Further programme reading

The seven prerequisite chapters used in this course are included with editable sources. The references below identify other programme chapters for further reading; their texts are outside this companion.

Fourier analysis on finite abelian groups

Characters and the dual group

The dual group as the Gelfand spectrum of \(L^1(G)\)

Functions of positive type

Raikov's theorem and the Gelfand–Raikov theorem

Bochner's theorem

The Fourier inversion theorem and the dual Haar measure

The Plancherel theorem

The Pontryagin duality theorem

Subgroups, quotients and annihilators

The Poisson summation formula

The structure of locally compact abelian groups

Unitary representations of abelian groups: the spectral theorem

Closed ideals of \(L^1(G)\) and Wiener's theorem

Tauberian theorems and spectral synthesis

Almost periodic functions and the Bohr compactification

Totally disconnected groups, the p-adic numbers and the adèles

Algebraic integers and rings of integers

Discriminants and integral bases

Discrete valuation rings and Dedekind domains

Norms of ideals, the ideal class group, and modules over Dedekind domains

Decomposition of primes in extensions

Hilbert's ramification theory in Galois extensions

Lattices, Minkowski's theorem and the Minkowski embedding

Finiteness of the class number

Dirichlet's unit theorem

Quadratic fields: ideal classes and binary quadratic forms

Orders in number fields and their Picard groups

Cyclotomic fields

Units and class numbers of cyclotomic fields; Kummer's theorem for regular primes

The different and the discriminant

Counting ideals of bounded norm

The Dedekind zeta function and the analytic class number formula

Abelian number fields and Dirichlet L-functions at s = 1

Flat morphisms

Flatness criteria, dimension and the flat locus

Unramified morphisms

Étale morphisms and their local structure

Smooth morphisms

Infinitesimal lifting and the invariance of étale morphisms under thickenings

Étale neighbourhoods, henselization and quasi-finite morphisms

Hahn–Banach, Baire and the basic theorems on Banach spaces

Cauchy's theorem for cycles and its consequences

Banach algebras, spectrum, holomorphic functional calculus and Gelfand theory

C\*-algebras: continuous functional calculus, automatic continuity, positive cones, approximate identities and quotients

The spectral theorem for bounded self-adjoint operators

Representations and positive functionals: the GNS construction and the Gelfand–Naimark theorem

AF-algebras

Compact and trace-class operators, the predual of B(H), and the operator topologies

The double commutant theorem

Kaplansky's density theorem and its consequences

Abelian operator algebras

The universal enveloping von Neumann algebra of a \(C^*\)-algebra, and \(W^*\)-algebras

Projections and types of von Neumann algebras

Polar decomposition of functionals and weak compactness in preduals

Completely positive maps

Integral representations of states

Polish spaces and standard Borel spaces

Numerical ranges, positive matrices and co-Souslin sets

Measurable fields of Hilbert spaces and their direct integrals

Decomposable operators and the diagonal algebra

Vector-valued functions, tensor products with \(L^p\), and preduals

The Effros Borel structure

Direct integrals of von Neumann algebras

Spatial tensor products of von Neumann algebras

Tensor products of Banach and Hilbert spaces, Jordan homomorphisms and isometries of \(C^*\)-algebras

