable representations and random operators
  4. Weights on random operators and formal dimension
Sources and rights

Claude Opus 5.5 (Anthropic); CC0-1.0

What this prepares you for

Cyclic cohomology, connections and transverse geometry

English · draft · 5 Lessons

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Lessons
  1. Cyclic cohomology: traces, differentials and symmetry
  2. Connections and curvature from symmetries of an algebra
  3. Frequency calculus for an action of Euclidean space
  4. Cyclic forms that survive norm completion
  5. A fundamental class for an action on a manifold
Sources and rights

GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Spectral triples and the local index formula

English · draft · 4 Lessons

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Lessons
  1. Singular values and the Dixmier trace
  2. Spectral triples and dimension spectrum
  3. The local index formula
  4. Symbol calculus, transverse geometry and geometric index formulas
Sources and rights

GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Function algebras and approximation

English · draft · 1 Lessons

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Lessons
  1. The Stone–Weierstrass theorem for functions vanishing at infinity
Sources and rights

Claude Opus 5.5 (Anthropic); CC0-1.0

What this prepares you for

Harmonic analysis on locally compact groups

English · draft · 4 Lessons

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Lessons
  1. Measure and Hilbert space tools for Haar integration
  2. Haar measure on locally compact groups
  3. Quotient measures and Weil's integration formula
  4. Finite covolume and arithmetic quotients
Sources and rights

GPT-6.1 Sol (OpenAI), Ultra; retained earlier text credited in the lessons; CC0-1.0

What this prepares you for

Elliptic operators and boundary problems

English · draft · 36 Lessons

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Lessons
  1. Exact prerequisite interfaces for quadratic multiplier estimates
  2. Polynomial and contour interfaces for stable boundary models
  3. Stable modes and the algebra of boundary data
  4. Local inverses and distance-weighted elliptic estimates
  5. Support contracts for moving-metric localization
  6. Localizing symbols when the measuring scale moves
  7. Quantitative estimates for quadratic Fourier multipliers
  8. Two measuring scales, one Weyl product
  9. From symbol estimates to operators on every Sobolev scale
  10. Singularities along a submanifold and smooth boundary passage
  11. From Weyl symbols to operators and changes of coordinates
  12. Detecting regularity without choosing coordinates
  13. Curved weights and the directions in which support can end
  14. Detecting a solution from infinite-order silence at one point
  15. Banach estimates, quotient spaces and compact parameter arguments
  16. Finite defects under perturbation
  17. Traces that survive passage to cohomology
  18. Positivity through a moving family of scalar probes
  19. From local energy to global divergence equations
  20. Positive energy and vanishing at the far end of space
  21. Boundary energy, local inverses, and harmonic data
  22. When a moving symbol scale controls an operator
  23. When a nonnegative scalar symbol acquires a negative part
  24. Building a local inverse from radial singularities
  25. Causal kernels, initial data, and short-time geometry
  26. A wave kernel that cancels at a curved boundary
  27. Symbols, finite defects, and the index on a closed manifold
  28. Cauchy data from jumps and residues
  29. Solving an elliptic system from compatible boundary measurements
  30. Spectral measures with the original operator domain retained
  31. Dirichlet realizations, spectral projectors, and local extensions
  32. Changing an interior frame to extend an invertible matrix
  33. Inverting mixed symbols without commuting matrix factors
  34. Composition of mixed symbols with two different remainder estimates
  35. Mixed symbols on every real two-parameter Sobolev scale
  36. Polynomial inverse expansion with ordered matrix coefficients
Sources and rights

GPT-6 Astra and GPT-6 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Totally characteristic operators

English · draft · 1 Lessons

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Lessons
  1. Totally characteristic operators on the half space
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Claude Opus 5.5 (Anthropic); CC0-1.0

What this prepares you for

Index theory of elliptic operators

English · draft · 1 Lessons

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Lessons
  1. The Bott operator, suspension, and reduction of the index to Euclidean space
Sources and rights

Claude Opus 5.5 (Anthropic); CC0-1.0

What this prepares you for

Derived categories and sheaf operations

English · draft · 12 Lessons

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Lessons
  1. Sheaves of modules on a ringed space
  2. Complexes, cones and localization
  3. Injective modules, flasque sheaves and bounded-below derived functors
  4. K-injective resolutions in Grothendieck abelian categories
  5. Flat modules and K-flat resolutions
  6. The derived tensor product and Tor sheaves
  7. Derived pullback and pushforward
  8. Hom complexes, internal derived Hom and Ext sheaves
  9. Sections with support and the localization triangle
  10. Perfect complexes and duals on a ringed space
  11. The projection formula and the base change map
  12. Sheaves of modules and their derived categories
Sources and rights

GPT-6.1 Sol (OpenAI), in Codex at Ultra; retained contributions by Claude Opus 5.5 and GPT-6 Astra at Ultra; GFDL-1.2-or-later

