Math Library

Mathematical definitions and deductions, with worked examples and counterexamples. Browse the complete pairs below, organized by subject.

Abstract Algebra

Reading order

78 pages

Page	Companion	Dependency level
Binary Operations, Monoids, Groups and Subgroups	Examples & counterexamples	0
Cosets, Index and Lagrange's Theorem	Examples & counterexamples	1
Rings, Subrings, Integral Domains and Fields	Examples & counterexamples	1
Normal Subgroups and Quotient Groups	Examples & counterexamples	2
Group Homomorphisms and the Isomorphism Theorems	Examples & counterexamples	3
Cyclic Groups and Direct Products	Examples & counterexamples	4
Free Groups and Presentations	Examples & counterexamples	4
Group Actions, Orbits, Stabilisers and Cayley's Theorem	Examples & counterexamples	4
Ideals, Quotient Rings and the Isomorphism Theorems for Rings	Examples & counterexamples	4
Composition Series, the Jordan–Hölder Theorem and Solvable Groups	Examples & counterexamples	5
Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation	Examples & counterexamples	5
Free Products and Amalgamation	Examples & counterexamples	5
Modules, Submodules, Quotient Modules and the Isomorphism Theorems	Examples & counterexamples	5
The Fundamental Theorem of Finite Abelian Groups	Examples & counterexamples	5
Free Modules, Exact Sequences, Projective and Injective Modules	Examples & counterexamples	6
Polynomial Rings, the Division Algorithm and Roots	Examples & counterexamples	6
Semidirect Products, Automorphism Groups and Split Extensions	Examples & counterexamples	7
Simple Field Extensions and the Construction of the Complex Numbers	Examples & counterexamples	7
Tensor Products of Modules	Examples & counterexamples	7
The Field of Fractions and Localisation	Examples & counterexamples	7
Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem	Examples & counterexamples	8
Heisenberg Lie Algebras and Polynomial Weyl Algebras	Examples & counterexamples	8
Jordan Algebras, Spin Factors, and Alternative Examples	Examples & counterexamples	8
Splitting Fields	Examples & counterexamples	8
Sylow's Theorems, p-Groups and Nilpotent Groups	Examples & counterexamples	8
Algebra, Modules, and Tensor Symmetry Interfaces	Examples & counterexamples	9
Algebraic Extensions, Extension Degree, and Finite Fields	Examples & counterexamples	9
Modules over a Principal Ideal Domain and the Canonical Forms	Examples & counterexamples	9
Symmetric Polynomials and the Fundamental Theorem of Symmetric Functions	Examples & counterexamples	9
The Group Algebra and Representations of Finite Groups	Examples & counterexamples	9
Algebraic Closure, Embeddings, and Separability	Examples & counterexamples	10
Maschke's Theorem, Complete Reducibility and the Structure of k[G]	Examples & counterexamples	10
The Galois Correspondence	Examples & counterexamples	11
Finite Fields and Cyclotomic Extensions	Examples & counterexamples	12
The Fundamental Theorem of Algebra	Examples & counterexamples	12
Finite Averaging and Character-Theory Prerequisites	Examples & counterexamples	13
Solvability by Radicals and Kummer Theory	Examples & counterexamples	13
Characters and the Orthogonality Relations	Examples & counterexamples	14
Induced Representations, Frobenius Reciprocity and Applications	Examples & counterexamples	15

Algebraic Geometry

Reading order

74 pages

Page	Companion	Dependency level
Affine Algebraic Sets and Coordinate Rings	Examples & counterexamples	0
Algebraic Differentials Separability and Smooth Local Presentations	Examples & counterexamples	0
Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps	Examples & counterexamples	0
Groups of Multiplicative Type and Arithmetic Tori	Examples & counterexamples	0
Mixed Hodge Structures And Strictness	Examples & counterexamples	0
Analytic Constructible Sheaf Foundations	Examples & counterexamples	1
Classical Complex Algebraic Actions and Affine Embeddings	Examples & counterexamples	1
Morphisms Local Rings and Rational Maps of Affine Varieties	Examples & counterexamples	1
Projective Algebraic Sets Projective Morphisms and Cones	Examples & counterexamples	1
Products Segre and Veronese Embeddings and Grassmannians	Examples & counterexamples	2
Dimension Constructible Images and Dimensions of Fibres	Examples & counterexamples	3
Zariski Tangent Spaces, Regular Points, Smoothness, and Bertini	Examples & counterexamples	4
Constructible Incidence And Cochain Models	Examples & counterexamples	5
Etale Covers and the Etale Fundamental Group	Examples & counterexamples	5
Intersection Products on Smooth Projective Surfaces	Examples & counterexamples	5
Nonaffine Algebraic Groups, Barsotti-Chevalley, and Abelian Varieties	Examples & counterexamples	5
Normal Varieties, Normalization, and Zariski's Main Theorem	Examples & counterexamples	5
Plane Curves, Local Intersection Multiplicity, and Bézout	Examples & counterexamples	5
Smooth-Projective Serre Duality and Flag-Variety Line Bundles	Examples & counterexamples	5
Abelian Varieties, Base Change, and Arithmetic Models	Examples & counterexamples	6
Chow Groups, Intersection Products, and Grothendieck-Riemann-Roch	Examples & counterexamples	6
Coherent Duality on Projective Cohen-Macaulay Schemes	Examples & counterexamples	6
Complex Algebraic Stratified Topology	Examples & counterexamples	6
Point Blowup Resolution on Arbitrary Regular Surfaces	Examples & counterexamples	6
Reductive Affine Invariant Theory and Geometric Quotients	Examples & counterexamples	6
Surface Riemann-Roch and the Hodge Index Theorem	Examples & counterexamples	6
Unipotent and Solvable Groups and Borel Fixed Points	Examples & counterexamples	6
Affine Perverse Vanishing And Universal Hyperplanes	Examples & counterexamples	7
Birational Morphisms, Contractions, and Surface Singularities	Examples & counterexamples	7
Higher-Dimensional Resolution of Singularities	Examples & counterexamples	7
Projective GIT from Linearized Line Bundles	Examples & counterexamples	7
Split Reductive Root Systems, Bruhat Cells, and Parabolics	Examples & counterexamples	7
Highest Weights and Rational Representations of Split Reductive Groups	Examples & counterexamples	8
Logarithmic Hodge Complexes And Smooth Open Weights	Examples & counterexamples	8
Geometric Fixed Parts And Gauss Manin Semisimplicity	Examples & counterexamples	9
Proper Hyperresolutions And Mixed Hodge Restriction Images	Examples & counterexamples	9
Projective Perverse Hard Lefschetz And Decomposition	Examples & counterexamples	10