Curriculum and full result contracts

Finite domains and representations of general weights

Weights and semicyclic representations: the exact opening package

Comparing weights through finite-energy vectors

A bounded-operator kernel for weight arguments

Finite domains, null directions, and support corners

Concrete preduals from Hilbert tensors

Detecting normal weights by finite observations

The predual-valued map of a normal weight

Countable amplification and ultraweak bicommutant closure

Normal positive maps and their preadjoints

The vector retained by a comparison map

A faithful weight on the GNS commutant

Spectral calculus with its domains retained

Banach holomorphy, generators, and resolvent limits

Recovering operators from energy forms

Closing energy domains and comparing resolvents

Compatible pairs and complex interpolation

Closing an involution and recovering its modular data

Building the two multiplication actions of a Hilbert algebra

Haar convolution as left and right Hilbert algebras

Polynomial cutoffs recover an operator's range

Right Hilbert algebra density and commutants

The full left Hilbert algebra obtained by dualizing twice

Weights and the Hilbert spaces of multiplication

Changing algebra coordinates and scaling a weight

Analytic kernels for unbounded modular operators

Approximating multiplication without losing its bound

The modular group and its analytic algebra

The positive cone of a standard representation

Recovering a representation from its positive cone

Supported GNS representations and the balanced cocycle

Contractive retractions and conditional expectations

Spatial comparison on an arbitrary representation

Constructing spatial energy from finite observations

Adding spatial energies and changing the reference weight

Spatial energy and modular time

The KMS boundary condition determines the modular group

From a C*-modular condition to the GNS von Neumann algebra

Operators recovered from small trace defects

Relative tensor products and fusion

Reconstructing a weight from a modular cocycle

Recovering a weight from spatial energy

Fixed elements and changes of density

Tracial multiplication actions and the preserving expectation

Conditional expectations from modular invariance

Corner weights tested on all positive energies

Trace densities and noncommutative integration

Recognizing invariant weights by their densities

Inner modular flow and rigidity of finite weight data

Weight domination and exact half-strip endpoints

Partial cocycles and their exact mixed KMS domains

Analytic generators and exact finite weight domains

Commuting modular flows and relative half-strip multipliers

Central cocycles and common finite domains

Normal-functional polar decomposition and invariant pairs

Finite corners, faithful traces, and affine group fixed points

Ergodic factors and trace normalization

Semifinite corners and bounded modular operators

Modular partitions and the full finite domain

Tensor operators and tensor weights

Compatible measurable GNS fields and both representation transports

Measurable weight fields and their integrals

The realization boundary in measurable weight fields

Separable GNS spaces and disintegration of C*-weights

Changing the reference of a weight cocycle

Compact-time continuity along automorphism orbits

Forced C*-unitization

Finite weight domains and GNS

Bounded functionals and cyclic vectors

States and the positive spanning family

Weak compactness and second adjoints

The universal C*-bidual

Lower semicontinuous C*-weights

A supported inverse weight cocycle

Comparing supported weights and transporting their cuts

Uniform geometry of faithful weights

Homogeneous operators in the continuous core

When countable sums detect extended positive energy

Finite calculus and composition of operator-valued weights

Detecting and determining operator-valued weights

Modular invariants of operator-valued weights

Commutant duality for operator-valued weights

Compatible modular weights and operator-valued reconstruction

Countable modular sums and infinite-weight boundaries

Joint spectral measures and commutator estimates in standard form

Measurable Hilbert fields and their diagonal commutant

Central decomposition from countable operator equations

Measurable algebra cores and exact selection

Direct integrals of Hilbert and Tomita algebras

Regular representations, Fourier algebras, and Borel group measures

Six laboratories on expectations and operator-valued averages

Left and right Hilbert algebras

Holomorphy in Banach spaces, Stone's theorem and resolvent convergence

Analytic elements and strip arguments

Traces on von Neumann algebras

Integration for a trace, the commutation theorem, and applications

Measurable operators for a trace: examples, convergence and the commutant

Multiplicity of a von Neumann algebra on a Hilbert space

Entry contracts and mathematical conventions

Transporting crossed products through action cocycles

Two crossed products and the surviving action

Building an intrinsic flow from modular coordinates

Central cocycles as obstructions to fixed implementers

Kernels, local fixed parts, and freeness

Closing a domain after spectral localization

Integrating covariance with left Haar measure

A coefficient domain for the dual GNS construction

Moving an action and a modular parameter together

Continuity of actions: scalar tests, preduals, and bounded nets

Modular time on a crossed-product coefficient algebra

Extending the dual weight beyond the common involution domain

How the dual weight moves crossed-product generators

Changing the Hilbert space of a regular crossed product

Recovering multiplicity from a Weyl system

Locating frequencies with Fourier filters

Detecting semifinite corners through central translation

Recovering a spectral coordinate from a scaled trace

Comparing dual weights through matrix corners

The coefficient algebra as the value space of a weight

Averaging a dual action and recovering its coefficients

Commuting with the original algebra inside its core

Recovering a group action from its integrated operators

Recovering covariance with nonunital coefficients

Realizing central cocycles by spectral averaging

Recovering a weight from dual invariance

Removing noncentral cocycles from a trace-scaling action

Recovering a crossed product from its eigenunitaries

Recognizing the continuous core of a fixed algebra

Haar coordinates and integration after recognition

Dual weights and their supports

The weight after crossing twice

Derivatives of actions: domains, perturbations, and implementation

Innerness and spectral tails

Ultraproducts and the asymptotic centralizer

Full factors

Almost periodic weights and the invariant \(Sd\)