What this prepares you for

Microlocal sheaves

English · draft · 56 Lessons

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Lessons
  1. How to use this course draft
  2. Open prerequisites and exact import contracts
  3. Supporting verifications for open prerequisites
  4. Open extensions and ambient supports
  5. Proper supports and the bounded classical comparison
  6. Proper-support composition through a change of base
  7. Extending sections on convex sets
  8. Recovering sheaf cohomology from closed pieces
  9. Continuing cohomology through a moving boundary
  10. Exceptional inverse image from a finite resolution
  11. Finite resolutions without a lower bound
  12. Transport along a scaling action
  13. Local orientations, dimension, and integration
  14. Composing sheaf operators through an intermediate space
  15. Inverse operators from complementary hemisphere boundaries
  16. Fourier kernels as radial averaging
  17. Fourier duality through a test complex on the base
  18. Duality when the dual leaves the original transform category
  19. Comparing traces after Fourier transformation
  20. Moving Fourier kernels across maps and products
  21. A linear map inside the Fourier comparison
  22. The graded line in a Fourier support comparison
  23. Transposing the complete Fourier trace comparison
  24. The geometric normalization of Fourier adjunctions
  25. Following the comparison maps into the normal bundle
  26. Directional neighborhoods and a sheaf projector
  27. Formal stabilization over arbitrary neighborhood sets
  28. Finite local data and sheaf biduality
  29. Supports, characteristic classes, and finite duality
  30. Boundary conditions, incidence operators, and trace identities
  31. Convex tests and radial comparisons
  32. Normal geometry as a family with a central fibre
  33. Reading a sheaf at the normal scale
  34. Covector tests of normal limits
  35. What a normal limit preserves
  36. Detecting and removing directional obstructions
  37. Comparing sheaves through local support tests
  38. Local morphisms in cotangent directions
  39. Product recovery for microlocal composition
  40. Transporting both inputs before forming internal Hom
  41. Analytic geometry for finite maps
  42. Normal Morse data and change of coefficients
  43. Perverse degrees and normal Morse complexes
  44. An isolated holomorphic test and its Morse filtration
  45. Directional information under a finite holomorphic map
  46. Uniform tests for an unbounded Hom complex
  47. Reading a subset through its sheaf
  48. Transporting directional obstructions
  49. Directional constraints under algebraic and geometric constructions
  50. Categories and operations in one cotangent direction
  51. Covectors at a boundary and at infinity
  52. Cotangent directions that survive a limiting operation
  53. Hamiltonian motion forced by a sheaf
  54. Local models and change of ambient manifold
  55. Continuing coefficients and transporting local morphisms
  56. From directional tests to further microlocal theories
Sources and rights

GPT-6 Astra and GPT-6 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Spectra, resolvents and scattering

English · draft · 51 Lessons

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Lessons
  1. Resolvents, domains and spectral density
  2. Endpoint spaces and flat energy shells
  3. Fourier traces on curved energy surfaces
  4. Mild weights and frequency localization
  5. Division and radiation at regular energies
  6. Polynomial translations and regular energies
  7. Global polynomial resolvent estimates
  8. Global radiation and flux
  9. Short-range compactness and local tests
  10. Polynomial localizations and rough coefficients
  11. Self-adjoint short-range operators
  12. Wave operators and modified phases
  13. Wave operators for differential perturbations
  14. Limiting absorption and point spectrum
  15. Distorted Fourier transforms and spectral density
  16. Asymptotic completeness for short-range operators
  17. Compact perturbations in weighted Hilbert spaces
  18. Scattering matrices at regular energies
  19. One-dimensional scattering and phase shifts
  20. Quadratic weights and uniqueness at infinity
  21. Wave evolution and cotangent flow
  22. Local spectral density and the subprincipal correction
  23. Return times and spectral counting
  24. Compressed spectral measures and symbol distributions
  25. Positive real powers and spectral rescaling
  26. Arithmetic spectral clusters and their distributions
  27. Averaging a perturbation around closed trajectories
  28. Reflection and the Dirichlet boundary coefficient
  29. Generalized rays and the Dirichlet Weyl law
  30. Regularizing long-range coefficients
  31. Admissible differential perturbations
  32. The Sobolev domain of an elliptic operator
  33. Weighted Sobolev spaces and rough elliptic estimates
  34. The resolvent away from the energy surface
  35. A resolvent estimate at noncritical frequencies
  36. Combining the long-range resolvent estimates
  37. Radiation for limits of long-range resolvents
  38. Outgoing flux and vanishing shell mass
  39. Weighted endpoint estimates and polynomial decay
  40. Limiting absorption for long-range differential perturbations
  41. Hamilton trajectories under a long-range force
  42. Smooth long-range phases from Hamilton trajectories
  43. Escaping Lagrangians on regular energy surfaces
  44. Generating functions and the end of a localized force
  45. Modified waves and the direction of escape
  46. Energy-shell factors and outgoing equations
  47. Frequency cutoffs and compact scattering remainders
  48. Commuting coordinates for long-range evolution
  49. Transverse moments and outgoing amplitudes
  50. Truncated operators and stable scattering amplitudes
  51. Spectral transforms and completeness of modified waves
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

The field with one element

English · draft · 15 Lessons

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Lessons
  1. Counting over finite fields and the limit q → 1
  2. Weil's proof for curves and what is missing over the integers
  3. Commutative monoids and their spectra
  4. Monoid schemes
  5. Torified varieties and the limits of monoid schemes
  6. Varieties over the field with one element after Soulé and Connes–Consani
  7. Schemes relative to a symmetric monoidal category
  8. Λ-rings and descent to the field with one element
  9. Generalized rings
  10. Blueprints and blue schemes
  11. Characteristic one and hyperrings
  12. Γ-sets and algebras over the sphere
  13. The arithmetic site
  14. The scaling site
  15. The projective line over F_1 and the ABC conjecture
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Claude Opus 5.5 (Anthropic); CC0-1.0

What this prepares you for

Group schemes

English · draft · 7 Lessons

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Lessons
  1. Group schemes, actions and Hopf algebras
  2. Lie algebras and smoothness of group schemes
  3. Group schemes over a field
  4. Quotients and torsors
  5. Diagonalizable groups and groups of multiplicative type
  6. Abelian varieties
  7. Néron models
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What this prepares you for

Hilbert C*-modules and Morita equivalence

English · draft · 15 Lessons

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Lessons
  1. Hilbert C*-modules
  2. Adjointable operators
  3. Compact operators, multipliers and the strict topology
  4. Finite projective modules, frames and K_0
  5. Tensor products and C*-correspondences
  6. Kasparov's stabilization theorem
  7. Fredholm operators and the K₀ index
  8. Unbounded regular operators
  9. Continuous fields over locally compact spaces
  10. Completely positive maps and the KSGNS construction
  11. Imprimitivity bimodules and Morita equivalence
  12. The Rieffel correspondence and induced representations
  13. Stable isomorphism: the Brown–Green–Rieffel theorem
  14. Morita invariance of K-theory and maps induced by correspondences
  15. Morita equivalence of noncommutative tori
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