Algebraic Topology

Reading order

52 pages

Page	Companion	Dependency level
Simplicial Complexes and Simplicial Homology	Examples & counterexamples	0
Simplicial Subdivision and Simplicial Approximation	Examples & counterexamples	1
Singular Chains and Singular Homology	Examples & counterexamples	1
Relative Homology Excision and Mayer Vietoris	Examples & counterexamples	2
Cw Complexes and Cellular Homology	Examples & counterexamples	3
Homology Axioms Degree and Classical Applications	Examples & counterexamples	4
Higher Homotopy Groups and Cofiber Sequences	Examples & counterexamples	5
Singular Cohomology and Coefficient Theorems	Examples & counterexamples	5
Cup Cap Cross Products and Cohomology Rings	Examples & counterexamples	6
Fibrations Fiber Bundles and Homotopy Exact Sequences	Examples & counterexamples	6
Hurewicz Whitehead Freudenthal and Cw Approximation	Examples & counterexamples	7
Orientations Poincare Lefschetz and Alexander Duality	Examples & counterexamples	7
Bocksteins Steenrod Squares and Cohomology Operations	Examples & counterexamples	8
Local Coefficients, Twisted Homology, and Duality	Examples & counterexamples	8
Simple Homotopy, Whitehead Groups, and Torsion	Examples & counterexamples	8
Spectra and Stable Homotopy Groups	Examples & counterexamples	8
Obstruction Theory, Postnikov Towers, and Classifying Spaces	Examples & counterexamples	9
The Serre Spectral Sequence and Applications	Examples & counterexamples	10
Topological Vector Bundles and Grassmannian Classification	Examples & counterexamples	10
Complex Topological K Theory and Bott Periodicity	Examples & counterexamples	11
Leray–Hirsch, the Thom Isomorphism, and Gysin Sequences	Examples & counterexamples	11
Generalized Cohomology and the Atiyah Hirzebruch Spectral Sequence	Examples & counterexamples	12
Stiefel Whitney and Euler Classes by Universal Constructions	Examples & counterexamples	12
Chern and Pontryagin Classes by Splitting and Complexification	Examples & counterexamples	13
Real Clifford Modules, KO Theory, and ABS Orientations	Examples & counterexamples	13
Thom Spectra and Unoriented Bordism Detection	Examples & counterexamples	14

Bayesian Computational Statistics

Reading order

118 pages

Page	Companion	Dependency level
Adaptive empirical-Bayes credible-set calibration and failure	Examples & counterexamples	0
Banach Gaussian priors, RKHS support, and concentration	Examples & counterexamples	0
Bayesian models, priors, and posterior laws	Examples & counterexamples	0
Computational foundations: factorization, KL, and optimization	Examples & counterexamples	0
Control variates, antithetics, and stratification	Examples & counterexamples	0
Cross-validation risk, model selection, and comparison uncertainty	Examples & counterexamples	0
Deterministic quasi-Monte Carlo and discrepancy	Examples & counterexamples	0
Exact permutation tests and random-permutation Monte Carlo	Examples & counterexamples	0
General-state Markov kernels, invariance, and reversibility	Examples & counterexamples	0
General-state MCMC ergodicity and mixing	Examples & counterexamples	0
Generalized Bayes loss updates and temperature	Examples & counterexamples	0
Gibbs sampling and scan schemes	Examples & counterexamples	0
Hellinger geometry and posterior consistency	Examples & counterexamples	0
IID bootstrap and jackknife for smooth statistics	Examples & counterexamples	0
IID Monte Carlo integration and error	Examples & counterexamples	0
IID posterior contraction by entropy and prior mass	Examples & counterexamples	0
Laplace approximation and deterministic quadrature	Examples & counterexamples	0
Parametric Bernstein–von Mises limits and misspecification	Examples & counterexamples	0
Posterior consistency at a KL projection	Examples & counterexamples	0
Posterior Prediction, Credible Sets, and Bayes Actions	Examples & counterexamples	0
Rejection sampling	Examples & counterexamples	0
Resampling schemes and the sampling-unit contract	Examples & counterexamples	0
State-space models and exact Bayesian filtering	Examples & counterexamples	0
Statistical, resampling, simulation, and numerical error ledger	Examples & counterexamples	0
Synthetic likelihood and indirect inference	Examples & counterexamples	0
Weak nonparametric Bernstein–von Mises limits	Examples & counterexamples	0
Adaptive MCMC convergence limits	Examples & counterexamples	1
Bootstrap failure, m-out-of-n methods, and subsampling	Examples & counterexamples	1
Feynman–Kac targets and sequential particle filters	Examples & counterexamples	1
Gaussian-process posterior contraction	Examples & counterexamples	1
Identifiability and approximation bias in computational inference	Examples & counterexamples	1
Latent-variable EM and likelihood ascent	Examples & counterexamples	1
Metropolis–Hastings kernels	Examples & counterexamples	1
Moving and tapered block bootstrap for dependent data	Examples & counterexamples	1
Multilevel Monte Carlo and cost theorem	Examples & counterexamples	1
Ordinary and self-normalized importance sampling	Examples & counterexamples	1
Posterior propriety and dominated Bayes formula	Examples & counterexamples	1
Randomized quasi-Monte Carlo and replicate error	Examples & counterexamples	1
Studentized permutation tests under weak parameter nulls	Examples & counterexamples	1
Approximate Bayesian computation and summary-based inference	Examples & counterexamples	2
Conjugate families and Gaussian conditioning	Examples & counterexamples	2
Finite-horizon SMC error, CLTs, and degeneracy	Examples & counterexamples	2
Particle MCMC: PMMH and particle Gibbs	Examples & counterexamples	2
ABC-MCMC and ABC-sequential Monte Carlo	Examples & counterexamples	3
Dirichlet-process priors and finite-partition updates	Examples & counterexamples	3
Exact linear-Gaussian Kalman filtering	Examples & counterexamples	3
Gaussian-process priors and finite posterior laws	Examples & counterexamples	3
Hamiltonian Monte Carlo and leapfrog correction	Examples & counterexamples	3
Hierarchical Bayes and partial pooling	Examples & counterexamples	3
MCMC diagnostics and finite-run limitations	Examples & counterexamples	3
Variational inference and the evidence lower bound	Examples & counterexamples	3
Dirichlet normal-mixture posterior contraction	Examples & counterexamples	4
Fixed-interval Gaussian smoothing and RTS recursion	Examples & counterexamples	4
Functional BvM limits and credible-set calibration	Examples & counterexamples	4
Langevin algorithms: ULA bias and MALA exactness	Examples & counterexamples	4
Marginal likelihoods, Bayes factors, and empirical Bayes	Examples & counterexamples	4
MCMC central limit theorems, MCSE, and long-run variance	Examples & counterexamples	4
Particle smoothing, backward simulation, and Rao–Blackwellized filters	Examples & counterexamples	4
Data assimilation: 3DVAR, 4DVAR, and ensemble filters	Examples & counterexamples	5