Full factors without almost periodic weights

Trace inequalities for finite von Neumann algebras

Property Γ and the algebra generated by a factor and its commutant

Approximately inner and centrally trivial automorphisms

Injective von Neumann algebras

Uniqueness of the injective II₁ factor

Classification of injective factors

Entropy of finite-dimensional subalgebras

The entropy of a trace-preserving automorphism

Entropy defect and abelian models

Dynamical entropy of C\*-algebras and von Neumann algebras

Entropy of lattice translations in quantum spin chains

The spatial derivative

Measured groupoids and transverse measures

Square-integrable representations and random operators

Weights on random operators and formal dimension

Cyclic cohomology: traces, differentials and symmetry

Connections and curvature from symmetries of an algebra

Frequency calculus for an action of Euclidean space

Cyclic forms that survive norm completion

A fundamental class for an action on a manifold

Singular values and the Dixmier trace

Spectral triples and dimension spectrum

The local index formula

Symbol calculus, transverse geometry and geometric index formulas

Localizing symbols when the measuring scale moves

Quantitative estimates for quadratic Fourier multipliers

Stable modes and the algebra of boundary data

Exact prerequisite interfaces for quadratic multiplier estimates

Two measuring scales, one Weyl product

Finite defects under perturbation

Polynomial and contour interfaces for stable boundary models

Local inverses and distance-weighted elliptic estimates

Traces that survive passage to cohomology

From symbol estimates to operators on every Sobolev scale

Support contracts for moving-metric localization

Singularities along a submanifold and smooth boundary passage

From Weyl symbols to operators and changes of coordinates

Positivity through a moving family of scalar probes

Detecting regularity without choosing coordinates

Curved weights and the directions in which support can end

Detecting a solution from infinite-order silence at one point

Banach estimates, quotient spaces and compact parameter arguments

From local energy to global divergence equations

Positive energy and vanishing at the far end of space

Boundary energy, local inverses, and harmonic data

When a moving symbol scale controls an operator

When a nonnegative scalar symbol acquires a negative part

Building a local inverse from radial singularities

Causal kernels, initial data, and short-time geometry

A wave kernel that cancels at a curved boundary

Symbols, finite defects, and the index on a closed manifold

Cauchy data from jumps and residues

Solving an elliptic system from compatible boundary measurements

Spectral measures with the original operator domain retained

Dirichlet realizations, spectral projectors, and local extensions

Changing an interior frame to extend an invertible matrix

Inverting mixed symbols without commuting matrix factors

Composition of mixed symbols with two different remainder estimates

Mixed symbols on every real two-parameter Sobolev scale

Polynomial inverse expansion with ordered matrix coefficients

Totally characteristic operators on the half space

The Bott operator, suspension, and reduction of the index to Euclidean space

Sheaves of modules and their derived categories

SH02-COURSE-CONTRACT — How to use this course draft

SH02-PREREQ-OPEN — Open prerequisites and exact import contracts

SH02-PREREQ-PROOFS — Supporting verifications for open prerequisites

Open extensions and ambient supports

SH02-SIX-BRIDGE — Proper supports and the bounded classical comparison

Proper-support composition through a change of base

SH02-CA — Extending sections on convex sets

SH02-SUP-EXH-001 — Recovering sheaf cohomology from closed pieces

SH02-NCD — Continuing cohomology through a moving boundary

SH02-EXCEPTIONAL-OPERATIONS — Exceptional inverse image from a finite resolution

SH02-UR — Finite resolutions without a lower bound

SH02-MD — Local orientations, dimension, and integration

Transport along a scaling action

Composing sheaf operators through an intermediate space

SH02-SPH — Inverse operators from complementary hemisphere boundaries

Fourier kernels as radial averaging

Fourier duality through a test complex on the base

SH02-UFD — Duality when the dual leaves the original transform category

SH02-FTC. Comparing traces after Fourier transformation

Moving Fourier kernels across maps and products

SH02-LFT. A linear map inside the Fourier comparison

SH02-FGC-UNIT. The graded line in a Fourier support comparison

SH02-FTE-UNIT. Transposing the complete Fourier trace comparison

SH02-NDF-UNIT. The geometric normalization of Fourier adjunctions

SH02-MEP-UNIT. Following the comparison maps into the normal bundle

SH02-GAM — Directional neighborhoods and a sheaf projector

SH02-FSB — Formal stabilization over arbitrary neighborhood sets