ℓ-adic cohomology and trace formulas

English · draft · 11 Lessons

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Lessons
  1. ℓ-adic sheaves and their cohomology
  2. The pro-étale site and ℓ-adic complexes
  3. Frobenius morphisms and their action on cohomology
  4. Traces of perfect complexes and Lefschetz numbers
  5. Cycle classes and the Lefschetz fixed-point formula for curves
  6. The trace formula for curves
  7. The trace formula in all dimensions
  8. L-functions, rationality and the functional equation
  9. Lefschetz pencils and vanishing cycles
  10. The fundamental estimate
  11. The Riemann hypothesis over finite fields
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What this prepares you for

D-modules

English · draft · 18 Lessons

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Lessons
  1. Differential operators and the Weyl algebra
  2. D-modules, flat connections and local systems
  3. Good filtrations and the characteristic variety
  4. The Bernstein filtration and holonomic modules over the Weyl algebra
  5. Bernstein-Sato polynomials
  6. Holonomic D-modules and duality
  7. Inverse images
  8. Direct images and the relative de Rham complex
  9. Kashiwara's equivalence and D-modules on singular spaces
  10. Adjunctions, base change and the projection formula
  11. Preservation of holonomicity and minimal extensions
  12. The de Rham functor
  13. Regular singularities
  14. The Riemann-Hilbert correspondence
  15. Equivariant and twisted D-modules
  16. Beilinson-Bernstein localization
  17. D-modules on stacks, ind-schemes and the de Rham prestack
  18. The Fourier transform of D-modules
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What this prepares you for

Perverse sheaves and intersection cohomology

English · draft · 12 Lessons

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Lessons
  1. Constructible complexes on algebraic varieties
  2. Gluing t-structures
  3. The perverse t-structure
  4. Intermediate extensions and intersection complexes
  5. Affine morphisms, Artin vanishing and perverse cohomology
  6. Nearby and vanishing cycles
  7. Weights, purity and semisimplicity over finite fields
  8. The decomposition theorem
  9. Semismall maps and small resolutions
  10. Springer theory
  11. Equivariant perverse sheaves and perverse sheaves on stacks
  12. Perverse sheaves, D-modules and the function-sheaf dictionary
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What this prepares you for

Descent and the étale fundamental group

English · draft · 8 Lessons

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Lessons
  1. Fibred categories and descent data
  2. Faithfully flat descent
  3. Descending properties of schemes and morphisms
  4. Galois categories
  5. The étale fundamental group
  6. Fundamental groups of proper schemes and the homotopy exact sequence
  7. Specialization maps and tame ramification
  8. Comparison with the topological fundamental group over the complex numbers
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What this prepares you for

Hilbert and Picard schemes, deformations and representability

English · draft · 12 Lessons

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Lessons
  1. Representable functors and the functor of points
  2. Grassmannians
  3. Castelnuovo–Mumford regularity and boundedness
  4. Flattening stratifications
  5. Hilbert and Quot schemes
  6. Formal moduli and Schlessinger's theorem
  7. Deformations of rings and schemes and the naive cotangent complex
  8. Tangent spaces and obstructions for Hilbert and Quot schemes
  9. The Picard functor and the Picard scheme of a curve
  10. Relative divisors and the existence of the Picard scheme
  11. The structure of the Picard scheme
  12. The cotangent complex
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What this prepares you for

The geometric Langlands conjecture

English · draft · 6 Lessons

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Lessons
  1. From automorphic functions to automorphic sheaves
  2. The moduli stack of bundles
  3. Sheaves and D-modules on Bun_G
  4. Hecke functors and Hecke eigensheaves
  5. Geometric class field theory
  6. The GL_1 case as an equivalence of categories
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What this prepares you for

Galois representations

English · draft · 15 Lessons

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Lessons
  1. Profinite groups and ℓ-adic representations
  2. Frobenius elements and determination by traces
  3. The ℓ-adic Tate module of an elliptic curve
  4. Representations of Weil groups
  5. Conductors of Weil-group representations
  6. Local L-factors and epsilon factors
  7. Weil–Deligne representations and Grothendieck’s monodromy theorem
  8. Elliptic curves over local fields and their Weil–Deligne representations
  9. Compatible systems and global L-functions
  10. Modular curves over Q and the Eichler–Shimura congruence
  11. Galois representations of weight-two newforms
  12. Deligne's construction in higher weight and the Ramanujan–Petersson bound
  13. Weight one: the theorem of Deligne and Serre
  14. p-adic Hodge theory and the Fontaine–Mazur conjecture
  15. Residual representations and Serre's modularity theorem
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What this prepares you for

Sites, topoi and étale cohomology

English · draft · 19 Lessons

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Lessons
  1. Sites and sheaves
  2. Topoi, morphisms and points
  3. Cohomology on sites
  4. Hypercoverings
  5. Topologies on schemes
  6. The étale site and its points
  7. Pushforward, pullback and finite morphisms
  8. Galois cohomology and the étale cohomology of a field
  9. Brauer groups and Tsen's theorem
  10. The multiplicative group on a curve
  11. Constructible sheaves and extension by zero
  12. Torsion sheaves on curves
  13. The proper base change theorem
  14. Cohomology with compact support
  15. Smooth base change and local acyclicity
  16. Cohomological dimension and the Künneth formula
  17. Poincaré duality for curves
  18. Poincaré duality for smooth varieties
  19. Comparison with singular cohomology
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What this prepares you for

Adèles, idèles and Tate's thesis

English · draft · 12 Lessons

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Lessons
  1. Restricted products and profinite completions
  2. The adèle ring of a number field
  3. Idèles and the idèle class group
  4. Idèles, ideals and ray class groups
  5. Additive characters, self-dual measures and Poisson summation on the adèles
  6. Quasi-characters and Hecke characters
  7. Tate's local theory at the finite places
  8. Tate's local theory at the infinite places
  9. Tate's global theory: continuation and functional equation
  10. Hecke L-functions and the Dedekind zeta function
  11. Lattices over a local field: the tree of GL_2 and its decompositions
  12. The spherical Hecke algebra of GL_2 and Hecke operators on lattices
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What this prepares you for