Braid Groups

Reading order

38 pages

Page	Companion	Dependency level
Garside Structure, Normal Forms, and the Center	Examples & counterexamples	0
Geometric Braids and Artin Generators	Examples & counterexamples	0
Graded Quiver Algebras and Derived Tensor Functors	Examples & counterexamples	0
Ordered and Unordered Configuration Spaces	Examples & counterexamples	0
Braids as Fundamental Groups of Configuration Spaces	Examples & counterexamples	1
Type-A Soergel Bimodules and Hecke Categorification	Examples & counterexamples	1
Punctured Disks, Mapping Classes, and Point Pushing	Examples & counterexamples	2
Pure Braids, Fadell–Neuwirth, and Asphericity	Examples & counterexamples	3
Artin Presentation Completeness and Braid Combing	Examples & counterexamples	4
The Artin Action on a Free Group	Examples & counterexamples	5
Lawrence–Krammer–Bigelow Representations and Linearity	Examples & counterexamples	6
Oriented Links, Braid Closures, and Markov Equivalence	Examples & counterexamples	6
The Burau Representations	Examples & counterexamples	6
Categorical Braid Actions and Decategorification	Examples & counterexamples	7
Hecke Markov Traces and Polynomial Link Invariants	Examples & counterexamples	7
Yang–Baxter Operators and Quantum Braid Representations	Examples & counterexamples	7
Rouquier Complexes and Categorical Braid Relations	Examples & counterexamples	8
Matrix Factorizations and Khovanov–Rozansky Link Homology	Examples & counterexamples	9
Hochschild Homology and Triply-Graded Link Homology	Examples & counterexamples	10

Category O and Geometric Representation Theory

Reading order

24 pages

Page	Companion	Dependency level
Equivariant Borel–Moore Correspondence Convolution and Constructible Ext	Examples & counterexamples	0
Kazhdan–Lusztig Bases, Polynomials, and Cells	Examples & counterexamples	0
Proper Cohomological Descent And Constructible Hypercovers	Examples & counterexamples	0
Rational Cherednik Algebras and Their Category O	Examples & counterexamples	0
Soergel Intersection Forms and Hodge Theory	Examples & counterexamples	1
Projective Functors and the Soergel Functor in Category O	Examples & counterexamples	2
Beilinson–Bernstein Localization and Category O	Examples & counterexamples	3
The Kazhdan–Lusztig Character and Multiplicity Theorem	Examples & counterexamples	3
Singular and Parabolic Kazhdan–Lusztig Theory	Examples & counterexamples	4
Parabolic Category O and Fock-Space Bases	Examples & counterexamples	5
Arakawa–Suzuki Functors and Higher-Level Schur–Weyl Duality	Examples & counterexamples	6
Category O, KLR, and Hecke Categorification Route	Examples & counterexamples	7

Category Theory

Reading order

42 pages

Page	Companion	Dependency level
Categories, Functors and Natural Transformations	Examples & counterexamples	0
Universal Properties, Representables and the Yoneda Lemma	Examples & counterexamples	1
Limits and Colimits	Examples & counterexamples	2
Adjunctions Units and Counits	Examples & counterexamples	3
Ends Coends and Weighted Limits	Examples & counterexamples	4
Preadditive and Additive Categories and Biproducts	Examples & counterexamples	4
Reflective Subcategories and the Adjoint Functor Theorems	Examples & counterexamples	4
Abelian Categories	Examples & counterexamples	5
Monads Comonads and Their Algebras	Examples & counterexamples	5
Kan Extensions Density and the Free Cocompletion	Examples & counterexamples	6
Monadicity and Beck's Theorem	Examples & counterexamples	6
Monoidal Categories and Monoidal Functors	Examples & counterexamples	6
Subobject Lattices Generators and the Grothendieck Axioms	Examples & counterexamples	6
Closed Monoidal Categories and the Internal Hom	Examples & counterexamples	7
Exactness and the Member Calculus	Examples & counterexamples	7
Strictification and Mac Lanes Coherence Theorem	Examples & counterexamples	7
Braided and Symmetric Monoidal Categories	Examples & counterexamples	8
The Diagram Lemmas in an Abelian Category	Examples & counterexamples	8
Duality and Rigidity in Monoidal Categories	Examples & counterexamples	9
Enriched Categories	Examples & counterexamples	9
Tensor and Fusion Categories	Examples & counterexamples	10