SH02-CB-UNIT — Finite local data and sheaf biduality

SH02-DA-UNIT — Supports, characteristic classes, and finite duality

Boundary conditions, incidence operators, and trace identities

Convex tests and radial comparisons

SH02-NG-UNIT — Normal geometry as a family with a central fibre

Reading a sheaf at the normal scale

Covector tests of normal limits

SH02-SA — What a normal limit preserves

SH02-MST — Detecting and removing directional obstructions

SH02-MSD — Comparing sheaves through local support tests

Local morphisms in cotangent directions

SH02-MHPR — Product recovery for microlocal composition

SH02-MHPC — Transporting both inputs before forming internal Hom

SH02-SUB-UNIT — Reading a subset through its sheaf

SH02-MO-UNIT — Transporting directional obstructions

SH02-FAG-UNIT — Analytic geometry for finite maps

SH02-NMC-UNIT — Normal Morse data and change of coefficients

SH02-PNM-UNIT — Perverse degrees and normal Morse complexes

SH02-IHM-UNIT — An isolated holomorphic test and its Morse filtration

SH02-FH-UNIT — Directional information under a finite holomorphic map

Directional constraints under algebraic and geometric constructions

SH02-MC — Categories and operations in one cotangent direction

SH02-AE-UNIT — Covectors at a boundary and at infinity

SH02-CHE-UNIT — Cotangent directions that survive a limiting operation

SH02-UCE — Uniform tests for an unbounded Hom complex

SH02-INV — Hamiltonian motion forced by a sheaf

SH02-LFI — Local models and change of ambient manifold

SH02-MA — Continuing coefficients and transporting local morphisms

SH02-RR — From directional tests to further microlocal theories

Resolvents, domains and spectral density

Endpoint spaces and flat energy shells

Fourier traces on curved energy surfaces

Mild weights and frequency localization

Division and radiation at regular energies

Polynomial translations and regular energies

Global polynomial resolvent estimates

Global radiation and flux

Short-range compactness and local tests

Polynomial localizations and rough coefficients

Self-adjoint short-range operators

Wave operators and modified phases

Wave operators for differential perturbations

Limiting absorption and point spectrum

Distorted Fourier transforms and spectral density

Asymptotic completeness for short-range operators

Compact perturbations in weighted Hilbert spaces

Scattering matrices at regular energies

One-dimensional scattering and phase shifts

Quadratic weights and uniqueness at infinity

Wave evolution and cotangent flow

Local spectral density and the subprincipal correction

Return times and spectral counting

Compressed spectral measures and symbol distributions

Positive real powers and spectral rescaling

Arithmetic spectral clusters and their distributions

Averaging a perturbation around closed trajectories

Reflection and the Dirichlet boundary coefficient

Regularizing long-range coefficients

Admissible differential perturbations

The Sobolev domain of an elliptic operator

Weighted Sobolev spaces and rough elliptic estimates

The resolvent away from the energy surface

A resolvent estimate at noncritical frequencies

Combining the long-range resolvent estimates

Radiation for limits of long-range resolvents

Outgoing flux and vanishing shell mass

Weighted endpoint estimates and polynomial decay

Limiting absorption for long-range differential perturbations

Hamilton trajectories under a long-range force

Smooth long-range phases from Hamilton trajectories

Escaping Lagrangians on regular energy surfaces

Generating functions and the end of a localized force

Modified waves and the direction of escape

Energy-shell factors and outgoing equations

Frequency cutoffs and compact scattering remainders

Commuting coordinates for long-range evolution

Transverse moments and outgoing amplitudes

Truncated operators and stable scattering amplitudes

Spectral transforms and completeness of modified waves

Counting over finite fields and the limit q → 1

Weil's proof for curves and what is missing over the integers

Commutative monoids and their spectra

Monoid schemes

Torified varieties and the limits of monoid schemes

Varieties over the field with one element after Soulé and Connes–Consani

Schemes relative to a symmetric monoidal category

Λ-rings and descent to the field with one element

Generalized rings

Blueprints and blue schemes

Characteristic one and hyperrings

Γ-sets and algebras over the sphere

The arithmetic site

The scaling site

The projective line over F_1 and the ABC conjecture

Group schemes, actions and Hopf algebras

Lie algebras and smoothness of group schemes

Group schemes over a field

Quotients and torsors

Diagonalizable groups and groups of multiplicative type

Abelian varieties

Néron models

Hilbert C*-modules

Adjointable operators

Compact operators, multipliers and the strict topology

Finite projective modules, frames and \(K_0\)