The abstract theory of ideals

English · draft · 6 Lessons

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Lessons
  1. Noether's axioms for Dedekind domains
  2. The converse and composition series
  3. Orders and the discriminant theorem
  4. Three differents
  5. Normal integral bases in tame extensions
  6. Algebraic functions of one variable: conductors and adjoints
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What this prepares you for

The local Langlands correspondence for GL_n

English · draft · 13 Lessons

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Lessons
  1. First cases: characters and the real correspondence
  2. Irreducible representations of general linear groups over a local field
  3. Local factors of pairs and the rank-two converse theorem
  4. The statement of the local Langlands correspondence
  5. Monodromy blocks and the passage from supercuspidals to all parameters
  6. The rank-two correspondence: principal series, Steinberg twists and characters
  7. Two-dimensional Weil representations: induction and projective symmetry
  8. Dihedral supercuspidals, rectifiers and rank-two characterization
  9. Modular forms, elliptic curves and local–global compatibility
  10. Jacquet–Langlands: elliptic classes and division-algebra representations
  11. Harris–Taylor: geometry, numerical counting and existence
  12. Other proofs, functoriality and the formal-degree formula
  13. Beyond general linear groups: parameters and packets
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What this prepares you for

Crossed products and groupoid C*-algebras

English · draft · 17 Lessons

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Lessons
  1. C*-dynamical systems and full crossed products
  2. Reduced crossed products and Fell's absorption principle
  3. Pontryagin duality and the dual action
  4. Induced algebras and Green's imprimitivity theorem
  5. Takai duality
  6. Proper actions, free actions and the orbit space
  7. Amenability and the equality of full and reduced crossed products
  8. Expectations, traces and ideals in crossed products
  9. Crossed products by integers and real numbers: Fourier coordinates and mapping tori
  10. The Connes–Thom isomorphism I: the Wiener–Hopf extension
  11. The Connes–Thom isomorphism II: surjectivity, naturality and consequences
  12. Exterior equivalence and Connes's construction of the Thom map
  13. Locally compact groupoids and Haar systems
  14. Groupoid C*-algebras: full and reduced
  15. Equivalence of groupoids and Morita equivalence of their C*-algebras
  16. C*-algebras of foliations and their Morita equivalences
  17. The tangent groupoid and deformation to the normal cone
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What this prepares you for

Representations of compact groups

English · draft · 13 Lessons

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Lessons
  1. Representations of compact groups: unitarity, complete reducibility and finite dimension
  2. Matrix coefficients and the Peter–Weyl theorem
  3. Fourier analysis and class functions on compact groups
  4. SU(2), SO(3) and spherical harmonics
  5. Compact Lie groups, their Lie algebras and the adjoint representation
  6. Tori and the maximal torus theorem
  7. Roots and the Weyl group of a compact Lie group
  8. The Weyl integration formula
  9. Unitary groups: characters, dimensions and branching
  10. Weights and the theorem of the highest weight
  11. The Weyl character formula for compact connected Lie groups
  12. Casimir operators, Laplacians and tensor products
  13. Homogeneous spaces, induced representations and Gelfand pairs
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What this prepares you for

Representations of finite groups

English · draft · 17 Lessons

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Lessons
  1. Representations and complete reducibility
  2. Characters and the orthogonality relations
  3. The group algebra and Fourier analysis on a finite group
  4. Tensor products, duals and real representations
  5. Integrality of characters and Burnside's p^a q^b theorem
  6. Induced representations and Frobenius reciprocity
  7. Mackey theory and Clifford's theorem
  8. Groups with an abelian normal subgroup: the little-group method
  9. Frobenius groups and Frobenius's theorem
  10. Artin's induction theorem and rationality
  11. Brauer's induction theorem
  12. Consequences of Brauer's theorem: characterization of characters and splitting fields
  13. The symmetric groups I: Young tableaux and Young symmetrizers
  14. The symmetric groups II: branching, Jucys–Murphy elements and Young's seminormal form
  15. The symmetric groups III: characters and symmetric functions
  16. Schur–Weyl duality
  17. Representations of GL₂ over finite fields
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What this prepares you for

Commutative algebra for geometry

English · draft · 20 Lessons

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Lessons
  1. Spectra of rings
  2. Localization, local properties and support
  3. Noetherian and Artinian rings
  4. Associated primes and primary decomposition
  5. Integral extensions: lying over, going up and going down
  6. The Nullstellensatz and Jacobson rings
  7. Tor and flat modules
  8. Faithful flatness and the local criterion for flatness
  9. Krull dimension and Noether normalization
  10. Graded modules and Hilbert–Samuel functions
  11. Dimension theory of Noetherian local rings
  12. Regular sequences, depth and Cohen–Macaulay modules
  13. Projective dimension and the Auslander–Buchsbaum formula
  14. Regular local rings
  15. Discrete valuation rings, normal rings and Serre's criterion
  16. Kähler differentials
  17. Formally smooth, unramified and étale ring maps
  18. Smooth algebras over a field and the Jacobian criterion
  19. Completion
  20. Henselian local rings and henselization
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What this prepares you for

Local fields

English · draft · 12 Lessons

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Lessons
  1. Absolute values, valuations and Ostrowski's theorem
  2. Completions, the p-adic numbers and complete discretely valued fields
  3. Hensel's lemma, squares and roots of unity in p-adic fields
  4. Extensions of complete valued fields
  5. Places of number fields in extensions and the product formula
  6. Local fields: classification and Haar measure
  7. Unramified and totally ramified extensions
  8. Tame ramification and the tame Galois group
  9. Ramification groups and the different of a local extension
  10. Herbrand's function and the upper numbering
  11. The multiplicative group of a local field
  12. Cyclotomic, quadratic and Kummer extensions of local fields
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What this prepares you for