Combinatorics

Reading order

104 pages

Page	Companion	Dependency level
Finite Counting, Factorials and Binomial Coefficients	Examples & counterexamples	0
Finite Abelian Characters for Combinatorics	Examples & counterexamples	1
Finite Probability Spaces and Random Variables	Examples & counterexamples	1
Formal Power Series	Examples & counterexamples	1
Inclusion–Exclusion, the Pigeonhole Principle and Double Counting	Examples & counterexamples	1
Skolem construction and design prerequisites	Examples & counterexamples	1
Chains, Antichains, Sperner and Dilworth	Examples & counterexamples	2
Combinatorial Classes and the Symbolic Method	Examples & counterexamples	2
Graphs, Walks and Connectivity	Examples & counterexamples	2
Linear Recurrences and Rational Generating Functions	Examples & counterexamples	2
Counting up to Symmetry: Burnside and Pólya	Examples & counterexamples	3
Eulerian and Hamiltonian Graphs	Examples & counterexamples	3
Graph Colouring	Examples & counterexamples	3
Incidence Algebras and Möbius Inversion	Examples & counterexamples	3
Induced Subgraphs and Hereditary Graph Classes	Examples & counterexamples	3
Lattice Paths and Catalan Numbers	Examples & counterexamples	3
Linear Algebra Methods in Combinatorics	Examples & counterexamples	3
Ramsey Theory	Examples & counterexamples	3
Set Partitions, Stirling Numbers and Exponential Generating Functions	Examples & counterexamples	3
Trees, Forests and Spanning Trees	Examples & counterexamples	3
Algebraic and Spectral Graph Theory	Examples & counterexamples	4
Block Designs and Finite Projective Planes	Examples & counterexamples	4
Extremal Graph Theory	Examples & counterexamples	4
Integer Partitions and the Twelvefold Way	Examples & counterexamples	4
Matchings, Covers, Menger and Network Flows	Examples & counterexamples	4
Finite Probability and the Probabilistic Method	Examples & counterexamples	5
Plane Graphs, Euler's Formula and the Five Colour Theorem	Examples & counterexamples	5
Regular Pairs and Induced Counting	Examples & counterexamples	6
The Erdős–Hajnal Property and Homogeneous Sets	Examples & counterexamples	6
Modules, Substitution and Prime Graphs	Examples & counterexamples	7
Sparse Restricted Subgraphs and the Rödl–Nikiforov Theorems	Examples & counterexamples	7
Blockades, Combs and Pattern Graphs	Examples & counterexamples	8
Bull-Free Graphs and the Erdős-Hajnal Property	Examples & counterexamples	8
Polynomial Rödl, Virality and Erdős–Hajnal Equivalence	Examples & counterexamples	8
Pure Pairs, Forests and Path–Antipath Classes	Examples & counterexamples	9
Quantitative Induced Density and the Log-Log Step	Examples & counterexamples	9
Quotient Blockades and Mixing Relations	Examples & counterexamples	9
Classical and Log-Log Erdős–Hajnal Bounds	Examples & counterexamples	10
Cographs, Perfect Patterns and Pure Pairs	Examples & counterexamples	10
The Five-Cycle and the Erdős-Hajnal Property	Examples & counterexamples	11
Iterative Sparsification and the Five-Vertex Path	Examples & counterexamples	12
Star Expansions and the Erdős-Hajnal Property	Examples & counterexamples	12
Iterative Restriction and Comb-Extraction Lemmas	Examples & counterexamples	13
Small-Graph Erdős-Hajnal Consequences	Examples & counterexamples	13
Leaf Reducibility and Wonderful Families	Examples & counterexamples	14
Comb Structure in co-Bird-Free Graphs	Examples & counterexamples	15
Generalized Niceness and Reduction Outcomes	Examples & counterexamples	15
From Generalized Niceness to Erdős-Hajnal	Examples & counterexamples	16
Property (*) and Comb Outcomes	Examples & counterexamples	16
The Structural Criterion for Property (*)	Examples & counterexamples	17
Comb Structure in co-E-Free Graphs	Examples & counterexamples	18
The Erdős–Hajnal Theorems for the E-Graph and Bird	Examples & counterexamples	19

Commutative Algebra

Reading order

44 pages

Page	Companion	Dependency level
Noetherian Rings and Hilbert Basis	Examples & counterexamples	0
Localisation of Modules and Support	Examples & counterexamples	1
Prime Spectra and Radicals	Examples & counterexamples	1
Artinian Rings and Length	Examples & counterexamples	2
Associated Primes and Primary Decomposition	Examples & counterexamples	2
Flatness and Faithful Flatness	Examples & counterexamples	2
Integral Extensions and Going Up	Examples & counterexamples	2
Koszul Complexes and Regular Sequences	Examples & counterexamples	2
Noether Normalisation and Nullstellensatz	Examples & counterexamples	3
Valuation Rings and Discrete Valuation Rings	Examples & counterexamples	3
Dedekind Domains and Ideal Classes	Examples & counterexamples	4
Krull Dimension and Height Theorems	Examples & counterexamples	4
Normalization Finiteness for Affine Domains	Examples & counterexamples	4
Zariski Topology on Prime Spectra	Examples & counterexamples	4
Algebraic Zariski Main for Quasi-Finite Morphisms	Examples & counterexamples	5
Rees Modules Artin Rees and Hilbert Samuel Theory	Examples & counterexamples	5
Homogeneous Resultants and Projective Intersection Length	Examples & counterexamples	6
Inverse Limits and Noetherian Completion	Examples & counterexamples	6
Koszul Euler Characteristics and Hilbert–Samuel Multiplicity	Examples & counterexamples	6
Depth and Cohen Macaulay Modules	Examples & counterexamples	7
Henselian Rings and Equicharacteristic Cohen Structure	Examples & counterexamples	7
Regular Local Rings and Homological Dimension	Examples & counterexamples	8