Tensor products and C*-correspondences

Kasparov's stabilization theorem

Fredholm operators and the K₀ index

Unbounded regular operators

Continuous fields over locally compact spaces

Completely positive maps and the KSGNS construction

Imprimitivity bimodules and Morita equivalence

The Rieffel correspondence and induced representations

Stable isomorphism: the Brown–Green–Rieffel theorem

Morita invariance of K-theory and maps induced by correspondences

Morita equivalence of noncommutative tori

ℓ-adic sheaves and their cohomology

The pro-étale site and ℓ-adic complexes

Frobenius morphisms and their action on cohomology

Traces of perfect complexes and Lefschetz numbers

Cycle classes and the Lefschetz fixed-point formula for curves

The trace formula for curves

The trace formula in all dimensions

L-functions, rationality and the functional equation

Lefschetz pencils and vanishing cycles

The fundamental estimate

The Riemann hypothesis over finite fields

Differential operators and the Weyl algebra

D-modules, flat connections and local systems

Good filtrations and the characteristic variety

The Bernstein filtration and holonomic modules over the Weyl algebra

Bernstein-Sato polynomials

Holonomic D-modules and duality

Inverse images

Direct images and the relative de Rham complex

Kashiwara's equivalence and D-modules on singular spaces

Adjunctions, base change and the projection formula

Preservation of holonomicity and minimal extensions

The de Rham functor

Regular singularities

The Riemann-Hilbert correspondence

Equivariant and twisted D-modules

Beilinson-Bernstein localization

D-modules on stacks, ind-schemes and the de Rham prestack

The Fourier transform of D-modules

Constructible complexes on algebraic varieties

Gluing t-structures

The perverse t-structure

Intermediate extensions and intersection complexes

Affine morphisms, Artin vanishing and perverse cohomology

Nearby and vanishing cycles

Weights, purity and semisimplicity over finite fields

The decomposition theorem

Semismall maps and small resolutions

Springer theory

Equivariant perverse sheaves and perverse sheaves on stacks

Perverse sheaves, D-modules and the function-sheaf dictionary

Fibred categories and descent data

Faithfully flat descent

Descending properties of schemes and morphisms

Galois categories

The étale fundamental group

Fundamental groups of proper schemes and the homotopy exact sequence

Specialization maps and tame ramification

Comparison with the topological fundamental group over the complex numbers

Representable functors and the functor of points

Grassmannians

Castelnuovo–Mumford regularity and boundedness

Flattening stratifications

Hilbert and Quot schemes

Formal moduli and Schlessinger's theorem

Deformations of rings and schemes and the naive cotangent complex

Tangent spaces and obstructions for Hilbert and Quot schemes

The Picard functor and the Picard scheme of a curve

Relative divisors and the existence of the Picard scheme

The structure of the Picard scheme

The cotangent complex

From automorphic functions to automorphic sheaves

The moduli stack of bundles

Sheaves and D-modules on Bun_G

Hecke functors and Hecke eigensheaves

Geometric class field theory

The GL_1 case as an equivalence of categories

Profinite groups and ℓ-adic representations

Frobenius elements and determination by traces

The ℓ-adic Tate module of an elliptic curve

Representations of Weil groups

Conductors of Weil-group representations

Local L-factors and epsilon factors

Weil–Deligne representations and Grothendieck’s monodromy theorem

Elliptic curves over local fields and their Weil–Deligne representations

Compatible systems and global L-functions

Modular curves over Q and the Eichler–Shimura congruence

Galois representations of weight-two newforms

Deligne's construction in higher weight and the Ramanujan–Petersson bound

Weight one: the theorem of Deligne and Serre

p-adic Hodge theory and the Fontaine–Mazur conjecture

Residual representations and Serre's modularity theorem

Sites and sheaves

Topoi, morphisms and points

Cohomology on sites

Hypercoverings

Topologies on schemes

The étale site and its points

Pushforward, pullback and finite morphisms

Galois cohomology and the étale cohomology of a field

Brauer groups and Tsen's theorem

The multiplicative group on a curve

Constructible sheaves and extension by zero

Torsion sheaves on curves

The proper base change theorem

Cohomology with compact support

Smooth base change and local acyclicity

Cohomological dimension and the Künneth formula

Poincaré duality for curves

Poincaré duality for smooth varieties

Comparison with singular cohomology

Idèles, ideals and ray class groups

Additive characters, self-dual measures and Poisson summation on the adèles

Quasi-characters and Hecke characters

Tate's local theory at the finite places

Tate's local theory at the infinite places

Tate's global theory: continuation and functional equation

Hecke L-functions and the Dedekind zeta function

Lattices over a local field: the tree of GL_2 and its decompositions

The spherical Hecke algebra of GL_2 and Hecke operators on lattices

Noether's axioms for Dedekind domains

The converse and composition series

Orders and the discriminant theorem

Three differents

Normal integral bases in tame extensions

Algebraic functions of one variable: conductors and adjoints

First cases: characters and the real correspondence

Irreducible representations of general linear groups over a local field

Local factors of pairs and the rank-two converse theorem

The statement of the local Langlands correspondence

Monodromy blocks and the passage from supercuspidals to all parameters

The rank-two correspondence: principal series, Steinberg twists and characters

Two-dimensional Weil representations: induction and projective symmetry

Dihedral supercuspidals, rectifiers and rank-two characterization

Modular forms, elliptic curves and local–global compatibility

Jacquet–Langlands: elliptic classes and division-algebra representations

Harris–Taylor: geometry, numerical counting and existence

Other proofs, functoriality and the formal-degree formula

Beyond general linear groups: parameters and packets

C*-dynamical systems and full crossed products

Reduced crossed products and Fell's absorption principle

Crossed products by discrete groups: expectations, traces and simplicity

Amenability and the equality of full and reduced crossed products

Crossed products by integers and real numbers: Fourier coordinates and mapping tori

Pontryagin duality and the dual action

Induced algebras and Green's imprimitivity theorem

Takai duality

Proper actions, free actions and the orbit space

The Connes–Thom isomorphism I: the Wiener–Hopf extension

The Connes–Thom isomorphism II: surjectivity, naturality and consequences

Exterior equivalence and Connes's construction of the Thom map

Locally compact groupoids and Haar systems

Groupoid C*-algebras: full and reduced

Equivalence of groupoids and Morita equivalence of their C*-algebras

C*-algebras of foliations and their Morita equivalences

The tangent groupoid and deformation to the normal cone

Representations of compact groups: unitarity, complete reducibility and finite dimension

Matrix coefficients and the Peter–Weyl theorem

Fourier analysis and class functions on compact groups

SU(2), SO(3) and spherical harmonics

Compact Lie groups, their Lie algebras and the adjoint representation

Tori and the maximal torus theorem

Roots and the Weyl group of a compact Lie group

The Weyl integration formula

Unitary groups: characters, dimensions and branching

Weights and the theorem of the highest weight

The Weyl character formula for compact connected Lie groups

Casimir operators, Laplacians and tensor products

Homogeneous spaces, induced representations and Gelfand pairs

Representations and complete reducibility

Characters and the orthogonality relations

The group algebra and Fourier analysis on a finite group

Tensor products, duals and real representations

Integrality of characters and Burnside's \(p^a q^b\) theorem

Induced representations and Frobenius reciprocity

Mackey theory and Clifford's theorem

Groups with an abelian normal subgroup: the little-group method

Frobenius groups and Frobenius's theorem

Artin's induction theorem and rationality

Brauer's induction theorem

Consequences of Brauer's theorem: characterization of characters and splitting fields