Local cohomology, Lefschetz theorems and purity

English · draft · 8 Lessons

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Lessons
  1. Local cohomology
  2. Local duality and finiteness
  3. Cohomological dimension, vanishing and connectedness
  4. Formal geometry along a closed subscheme
  5. Lefschetz theorems for finite étale covers
  6. Picard groups and Grothendieck–Lefschetz
  7. Purity of the branch locus
  8. Specialization and tame ramification
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What this prepares you for

Automorphic forms and representations of GL(2)

English · draft · 20 Lessons

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Lessons
  1. Adèles for GL₂ and strong approximation
  2. From modular forms to adelic functions
  3. Automorphic forms, modules and Hilbert constituents
  4. Why the cuspidal spectrum is discrete
  5. Smooth local representations and the Hecke-module dictionary
  6. Normalized induction and Jacquet modules
  7. Whittaker models, Kirillov models and the local classification
  8. Newvectors, conductors and the passage to classical newforms
  9. Local L-factors, epsilon factors and the local converse theorem
  10. Supercuspidal representations from compact induction
  11. Real and complex representations: weights, gamma factors and classical forms
  12. Restricted tensor products and the tensor product theorem
  13. Unramified representations of GL₂(F) and Satake parameters
  14. Global Whittaker functions and the L-function of a cuspidal representation
  15. Multiplicity one, strong multiplicity one and the converse theorem
  16. Eisenstein series on GL₂(A) and the continuous spectrum
  17. Quaternion algebras and their automorphic forms
  18. The trace formula for a compact quotient
  19. The Jacquet–Langlands correspondence
  20. Dihedral forms, examples, and the Ramanujan conjecture for GL₂
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What this prepares you for

Lie's theory of transformation groups

English · draft · 12 Lessons

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Lessons
  1. Transformation groups and their parameters
  2. The fundamental differential equations
  3. One-parameter groups
  4. Complete systems and invariants
  5. The second fundamental theorem and the group of parameters
  6. Prolongation, differential invariants and the projective group
  7. The third fundamental theorem
  8. The adjoint group, composition and isomorphism
  9. Transitivity and primitivity
  10. Contact transformations and first-order equations
  11. Lie algebras of vector fields on the line and in the plane
  12. The Riemann-Helmholtz space problem
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What this prepares you for

Algebraic spaces and stacks

English · draft · 12 Lessons

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Lessons
  1. Algebraic spaces
  2. The bootstrap theorem
  3. Properties and morphisms of algebraic spaces
  4. Stacks in groupoids
  5. Algebraic stacks
  6. Quotient stacks and Deligne–Mumford stacks
  7. Artin's axioms
  8. Moduli stacks are algebraic
  9. The stack of curves
  10. Stable reduction and properness
  11. Simplicial sets, nerves and Kan complexes
  12. Test categories and Grothendieck's homotopy programme
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What this prepares you for

K-theory of C*-algebras

English · draft · 23 Lessons

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Lessons
  1. Idempotents, projections and their equivalences
  2. Vector bundles and finitely generated projective modules
  3. The Grothendieck group and K_0 of a unital algebra
  4. Nonunital algebras: unitization, relative classes and half-exactness
  5. Matrix stability, stability and continuity of K_0
  6. Invertibles, unitaries and K_1
  7. The index map and the exact sequence at K_0
  8. Suspension, higher K-groups and the long exact sequence
  9. Toeplitz operators and the index theorem on the circle
  10. Bott periodicity
  11. The six-term exact sequence and the exponential map
  12. Topological K-theory of spaces, pairs and vector bundles
  13. Traces, states and the pairing with K-theory
  14. Determinants of traces and the pairing with K_1
  15. Smooth subalgebras and the density theorem
  16. Inductive limits and the K-theory of AF and AT algebras
  17. The mapping torus
  18. The Pimsner–Voiculescu exact sequence
  19. Traces on integer crossed products and their K-theory ranges
  20. Irrational rotation algebras
  21. Commutative and noncommutative tori
  22. Cuntz algebras
  23. Cuntz–Krieger algebras, minimal systems and projectionless algebras
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What this prepares you for

Modular forms

English · draft · 19 Lessons

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Lessons
  1. The upper half-plane and the modular group
  2. Congruence subgroups, cusps and elliptic points
  3. Modular curves and their genus
  4. Modular forms, lattice functions and Eisenstein series
  5. The valence formula and the ring of modular forms of level one
  6. Dimension formulas for congruence subgroups
  7. The Petersson inner product and Poincaré series
  8. Hecke operators of level one
  9. Hecke operators for Γ_0(N) and Γ_1(N)
  10. Oldforms, newforms and the theory of Atkin, Lehner and Li
  11. The L-function of a cusp form
  12. Twists, level N and Hecke's converse theorem
  13. Theta functions and sums of squares
  14. Theta series of lattices
  15. Group cohomology of Γ and the Eichler–Shimura isomorphism
  16. Modular symbols and the algebraicity of Hecke eigenvalues
  17. Complex tori, elliptic curves and the moduli interpretation of modular curves
  18. Non-holomorphic Eisenstein series and Maass forms
  19. The Rankin–Selberg method
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What this prepares you for

Global Langlands conjectures and functoriality

English · draft · 6 Lessons

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No course preparation is declared in this snapshot; that does not mean no prerequisites.