Complex Analysis

Reading order

100 pages

Page	Companion	Dependency level
Complex Coordinate and Path Transport	Examples & counterexamples	0
Complex Differentiability and the Cauchy–Riemann Equations	Examples & counterexamples	1
Complex Power Series and Analytic Functions	Examples & counterexamples	2
Contour Integration	Examples & counterexamples	2
Goursat's Theorem and Cauchy's Theorem in a Convex Domain	Examples & counterexamples	3
Analyticity of Holomorphic Functions; Liouville and Morera	Examples & counterexamples	4
The Identity Theorem, the Maximum Principle and the Open Mapping Theorem	Examples & counterexamples	5
The Winding Number and the Global Cauchy Theorem	Examples & counterexamples	6
Holomorphic Functions of Several Complex Variables	Examples & counterexamples	7
Isolated Singularities and Laurent Series	Examples & counterexamples	7
Harmonic Functions and the Poisson Integral	Examples & counterexamples	8
Infinite Products and the Weierstrass Factorisation Theorem	Examples & counterexamples	8
The Residue Theorem and the Evaluation of Real Integrals	Examples & counterexamples	8
Harmonic Hardy Classes and Fatou Boundary Limits	Examples & counterexamples	9
Jensen Theory and Nevanlinna's First Main Theorem	Examples & counterexamples	9
Mittag-Leffler and Runge's Theorem	Examples & counterexamples	9
Subharmonic Functions and the Dirichlet Problem	Examples & counterexamples	9
The Argument Principle and Rouché's Theorem	Examples & counterexamples	9
The Gamma Function	Examples & counterexamples	9
The Hartogs Phenomena	Examples & counterexamples	10
The Holomorphic Inverse Function Theorem and Weierstrass Preparation	Examples & counterexamples	10
The Riemann Sphere and Möbius Transformations	Examples & counterexamples	10
The Riemann Zeta Function	Examples & counterexamples	10
Analytic Hypersurfaces and Local Parametrisation	Examples & counterexamples	11
Conformal Mapping, Branches, and the Schwarz Lemma	Examples & counterexamples	11
Domains of Holomorphy, Plurisubharmonicity and Pseudoconvexity	Examples & counterexamples	11
Normal Families and Montel's Theorem	Examples & counterexamples	11
The Dolbeault Complex and Integral Solutions	Examples & counterexamples	11
Analytic Hardy Spaces and Canonical Factorisation	Examples & counterexamples	12
Hörmander Estimates and the Levi Problem	Examples & counterexamples	12
The Riemann Mapping Theorem	Examples & counterexamples	12
Bergman and Szegő Kernels	Examples & counterexamples	13
Bloch, Schottky, and the Picard Theorems	Examples & counterexamples	13
Green Functions, Harmonic Measure, and Conformal Invariance	Examples & counterexamples	13
Simply Connected Plane Domains: the Grand Equivalence	Examples & counterexamples	13
Analytic Continuation, Monodromy, and Riemann Surfaces	Examples & counterexamples	14
Extremal Length and Planar Quasiconformality	Examples & counterexamples	14
Logarithmic Potential, Capacity, and Riesz Decomposition	Examples & counterexamples	14
Nevanlinna's Second Main Theorem and Defects	Examples & counterexamples	14
Riemann Surfaces, Branched Maps, and Differentials	Examples & counterexamples	14
Elliptic Functions and Complex Tori	Examples & counterexamples	15
Hodge Theory on Compact Riemann Surfaces	Examples & counterexamples	15
Hyperbolic Riemann Surfaces and Uniformization	Examples & counterexamples	15
Divisors, Riemann--Roch, and Duality	Examples & counterexamples	16
Level-One Modular Forms and the j-Invariant	Examples & counterexamples	16
The Beltrami Equation and Measurable Riemann Mapping	Examples & counterexamples	16
Compact Kahler Hodge Foundations	Examples & counterexamples	17
Periods, Jacobians, and Abel--Jacobi Theory	Examples & counterexamples	17
Quasisymmetry, Welding, and Conformal Removability	Examples & counterexamples	17
Projective Hodge Polarization Foundations	Examples & counterexamples	18

Computability Theory

Reading order

86 pages

Page	Companion	Dependency level
Expander Graphs and Constraint Graphs	Examples & counterexamples	0
Formal Languages, Encodings, and Decision Problems	Examples & counterexamples	0
Context-Free Grammars and Normal Forms	Examples & counterexamples	1
Deterministic Finite Automata and Regular Languages	Examples & counterexamples	1
Turing Machines, Configurations, and Computation	Examples & counterexamples	1
Turing-Machine Configuration Boundary Interface	Examples & counterexamples	1
Finite-Automata Transition and Determinisation Interfaces	Examples & counterexamples	2
Myhill--Nerode Theory and DFA Minimization	Examples & counterexamples	2
One-Tape Simulation Normal Forms	Examples & counterexamples	2
Pushdown Automata and Context-Free Languages	Examples & counterexamples	2
Robust Machine Models and Universal Computation	Examples & counterexamples	2
Context-Free Pumping, Ogden's Lemma, and Parsing	Examples & counterexamples	3
Decidable, Recognizable, and Enumerable Languages	Examples & counterexamples	3
Effective Numberings, Reductions, Resources, and Randomness	Examples & counterexamples	3
Nondeterministic Finite Automata and Subset Construction	Examples & counterexamples	3
Primitive Recursive and Partial Computable Functions	Examples & counterexamples	3
Resource Bounds and Machine Invariance	Examples & counterexamples	3
Acceptable Numberings, the s-m-n Theorem, and Recursion	Examples & counterexamples	4
Diagonalization and the Halting Problem	Examples & counterexamples	4
Nondeterministic Recursive Padding and Time Separation	Examples & counterexamples	4
P, NP, coNP, and Polynomial Reductions	Examples & counterexamples	4
Randomized Complexity and Amplification	Examples & counterexamples	4
Regular Expressions and Kleene's Theorem	Examples & counterexamples	4
SAT Tableaux and Quantified-Space Completeness	Examples & counterexamples	4
Computable Reductions and Rice's Theorem	Examples & counterexamples	5
Interactive Proof Systems and Public Coins	Examples & counterexamples	5
Kolmogorov Complexity and Algorithmic Randomness	Examples & counterexamples	5
Space Complexity, Savitch's Theorem, and TQBF	Examples & counterexamples	5
The Arithmetical Hierarchy and Post's Theorem	Examples & counterexamples	5
The Cook--Levin Theorem	Examples & counterexamples	5
Arithmetization and the Sum-Check Protocol	Examples & counterexamples	6
Classical NP-Completeness Reductions	Examples & counterexamples	6
Counting Complexity and Sharp-P	Examples & counterexamples	6
Oracle Computability, the Jump, and Turing Degrees	Examples & counterexamples	6
Post Correspondence and Language Undecidability	Examples & counterexamples	6
Time and Space Hierarchy Theorems	Examples & counterexamples	6
Logarithmic Space, NL, and Reachability	Examples & counterexamples	7
The IP = PSPACE Theorem	Examples & counterexamples	7
Boolean Circuits and Nonuniform Complexity	Examples & counterexamples	8
Gap Amplification and Assignment Testing	Examples & counterexamples	9
The Polynomial Hierarchy and Relativization	Examples & counterexamples	9
Alphabet Reduction and the PCP Theorem	Examples & counterexamples	10
Approximation Algorithms and Gap Reductions	Examples & counterexamples	11