The symmetric groups I: Young tableaux and Young symmetrizers

The symmetric groups II: branching, Jucys–Murphy elements and Young's seminormal form

The symmetric groups III: characters and symmetric functions

Schur–Weyl duality

Representations of GL₂ over finite fields

Spectra of rings

Localization, local properties and support

Noetherian and Artinian rings

Associated primes and primary decomposition

Integral extensions: lying over, going up and going down

The Nullstellensatz and Jacobson rings

Tor and flat modules

Faithful flatness and the local criterion for flatness

Krull dimension and Noether normalization

Graded modules and Hilbert–Samuel functions

Dimension theory of Noetherian local rings

Regular sequences, depth and Cohen–Macaulay modules

Projective dimension and the Auslander–Buchsbaum formula

Regular local rings

Discrete valuation rings, normal rings and Serre's criterion

Kähler differentials

Formally smooth, unramified and étale ring maps

Smooth algebras over a field and the Jacobian criterion

Completion

Henselian local rings and henselization

Absolute values, valuations and Ostrowski's theorem

Hensel's lemma, squares and roots of unity in p-adic fields

Extensions of complete valued fields

Places of number fields in extensions and the product formula

Local fields: classification and Haar measure

Unramified and totally ramified extensions

Tame ramification and the tame Galois group

Ramification groups and the different of a local extension

Herbrand's function and the upper numbering

The multiplicative group of a local field

Cyclotomic, quadratic and Kummer extensions of local fields

Local cohomology

Local duality and finiteness

Cohomological dimension, vanishing and connectedness

Formal geometry along a closed subscheme

Lefschetz theorems for finite étale covers

Picard groups and Grothendieck–Lefschetz

Purity of the branch locus

Specialization and tame ramification

Adèles for GL₂ and strong approximation

From modular forms to adelic functions

Automorphic forms, modules and Hilbert constituents

Why the cuspidal spectrum is discrete

Smooth local representations and the Hecke-module dictionary

Normalized induction and Jacquet modules

Whittaker models, Kirillov models and the local classification

Newvectors, conductors and the passage to classical newforms

Local L-factors, epsilon factors and the local converse theorem

Supercuspidal representations from compact induction

Real and complex representations: weights, gamma factors and classical forms

Restricted tensor products and the tensor product theorem

Unramified representations of GL₂(F) and Satake parameters

Global Whittaker functions and the L-function of a cuspidal representation

Multiplicity one, strong multiplicity one and the converse theorem

Eisenstein series on GL₂(A) and the continuous spectrum

Quaternion algebras and their automorphic forms

The trace formula for a compact quotient

The Jacquet–Langlands correspondence

Transformation groups and their parameters

The fundamental differential equations

One-parameter groups

Complete systems and invariants

The second fundamental theorem and the group of parameters

The third fundamental theorem

The adjoint group, composition and isomorphism

Transitivity and primitivity

Prolongation, differential invariants and the projective group

Contact transformations and first-order equations

Lie algebras of vector fields on the line and in the plane

The Riemann-Helmholtz space problem

Algebraic spaces

The bootstrap theorem

Properties and morphisms of algebraic spaces

Stacks in groupoids

Algebraic stacks

Quotient stacks and Deligne–Mumford stacks

Artin's axioms

Moduli stacks are algebraic

The stack of curves

Stable reduction and properness

Simplicial sets, nerves and Kan complexes

Test categories and Grothendieck's homotopy programme

Idempotents, projections and their equivalences

Vector bundles and finitely generated projective modules

The Grothendieck group and \(K_0\) of a unital algebra

Nonunital algebras: unitization, relative classes and half-exactness

Matrix stability, stability and continuity of \(K_0\)

Invertibles, unitaries and \(K_1\)

The index map and the exact sequence at \(K_0\)

Suspension, higher K-groups and the long exact sequence

Toeplitz operators and the index theorem on the circle

Bott periodicity

The six-term exact sequence and the exponential map

Topological K-theory of spaces, pairs and vector bundles

Traces, states and the pairing with K_0

Determinants of traces and the pairing with K_1

Smooth subalgebras and the density theorem

Inductive limits and the K-theory of AF and AT algebras

The mapping torus

The Pimsner–Voiculescu exact sequence

Traces on integer crossed products and their K-theory ranges

Irrational rotation algebras

Commutative and noncommutative tori

Cuntz algebras

Cuntz–Krieger algebras, minimal systems and projectionless algebras

The upper half-plane and the modular group

Congruence subgroups, cusps and elliptic points

Modular curves and their genus

Modular forms, lattice functions and Eisenstein series

The valence formula and the ring of modular forms of level one

Dimension formulas for congruence subgroups

The Petersson inner product and Poincaré series

Hecke operators of level one

Hecke operators for Γ_0(N) and Γ_1(N)