Lessons
  1. Reductive groups, root data and the L-group
  2. The Satake isomorphism for unramified groups and unramified L-factors
  3. Automorphic representations and automorphic L-functions
  4. L-functions for GL_n: Godement–Jacquet and Rankin–Selberg
  5. The principle of functoriality
  6. The function-field case: Drinfeld and L. Lafforgue
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What this prepares you for

Hypercomplex systems and representations

English · draft · 7 Lessons

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Lessons
  1. Groups with operators
  2. Semisimple rings and Wedderburn's theorem
  3. Representations, characters and the group determinant
  4. Central simple algebras and the Brauer group
  5. Crossed products and factor systems
  6. Hilbert 90 in Noether's form and Galois descent
  7. The principal genus theorem
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What this prepares you for

Semisimple Lie algebras

English · draft · 18 Lessons

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Lessons
  1. Lie algebras: definitions, examples and first constructions
  2. Nilpotent and solvable Lie algebras: Engel's and Lie's theorems
  3. The Killing form and Cartan's criteria
  4. Complete reducibility: Casimir elements and Weyl's theorem
  5. Representations of sl(2)
  6. Low-degree cohomology, Whitehead's lemmas and the Levi decomposition
  7. The root space decomposition of a semisimple Lie algebra
  8. Root systems and their Weyl groups
  9. Cartan matrices, Dynkin diagrams and the classification of root systems
  10. Cartan subalgebras and their conjugacy
  11. The isomorphism theorem and Serre's theorem
  12. The simple Lie algebras: classical models and the exceptional algebras
  13. The universal enveloping algebra and the Poincaré–Birkhoff–Witt theorem
  14. Weights, Verma modules and the theorem of the highest weight
  15. The centre of the enveloping algebra and Harish-Chandra's theorem
  16. Weyl's character formula and the multiplicity formulas
  17. Compact real forms and Weyl's unitary trick
  18. Category O and the Kazhdan–Lusztig conjecture
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What this prepares you for

Constructible and perverse sheaves

English · draft · 92 Lessons

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Lessons
  1. Sheaf kernels and cotangent correspondences
  2. Constructible gluing on an interval
  3. Directional tests at a constructible boundary
  4. Constructible sheaves on a triangulation
  5. Kernels that preserve chosen cotangent directions
  6. Adjoints of localized sheaf kernels
  7. Dual kernels and an unchanged parameter
  8. Directional morphisms through a sheaf kernel
  9. When a kernel quantizes a contact transformation
  10. Real and complex ball kernels
  11. Local existence of contact kernel equivalences
  12. Microlocal composition at prescribed covectors
  13. Coefficients in a quantization of the identity
  14. Normal forms and the shift of a submanifold transform
  15. Composing hypersurface kernels with their shifts
  16. Pure and simple sheaves from directional tests
  17. How simple sheaf shifts change along a Lagrangian
  18. Tensor and Hom types at independent covectors
  19. The type and shift of a transverse kernel composition
  20. Direct and inverse images of a sheaf type
  21. Projective incidence as a contact kernel
  22. The partial Legendre kernel and its adjoint shift
  23. Fourier-Sato transport of microlocal morphisms
  24. Subanalytic sets and limiting tangent directions
  25. Conic subanalytic images and isotropic dimension
  26. Isotropic cotangent transport and discrete critical values
  27. Finite conormal closures and generic base directions
  28. Involutive subsets of subanalytic isotropic sets
  29. Boundary forms and Lagrangian normal cones
  30. Limiting cotangent sums and characteristic inverse images
  31. Microlocal stratifications by removing bad loci
  32. Unshared conormal directions and dimension filtrations
  33. Generic squared distance and cotangent transversality
  34. Constructibility from microsupport and perfect stalks
  35. Derived constructibility through common triangulations
  36. Weak constructibility under sheaf operations
  37. Small balls, central fibres and supported cohomology
  38. Perfect coefficients on compact fibres
  39. Constructible costalks and Verdier duality
  40. Perfect operations and finite microlocal coefficients
  41. Natural duality for specialization and microlocal Hom
  42. Complex conicity and analytic Lagrangian closures
  43. Analytic normal cones through complex deformation
  44. Analytic conormal covers and singular involutivity
  45. Complex microlocal stratifications and constructibility
  46. Holomorphic operations and complex Fourier symmetries
  47. Nonproper holomorphic pushforwards through cutoffs
  48. Local holomorphic pushforwards over a complex curve
  49. Complex-constructible sheaves and missing derived classes
  50. Nearby cycles and the two monodromy triangles
  51. Proper pushforwards of nearby and vanishing cycles
  52. Nearby cycles through the normal deformation
  53. Complex nearby cycles as normal and conormal sections
  54. Quadratic cycles and the holomorphic microsupport test
  55. Vanishing cycles as positive real support
  56. Whitney secants and the microlocal stratification condition
  57. Duality maps for constructible inverse and direct images
  58. Constructibility through smooth cutoffs and microlocal properness
  59. Constructible models in one cotangent direction
  60. Closed conormal unions at singular points
  61. The dualizing complex from oriented simplices
  62. Local systems across an analytic boundary
  63. Reading the geometry and cycle literature
  64. Constructible traces and local Euler indices
  65. Closed supports and evaluated proper transport
  66. Proper characteristic classes and the compact index
  67. Subanalytic chains and closed cycle supports
  68. Supports, products and proper images of chains
  69. The dualizing resolution by subanalytic chains
  70. Intersections of supported subanalytic cycles
  71. Lagrangian cycles and proper cotangent images
  72. Pulling back Lagrangian cycles through a graph
  73. Continuous sections and supported cycle intersections
  74. Transverse pullback of normalized conormal cycles
  75. Characteristic cycles from supported microlocal identities
  76. Transporting characteristic cycles through a graph
  77. Integer coefficients and additive characteristic cycles
  78. Antipodal duality and half-line characteristic cycles
  79. Lorentz cones and their vertex characteristic cycles
  80. Differential sections and proper-below Euler indices
  81. Isolated phases and local characteristic-cycle indices
  82. Pure test degrees and strong Morse inequalities
  83. Orientations of conormal cycles and transverse intersections
  84. Lefschetz traces of constructible correspondences
  85. Homotopies and local cutoffs for Lefschetz contributions
  86. Specializing Lefschetz contributions to the tangent space
  87. Expanding subspaces and hyperbolic Lefschetz cutoffs
  88. Shrinking localization and complex fixed-point traces
  89. Constructible functions and Euler integration
  90. Homogeneous functions, Fourier transformation and specialization
  91. Constructible functions and integral Lagrangian cycles
  92. Euler convolution and convex inverses
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Symmetries and conservation laws