Coxeter Groups

Reading order

66 pages

Page	Companion	Dependency level
Coxeter Polyhedral Gluings and Intrinsic Metrics	Examples & counterexamples	0
Finite Lattice Projections and Coxeter Chain Labels	Examples & counterexamples	0
Symmetric Groups, Cycle Decomposition and the Sign Homomorphism	Examples & counterexamples	0
Conjugacy in Sₙ, Generation, and the Simplicity of Aₙ	Examples & counterexamples	1
Permutation Statistics, Inversions and Eulerian Numbers	Examples & counterexamples	1
Spherical Simplex Metrics, Angular Links, and Cones	Examples & counterexamples	1
CAT Comparison, Link Criteria, and Local Globalization	Examples & counterexamples	2
Coxeter Presentations, Exchange, and Reduced Word Theorems	Examples & counterexamples	2
Real Forms and Reflection Geometry	Examples & counterexamples	3
Short Loop Polygons and Quantitative Energy Decrease	Examples & counterexamples	3
Canonical Roots, Signs, and Faithful Reflections	Examples & counterexamples	4
Coxeter, Artin, and Hecke Interfaces	Examples & counterexamples	5
Parabolic Subgroups and Double Coset Geometry	Examples & counterexamples	5
Tits Cones, Chambers, and Parabolic Stabilizers	Examples & counterexamples	5
Bruhat Subword Order and Lifting	Examples & counterexamples	6
Finite Coxeter Diagrams and Complete Classification	Examples & counterexamples	6
Bruhat Interval Labels, Shellings, and Möbius Functions	Examples & counterexamples	7
Crystallographic Root Lattices and Weyl Group Interfaces	Examples & counterexamples	7
Finite Reflection Arrangements and Spherical Coxeter Complexes	Examples & counterexamples	7
Large Spherical Metric Flags and the Moussong Girth Theorem	Examples & counterexamples	7
Affine Reflections, Coroot Translations, and Alcoves	Examples & counterexamples	8
Finite Reflection Length and Orthogonal Moved Spaces	Examples & counterexamples	8
Spherical Parabolic Cosets and the Davis Complex	Examples & counterexamples	8
Weak Order, Inversions, and Lattice Operations	Examples & counterexamples	8
Affine Coxeter Diagrams and Semidefinite Classification	Examples & counterexamples	9
Bipartite Coxeter Elements and Ordered Root Complexes	Examples & counterexamples	9
Coxeter Euler Forms and Sortable Chamber Cones	Examples & counterexamples	9
Davis CAT(0) Geometry and Finite Subgroup Fixed Points	Examples & counterexamples	9
Heaps, Commutation Classes, and Fully Commutative Elements	Examples & counterexamples	9
Finite Coxeter Invariants and Coinvariant Gradings	Examples & counterexamples	10
Noncrossing Partition Lattices and Kreweras Complements	Examples & counterexamples	10
Sortable Projections and Finite Cambrian Lattices	Examples & counterexamples	10
Coxeter Descents, Poincaré Polynomials, and Growth	Examples & counterexamples	11

Differential Geometry

Reading order

66 pages

Page	Companion	Dependency level
Euclidean Ordinary Differential Equations with Smooth Dependence	Examples & counterexamples	0
Smooth Manifolds and Smooth Maps	Examples & counterexamples	0
Smooth Partitions of Unity and Exhaustions	Examples & counterexamples	1
Tangent Cotangent and the Differential	Examples & counterexamples	1
Formal Deformation Quantization and Hochschild Obstructions	Examples & counterexamples	2
Rank Theorems and Embedded Submanifolds	Examples & counterexamples	2
Smooth Vector Bundles and Sections	Examples & counterexamples	3
Sard Theorem and Transversality	Examples & counterexamples	4
Tensor Fields Exterior Algebra and Differential Forms	Examples & counterexamples	4
Whitney Embedding Tubular Neighbourhoods and Approximation	Examples & counterexamples	5
Vector Fields Flows and Lie Derivatives	Examples & counterexamples	6
Distributions Integral Manifolds and the Frobenius Theorem	Examples & counterexamples	7
Manifolds with Boundary Collars and Orientations	Examples & counterexamples	7
The Exterior Derivative and Cartan Calculus	Examples & counterexamples	8
Integration of Forms and the General Stokes Theorem	Examples & counterexamples	9
Measurable Densities and Radon Volume on Manifolds	Examples & counterexamples	10
The De Rham Complex Homotopy and Mayer Vietoris	Examples & counterexamples	10
Riemannian Metrics Length Distance and Volume	Examples & counterexamples	11
Singular Cochains Mayer Vietoris and Smooth Singular Comparison	Examples & counterexamples	11
Connections Levi Civita and Parallel Transport	Examples & counterexamples	12
The de Rham Theorem and Degree	Examples & counterexamples	12
Geodesics, the Exponential Map, Completeness, and Hopf–Rinow	Examples & counterexamples	13
Riemann Curvature and Riemannian Submanifolds	Examples & counterexamples	14
Symplectic Manifolds, Moser Stability, and Darboux–Weinstein Theory	Examples & counterexamples	14
Chern–Weil Theory and Characteristic Forms	Examples & counterexamples	15
Hamiltonian Mechanics and Completely Integrable Systems	Examples & counterexamples	15
Jacobi Fields, Conjugate Points, and the Cut Locus	Examples & counterexamples	15
The Gauss Bonnet Theorem for Riemannian Surfaces	Examples & counterexamples	15
Compact Laplace-type operators and harmonic representatives	Examples & counterexamples	16
Moment Maps and Symplectic Reduction	Examples & counterexamples	16
Riemannian Comparison Theorems	Examples & counterexamples	16
Clifford Bundles, Dirac Operators, and Index Interfaces	Examples & counterexamples	17
Poisson Algebras, Symplectic Brackets, and Hamiltonian Actions	Examples & counterexamples	17