Oldforms, newforms and the theory of Atkin, Lehner and Li

The L-function of a cusp form

Twists, level N and Hecke's converse theorem

Theta functions and sums of squares

Theta series of lattices

Group cohomology of Γ and the Eichler–Shimura isomorphism

Modular symbols and the algebraicity of Hecke eigenvalues

Complex tori, elliptic curves and the moduli interpretation of modular curves

Reductive groups, root data and the L-group

The Satake isomorphism for unramified groups and unramified L-factors

Automorphic representations and automorphic L-functions

L-functions for GL_n: Godement–Jacquet and Rankin–Selberg

The principle of functoriality

The function-field case: Drinfeld and L. Lafforgue

Groups with operators

Semisimple rings and Wedderburn's theorem

Representations, characters and the group determinant

Central simple algebras and the Brauer group

Crossed products and factor systems

Hilbert 90 in Noether's form and Galois descent

The principal genus theorem

Lie algebras: definitions, examples and first constructions

Nilpotent and solvable Lie algebras: Engel's and Lie's theorems

The Killing form and Cartan's criteria

Complete reducibility: Casimir elements and Weyl's theorem

Representations of sl(2)

Low-degree cohomology, Whitehead's lemmas and the Levi decomposition

The root space decomposition of a semisimple Lie algebra

Root systems and their Weyl groups

Cartan matrices, Dynkin diagrams and the classification of root systems

Cartan subalgebras and their conjugacy

The isomorphism theorem and Serre's theorem

The simple Lie algebras: classical models and the exceptional algebras

The universal enveloping algebra and the Poincaré–Birkhoff–Witt theorem

Weights, Verma modules and the theorem of the highest weight

The centre of the enveloping algebra and Harish-Chandra's theorem

Weyl's character formula and the multiplicity formulas

Compact real forms and Weyl's unitary trick

Category O and the Kazhdan–Lusztig conjecture

Sheaf kernels and cotangent correspondences

Constructible gluing on an interval

Directional tests at a constructible boundary

Constructible sheaves on a triangulation

Kernels that preserve chosen cotangent directions

Adjoints of localized sheaf kernels

Dual kernels and an unchanged parameter

Directional morphisms through a sheaf kernel

When a kernel quantizes a contact transformation

Real and complex ball kernels

Local existence of contact kernel equivalences

Microlocal composition at prescribed covectors

Coefficients in a quantization of the identity

Normal forms and the shift of a submanifold transform

Composing hypersurface kernels with their shifts

Pure and simple sheaves from directional tests

How simple sheaf shifts change along a Lagrangian

Tensor and Hom types at independent covectors

The type and shift of a transverse kernel composition

Direct and inverse images of a sheaf type

Projective incidence as a contact kernel

The partial Legendre kernel and its adjoint shift

Fourier-Sato transport of microlocal morphisms

Subanalytic sets and limiting tangent directions

Conic subanalytic images and isotropic dimension

Isotropic cotangent transport and discrete critical values

Finite conormal closures and generic base directions

Involutive subsets of subanalytic isotropic sets

Boundary forms and Lagrangian normal cones

Limiting cotangent sums and characteristic inverse images

Microlocal stratifications by removing bad loci

Unshared conormal directions and dimension filtrations

Generic squared distance and cotangent transversality

Constructibility from microsupport and perfect stalks

Derived constructibility through common triangulations

Weak constructibility under sheaf operations

Small balls, central fibres and supported cohomology

Perfect coefficients on compact fibres

Constructible costalks and Verdier duality

Perfect operations and finite microlocal coefficients

Natural duality for specialization and microlocal Hom

Complex conicity and analytic Lagrangian closures

Analytic normal cones through complex deformation

Analytic conormal covers and singular involutivity

Complex microlocal stratifications and constructibility

Holomorphic operations and complex Fourier symmetries

Nonproper holomorphic pushforwards through cutoffs

Local holomorphic pushforwards over a complex curve

Complex-constructible sheaves and missing derived classes

Nearby cycles and the two monodromy triangles

Proper pushforwards of nearby and vanishing cycles

Nearby cycles through the normal deformation

Complex nearby cycles as normal and conormal sections

Quadratic cycles and the holomorphic microsupport test

Vanishing cycles as positive real support

Whitney secants and the microlocal stratification condition

Duality maps for constructible inverse and direct images

Constructibility through smooth cutoffs and microlocal properness

Constructible models in one cotangent direction

Closed conormal unions at singular points

The dualizing complex from oriented simplices

Local systems across an analytic boundary

Reading the geometry and cycle literature

Constructible traces and local Euler indices

Closed supports and evaluated proper transport

Proper characteristic classes and the compact index

Subanalytic chains and closed cycle supports

Supports, products and proper images of chains

The dualizing resolution by subanalytic chains

Intersections of supported subanalytic cycles

Lagrangian cycles and proper cotangent images

Pulling back Lagrangian cycles through a graph

Continuous sections and supported cycle intersections

Transverse pullback of normalized conormal cycles

Characteristic cycles from supported microlocal identities

Transporting characteristic cycles through a graph

Integer coefficients and additive characteristic cycles

Variational symmetries and Noether's first theorem

Infinite symmetry groups and Noether's second theorem

The formal calculus of variations and differential invariants

The explicit formula with general test functions

Explicit formulas for Dirichlet L-functions and primes in progressions

Guinand's formula

Li's criterion

Extensions of C*-algebras and the Busby invariant

Ext groups, absorption and Brown-Douglas-Fillmore theory

Fredholm modules and analytic K-homology

The index pairing between K-theory and K-homology

Graded C*-algebras, Clifford algebras and graded Hilbert modules

Dirichlet series and Euler products

Counting primes by elementary means: Chebyshev and Mertens

The Gamma function and Stirling's formula

Poisson summation, theta, and the functional equation

Entire functions of order one and the Hadamard product of ξ

Growth in the critical strip: convexity and the Lindelöf hypothesis

Exponential sums and a subconvex bound

Nonvanishing on the line σ = 1 and the zero-free region

The prime number theorem

The Riemann–von Mangoldt formula

Perron's formula and the explicit formula for ψ(x)