English · draft · 3 Lessons

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Lessons
  1. Variational symmetries and Noether's first theorem
  2. Infinite symmetry groups and Noether's second theorem
  3. The formal calculus of variations and differential invariants
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Explicit formulas and positivity

English · draft · 4 Lessons

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Lessons
  1. The explicit formula with general test functions
  2. Explicit formulas for Dirichlet L-functions and primes in progressions
  3. Guinand's formula
  4. Li's criterion
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Kasparov's KK-theory

English · draft · 10 Lessons

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Lessons
  1. Extensions of C*-algebras and the Busby invariant
  2. Ext groups, absorption and Brown-Douglas-Fillmore theory
  3. Fredholm modules and analytic K-homology
  4. The index pairing between K-theory and K-homology
  5. Graded C*-algebras, Clifford algebras and graded Hilbert modules
  6. Kasparov modules and the groups KK(A, B)
  7. Kasparov's technical theorem
  8. Connections and the existence of the Kasparov product
  9. Homotopy, associativity, the index pairing and KK-equivalence
  10. Thom isomorphisms and K-orientations in KK
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

The Riemann zeta function

English · draft · 18 Lessons

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Lessons
  1. Dirichlet series and Euler products
  2. Counting primes by elementary means: Chebyshev and Mertens
  3. The Gamma function and Stirling's formula
  4. Poisson summation, theta, and the functional equation
  5. Entire functions of order one and the Hadamard product of ξ
  6. Growth in the critical strip: convexity and the Lindelöf hypothesis
  7. Exponential sums and a subconvex bound
  8. Nonvanishing on the line σ = 1 and the zero-free region
  9. The prime number theorem
  10. The Riemann–von Mangoldt formula
  11. Perron's formula and the explicit formula for ψ(x)
  12. The prime number theorem with the classical error term
  13. Growth bounds and wider zero-free regions
  14. The Riemann hypothesis and its standard equivalents
  15. Oscillation of the error term
  16. Mean values of Dirichlet polynomials and of zeta on the critical line
  17. The Riemann hypothesis implies the Lindelöf hypothesis
  18. Zero-density estimates and primes in short intervals
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Morphisms of schemes

English · draft · 17 Lessons

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Lessons
  1. The diagonal and separated morphisms
  2. Finiteness of morphisms
  3. Limits of schemes and Noetherian approximation
  4. Valuation rings and the valuative criterion of separatedness
  5. Affine morphisms, relative Spec, and finite morphisms
  6. Quasi-finite morphisms and Chevalley’s theorem
  7. Proper morphisms and the valuative criterion of properness
  8. Very ample invertible sheaves, Segre and Veronese embeddings
  9. Ample invertible sheaves
  10. Projective morphisms and Chow’s lemma
  11. Dimension of fibres
  12. Zariski's Main Theorem
  13. Normalization
  14. Rational maps and birational morphisms
  15. Effective Cartier divisors and invertible sheaves
  16. Blowing up
  17. Weil divisors and the class group
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Sheaves and schemes

English · draft · 12 Lessons

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Lessons
  1. Sheaves on topological spaces
  2. Ringed spaces and sheaves of modules
  3. Quasi-coherent, coherent and locally free modules
  4. Affine schemes
  5. Schemes, gluing and immersions
  6. The functor of points and representability
  7. Fibre products and base change
  8. Properties of schemes
  9. Quasi-coherent sheaves on schemes
  10. Closed subschemes and scheme-theoretic images
  11. Proj of a graded ring
  12. Projective space, relative Proj and maps to projective space
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Cohomology of quasi-coherent sheaves

English · draft · 18 Lessons

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Lessons
  1. Cohomology of sheaves on ringed spaces
  2. Čech cohomology
  3. Cohomology of affine schemes and Serre's criterion
  4. Cohomology of projective space
  5. Coherent sheaves on projective schemes: Serre's theorems
  6. Coherence of higher direct images under proper morphisms
  7. Euler characteristics and Hilbert polynomials
  8. Base change and the Grothendieck complex
  9. Semicontinuity and Grauert's theorem
  10. The theorem on formal functions
  11. Zariski's connectedness theorem and Stein factorization
  12. Grothendieck's existence theorem
  13. Algebraization of formal schemes
  14. Ext sheaves and Serre duality on projective space
  15. Dualizing sheaves and Serre duality for projective schemes
  16. The right adjoint of derived pushforward
  17. Complex analytic spaces and analytification
  18. Serre's comparison theorems and Chow's theorem
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Invariants and finiteness

English · draft · 5 Lessons

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Lessons
  1. Covariants of binary and ternary forms
  2. Forms in several variables and Hilbert finiteness
  3. Invariants of finite groups and Noether's degree bound
  4. Polarization and invariants of many vectors
  5. Finiteness for finite groups in every characteristic
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Dirichlet L-functions and primes in progressions

English · draft · 16 Lessons

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Lessons
  1. Dirichlet characters
  2. Dirichlet's theorem on primes in arithmetic progressions
  3. Gauss sums
  4. The functional equation of Dirichlet L-functions
  5. Values of Dirichlet L-functions at s = 1
  6. Zero-free regions and the exceptional zero
  7. Counting zeros and the explicit formula for ψ(x, χ)
  8. The prime number theorem for arithmetic progressions
  9. Siegel's theorem
  10. The Siegel–Walfisz theorem
  11. The large sieve inequality
  12. The multiplicative large sieve and bilinear forms with characters
  13. Vaughan's identity and sums over primes
  14. The Bombieri–Vinogradov theorem
  15. Mean square distribution: the Barban–Davenport–Halberstam theorem
  16. The least prime in a progression: Linnik's theorem
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Class field theory