Differential Topology

Reading order

68 pages

Page	Companion	Dependency level
Foliation Holonomy and the Holonomy Groupoid	Examples & counterexamples	0
Morse Critical Points Hessians and Indices	Examples & counterexamples	0
Oriented and Mod Two Intersection Numbers	Examples & counterexamples	0
Relative smooth triangulation and bundle fitting	Examples & counterexamples	0
Smooth Cobordism Relations Groups and Rings	Examples & counterexamples	0
Thom Spaces Normal Data and Collapse Maps	Examples & counterexamples	0
Characteristic Numbers and Cobordism Obstructions	Examples & counterexamples	1
Gradient Like Vector Fields and Morse Trajectories	Examples & counterexamples	1
Intersection Pairings Self Intersection and Euler Classes	Examples & counterexamples	1
Morse Functions Critical Values and Genericity	Examples & counterexamples	1
Pontryagin Thom and Framed Cobordism	Examples & counterexamples	1
Stable Unstable Manifolds and Morse Smale Transversality	Examples & counterexamples	2
Sublevel Deformation and the Handle Attachment Theorem	Examples & counterexamples	2
The Hirzebruch Signature Theorem	Examples & counterexamples	2
The Hopf Degree Theorem	Examples & counterexamples	2
Handle Decompositions Duality and Rearrangement	Examples & counterexamples	3
Morse Trajectory Moduli Spaces and the Morse Differential	Examples & counterexamples	3
Formal Immersions and the Smale Hirsch Theorem	Examples & counterexamples	4
Handle Cancellation Slides and Elementary Moves	Examples & counterexamples	4
Smooth Surgery Traces and Handle Trading	Examples & counterexamples	4
Morse Inequalities and the Handle Chain Complex	Examples & counterexamples	5
Regular Homotopy and Sphere Eversion	Examples & counterexamples	5
The Whitney Trick and Surgery Below the Middle Dimension	Examples & counterexamples	5
Isotopy Extension and Embedding Theory Beyond Whitney	Examples & counterexamples	6
Morse Homology Continuation and Comparison	Examples & counterexamples	6
The Smooth H Cobordism Theorem	Examples & counterexamples	6
Vector Field Index Euler Characteristic and Poincare Hopf	Examples & counterexamples	6
Characteristic Class Obstructions to Immersions and Embeddings	Examples & counterexamples	7
Exotic Smooth Structures and Milnor Spheres	Examples & counterexamples	7
Fixed Point Index and the Lefschetz Theorem	Examples & counterexamples	7
Whitehead Torsion and the S Cobordism Theorem	Examples & counterexamples	7
Reeb Stability and Global Foliation Constructions	Examples & counterexamples	8
Codimension One Foliations, Secondary Classes and Characteristic Disk Foundations	Examples & counterexamples	9
Vanishing Cycles, Novikov and Taut Foliations	Examples & counterexamples	10

Fourier Analysis

Reading order

44 pages

Page	Companion	Dependency level
Character Groups and Elementary LCA Duals	Examples & counterexamples	0
Dirichlet Kernel Localisation and Pointwise Fourier Convergence	Examples & counterexamples	0
Finite Fourier Analysis and the Fast Fourier Transform	Examples & counterexamples	0
Fourier Multipliers and Sobolev Characterisations	Examples & counterexamples	0
Riesz Potentials and the Hardy–Littlewood–Sobolev Inequality	Examples & counterexamples	0
Bochner Inversion and Plancherel on LCA Groups	Examples & counterexamples	1
Fejer and Poisson Summability of Fourier Series	Examples & counterexamples	1
Fourier Restriction and the Stein–Tomas Theorem	Examples & counterexamples	1
Poisson Summation Sampling and Lattice Duality	Examples & counterexamples	1
Uncertainty Principles for Fourier Analysis	Examples & counterexamples	1
Absolute Convergence and the Wiener Algebra	Examples & counterexamples	2
Kolmogorov’s Block Construction and Almost-Everywhere Divergence	Examples & counterexamples	2
Lacunary Fourier Series and Sidon Sets	Examples & counterexamples	2
Pontryagin Duality for Locally Compact Abelian Groups	Examples & counterexamples	2
The Carleson–Hunt Time–Frequency Theorem	Examples & counterexamples	2
Divergence and Almost Everywhere Convergence of Fourier Series	Examples & counterexamples	3
Hilbert and Riesz Transforms	Examples & counterexamples	4
Calderón–Zygmund Decomposition and Singular Integrals	Examples & counterexamples	5
Muckenhoupt Weights and Weighted Estimates	Examples & counterexamples	6
Real Hardy Spaces Maximal Functions and Atoms	Examples & counterexamples	6
BMO, John-Nirenberg, and H1 Duality	Examples & counterexamples	7
Littlewood Paley Theory and Square Functions	Examples & counterexamples	8