The diagonal and separated morphisms

Finiteness of morphisms

Limits of schemes and Noetherian approximation

Valuation rings and the valuative criterion of separatedness

Affine morphisms, relative Spec, and finite morphisms

Quasi-finite morphisms and Chevalley’s theorem

Proper morphisms and the valuative criterion of properness

Very ample invertible sheaves, Segre and Veronese embeddings

Ample invertible sheaves

Projective morphisms and Chow’s lemma

Dimension of fibres

Zariski's Main Theorem

Normalization

Rational maps and birational morphisms

Effective Cartier divisors and invertible sheaves

Blowing up

Sheaves on topological spaces

Ringed spaces and sheaves of modules

Quasi-coherent, coherent and locally free modules

Affine schemes

Schemes, gluing and immersions

The functor of points and representability

Fibre products and base change

Properties of schemes

Quasi-coherent sheaves on schemes

Closed subschemes and scheme-theoretic images

Proj of a graded ring

Projective space, relative Proj and maps to projective space

Cohomology of sheaves on ringed spaces

Čech cohomology

Cohomology of affine schemes and Serre's criterion

Cohomology of projective space

Coherent sheaves on projective schemes: Serre's theorems

Coherence of higher direct images under proper morphisms

Euler characteristics and Hilbert polynomials

Base change and the Grothendieck complex

Semicontinuity and Grauert's theorem

The theorem on formal functions

Covariants of binary and ternary forms

Forms in several variables and Hilbert finiteness

Invariants of finite groups and Noether's degree bound

Polarization and invariants of many vectors

Finiteness for finite groups in every characteristic

Dirichlet characters

Dirichlet's theorem on primes in arithmetic progressions

Gauss sums

Profinite groups and infinite Galois theory

Cohomology of cyclic groups and the Herbrand quotient

Hilbert's Theorem 90 and Kummer theory

Frobenius lifts and abstract reciprocity

The reciprocity law and the class field correspondence

Local reciprocity and norm groups

Formal groups and Lubin–Tate modules

Lubin–Tate division fields

Explicit local reciprocity and the existence theorem

Abelian ramification, conductors and Hasse–Arf

Hilbert symbols and local conics

Chain conditions and irreducible decompositions

Primary ideals and isolated components

Modules, staircases and elementary divisors

Generic zeros and absolute primality

Resultant forms and elimination

Spherical Hecke algebras as functions on the affine Grassmannian

Linear forms in logarithms: the problem, the trivial bounds, and effectivity

Baker's theorem I: building an auxiliary function

Baker's theorem II: extrapolation and the end of the proof

Effective lower bounds I: Baker's theorem and its arithmetic tools

Effective lower bounds II: proof of Baker's theorem

Two logarithms I: interpolation determinants

Algebraic and transcendental numbers; Liouville's theorem

Heights of algebraic numbers

Hermite and the transcendence of e

Lindemann, pi and the Lindemann–Weierstrass theorem

Siegel's lemma, analytic estimates and the six exponentials theorem

Gelfond–Schneider and Hilbert's seventh problem

The Schneider–Lang criterion

Weierstrass functions, elliptic values and periods

Subfields and subrings of rational function fields

Noether's problem and generic polynomials

Absolute irreducibility and Noether forms

Vector bundles and their constructions

Grassmannians and classifying maps

Local tools for bundles and transport

Principal bundles and associated bundles

Connections and parallel transport

Curvature and holonomy groups

dependencies/foundations-of-von-neumann-algebras/src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md

dependencies/foundations-of-von-neumann-algebras/src/../../../course/measure-and-hilbert-space-tools.md

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dependencies/foundations-of-von-neumann-algebras/src/the-spectral-theorem-for-bounded-self-adjoint-operators.md

dependencies/foundations-of-von-neumann-algebras/src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md

dependencies/foundations-of-von-neumann-algebras/src/../../../course/haar-measure-on-locally-compact-groups.md

dependencies/foundations-of-von-neumann-algebras/src/abelian-operator-algebras.md

dependencies/NT-ADL/src/../../NT-LOC/src/NT-LOC-02.md

dependencies/NT-ADL/src/../../../course/haar-measure-on-locally-compact-groups.md

dependencies/NT-ADL/src/../../foundations-of-von-neumann-algebras/src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md

dependencies/NT-ADL/src/../../function-algebras-and-approximation/src/the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity.md