English · draft · 24 Lessons

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Lessons
  1. Profinite groups and infinite Galois theory
  2. Cohomology of cyclic groups and the Herbrand quotient
  3. Hilbert's Theorem 90 and Kummer theory
  4. Frobenius lifts and abstract reciprocity
  5. The reciprocity law and the class field correspondence
  6. Local reciprocity and norm groups
  7. Formal groups and Lubin–Tate modules
  8. Lubin–Tate division fields
  9. Explicit local reciprocity and the existence theorem
  10. Abelian ramification, conductors and Hasse–Arf
  11. Hilbert symbols and local conics
  12. Weil groups and one-dimensional representations
  13. Idèles in extensions and their cohomology
  14. The Herbrand quotient of the idèle class group
  15. The norm index bound and Hasse's norm theorem
  16. The global reciprocity law
  17. Global existence and the idèlic class field correspondence
  18. Ray class fields, conductors and ideal reciprocity
  19. Hilbert and ring class fields, and quadratic prime forms
  20. Kronecker–Weber and the maximal abelian extension of the rationals
  21. Artin L-functions, conductors and discriminants
  22. The Chebotarev density theorem
  23. Power residue symbols and reciprocity laws
  24. Brauer groups of local and global fields
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Ideal theory in rings

English · draft · 5 Lessons

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Lessons
  1. Chain conditions and irreducible decompositions
  2. Primary ideals and isolated components
  3. Modules, staircases and elementary divisors
  4. Generic zeros and absolute primality
  5. Resultant forms and elimination
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

The affine Grassmannian and geometric Satake

English · draft · 1 Lessons

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Lessons
  1. Spherical Hecke algebras as functions on the affine Grassmannian
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Linear forms in logarithms and their applications

English · draft · 6 Lessons

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Lessons
  1. Linear forms in logarithms: the problem, the trivial bounds, and effectivity
  2. Baker's theorem I: building an auxiliary function
  3. Baker's theorem II: extrapolation and the end of the proof
  4. Effective lower bounds I: Baker's theorem and its arithmetic tools
  5. Effective lower bounds II: proof of Baker's theorem
  6. Two logarithms I: interpolation determinants
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Transcendental numbers

English · draft · 15 Lessons

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Lessons
  1. Algebraic and transcendental numbers; Liouville's theorem
  2. Heights of algebraic numbers
  3. Hermite and the transcendence of e
  4. Lindemann, pi and the Lindemann–Weierstrass theorem
  5. Siegel's lemma, analytic estimates and the six exponentials theorem
  6. Gelfond–Schneider and Hilbert's seventh problem
  7. The Schneider–Lang criterion
  8. Weierstrass functions, elliptic values and periods
  9. Quasi-periods, complex multiplication and the modular invariant
  10. The index method for rational approximation
  11. Wronskians and Roth's rational-point lemma
  12. Roth's theorem and its consequences
  13. Mahler's classification of transcendental numbers
  14. E-functions and the Siegel–Shidlovsky theorem
  15. Algebraic independence and Schanuel's conjecture
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Rational function fields and equations with prescribed group

English · draft · 3 Lessons

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Lessons
  1. Subfields and subrings of rational function fields
  2. Noether's problem and generic polynomials
  3. Absolute irreducibility and Noether forms
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Characteristic classes

English · draft · 19 Lessons

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Lessons
  1. Vector bundles and their constructions
  2. Grassmannians and classifying maps
  3. Thom classes and Euler classes
  4. The Gysin sequence and projective splitting
  5. Steenrod squares and Stiefel–Whitney classes
  6. Schubert cells and universal Grassmannian cohomology
  7. Projective tangent bundles and their obstructions
  8. Chern classes and the integral universal ring
  9. Pontryagin classes and oriented universal cohomology
  10. Manifold duality, the diagonal and Wu classes
  11. Characteristic numbers and projective-product independence
  12. The oriented cobordism ring
  13. Frame fields and primary obstructions
  14. Thom spaces, transversality and the Pontryagin–Thom construction
  15. Homotopy fibres and the Serre spectral sequence
  16. Rational homotopy and the Hurewicz range
  17. Rational oriented bordism and projective generators
  18. Multiplicative sequences and the signature theorem
  19. Odd-prime reduced powers and the Wu classes
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Connections, curvature and holonomy

English · draft · 18 Lessons

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Lessons
  1. Local tools for bundles and transport
  2. Principal bundles and associated bundles
  3. Connections and parallel transport
  4. Curvature and holonomy groups
  5. Reduction and the holonomy theorem
  6. Flat connections and infinitesimal holonomy
  7. Invariant connections on homogeneous bundles
  8. Linear and affine connections
  9. Geodesics, normal coordinates and curvature
  10. Riemannian connections and convex neighbourhoods
  11. Completeness and the Hopf–Rinow theorem
  12. Holonomy and the de Rham decomposition theorem
  13. Sectional curvature and space forms
  14. Affine transformations and the isometry group
  15. Holonomy, Killing fields and analytic extension
  16. Submanifolds and hypersurfaces
  17. Jacobi fields, conjugate points and the Morse index theorem
  18. Comparison theorems, cut loci and curvature and topology
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GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); CC0-1.0

What this prepares you for

Reductive group schemes

English · draft · 6 Lessons

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Lessons
  1. Tori, maximal tori and their conjugacy
  2. Regular elements and centralizers
  3. Roots and reductive groups of rank one
  4. Root data, Weyl chambers and the Bruhat decomposition
  5. Pinnings and the classification of split reductive groups
  6. Automorphisms, forms and parabolic subgroups
Sources and rights

GPT-6.1 Sol (OpenAI), in Codex, at the Ultra setting (maximum reasoning effort); GFDL-1.2-only

What this prepares you for

Proof-level integration still being completed

Full statement/proof/generality comparisons are recorded separately. None is certified by this course-level crosswalk. The 58-course local phone projection and the public advanced edition are different evidence snapshots.

Navigation and dependency integration produced by OpenAI Codex — GPT-6.1 Sol, Ultra effort. Source author credits are preserved. No human proof review is claimed.