Functional Analysis

Reading order

70 pages

Page	Companion	Dependency level
Complex Norms and Realification	Examples & counterexamples	0
Fourier Transform Convolution and Approximate Identities	Examples & counterexamples	0
Normed and Banach Spaces	Examples & counterexamples	1
Bounded Linear Operators and Quotient Spaces	Examples & counterexamples	2
Complex Subspace Realification	Examples & counterexamples	2
Finite Dimensional Normed Spaces and Riesz Lemma	Examples & counterexamples	3
Schwartz Space and the Plancherel Theorem	Examples & counterexamples	3
Sequential Uniform Boundedness with Countable Choice	Examples & counterexamples	3
The Analytic Hahn Banach Theorem	Examples & counterexamples	3
The Baire Principles of Functional Analysis	Examples & counterexamples	3
Distributions Test Functions and Differentiation	Examples & counterexamples	4
Geometric Hahn Banach and Convex Separation	Examples & counterexamples	4
Norming and Separation under Hahn–Banach	Examples & counterexamples	4
Dual Spaces Adjoint Operators and Annihilators	Examples & counterexamples	5
Locally Convex Spaces and Continuous Separation	Examples & counterexamples	5
Tempered Distributions and the Fourier Transform	Examples & counterexamples	5
Weak and Weak Star Topologies	Examples & counterexamples	6
Banach Alaoglu Goldstine and Krein Milman	Examples & counterexamples	7
Reflexivity and Eberlein Smulian	Examples & counterexamples	8
Banach Valued Integration and the Radon Nikodym Property	Examples & counterexamples	9
Schauder Bases Approximation and Banach Space Pathologies	Examples & counterexamples	9
Compact Operators and Riesz Schauder Theory	Examples & counterexamples	10
Hilbert Space Geometry and Riesz Representation	Examples & counterexamples	10
Banach Algebras Spectrum and Holomorphic Functional Calculus	Examples & counterexamples	11
Banach-Space Differential Calculus and Banach Manifolds	Examples & counterexamples	11
Orthonormal Bases, Parseval and Fourier Series	Examples & counterexamples	11
Gelfand Theory and Commutative C Star Algebras	Examples & counterexamples	12
Square-Integrable Kernels and Hilbert–Schmidt Compactness	Examples & counterexamples	12
Compact Self Adjoint Hilbert Schmidt and Trace Class Operators	Examples & counterexamples	13
Continuous Functional Calculus for Self Adjoint and Normal Operators	Examples & counterexamples	13
Operator-Algebraic States and GNS Representations	Examples & counterexamples	13
Fredholm Determinants and the Lidskii Trace Formula	Examples & counterexamples	14
Spectral Measures and Borel Functional Calculus	Examples & counterexamples	14
Measurable Hilbert Fields and Direct-Integral Operators	Examples & counterexamples	15
Unbounded Self Adjoint Operators and Stones Theorem	Examples & counterexamples	15

Group Theory

Reading order

62 pages

Page	Companion	Dependency level
An elementary simple group on the projective line	Examples & counterexamples	0
Blocks Primitivity and Multiple Transitivity	Examples & counterexamples	0
Cayley Graphs, Word Metrics and Quasi-Isometry	Examples & counterexamples	0
Crossed Homomorphisms Complements and First Cohomology	Examples & counterexamples	0
Decision Problems for Finitely Presented Groups	Examples & counterexamples	0
Frattini Subgroups and the Burnside Basis Theorem	Examples & counterexamples	0
Group Extensions Complements and Schur Zassenhaus	Examples & counterexamples	0
Hnn Extensions and Brittons Lemma	Examples & counterexamples	0
Modular Representations and Projective Covers	Examples & counterexamples	0
Small-Cancellation Disc Diagrams and the Torsion Toolkit	Examples & counterexamples	0
Subgroups of Free Groups and Schreier Rewriting	Examples & counterexamples	0
Asymptotic Cones and the Sublinear Triangle Criterion	Examples & counterexamples	1
Boone Machine Simulation and Fixed-Presentation Undecidability	Examples & counterexamples	1
Brauer Characters and Decomposition Matrices	Examples & counterexamples	1
Extraspecial p-Groups and Central Products	Examples & counterexamples	1
Group Homology Transfer and Low-Degree Exact Sequences	Examples & counterexamples	1
Hall–Mal’cev Coordinates and Bass–Guivarc’h Growth	Examples & counterexamples	1
Inverse Systems Profinite Groups and Completion	Examples & counterexamples	1
Modular Traces and Brauer-Character Independence	Examples & counterexamples	1
Second Cohomology and Abelian Kernel Extensions	Examples & counterexamples	1
Simplicial Trees and Group Actions	Examples & counterexamples	1
Small Cancellation and Dehn Algorithms	Examples & counterexamples	1
Socles and the Onan Scott Landscape	Examples & counterexamples	1
Geometric Actions Svarc Milnor and Growth	Examples & counterexamples	2
Graphs of Groups and Bass Serre Theory	Examples & counterexamples	2
Pro P Groups and the P Adic Integers	Examples & counterexamples	2
Quantitative Hyperbolic Geometry Toolkit	Examples & counterexamples	2
Schur Multipliers and Universal Central Extensions	Examples & counterexamples	2
The Finite Simple Group Classification Landscape	Examples & counterexamples	2
Amenable Groups and Folner Criteria	Examples & counterexamples	3
Hyperbolic Spaces and Hyperbolic Groups	Examples & counterexamples	3

Hall Algebras and Tensor Categories

Reading order

22 pages

Page	Companion	Dependency level
Cohomological Hall Algebras and Quiver Correspondences	Examples & counterexamples	0
Finitary Ringel–Hall Algebras	Examples & counterexamples	0
Frobenius Algebras and Oriented Two-Dimensional Sewing	Examples & counterexamples	0
Modular Categories, Verlinde Formula, and TQFT	Examples & counterexamples	0
Noninvertible Symmetries and Anyonic Chains	Examples & counterexamples	0
Derived Hall Algebras from dg-Enhanced Categories	Examples & counterexamples	1
Euler-Twisted Hall Products, Green Coproducts, and Quantum Groups	Examples & counterexamples	1
Hall Algebra Modules, Fock Spaces, and Geometric Actions	Examples & counterexamples	1
Higher Categorical Symmetries and Defect Algebras	Examples & counterexamples	1
Motivic Hall Algebras and Integration Maps	Examples & counterexamples	2
Hall Integration, Stability, DT/BPS Invariants, and Wall Crossing	Examples & counterexamples	3

Hecke Algebras and Categorification

Reading order

16 pages

Page	Companion	Dependency level
KLR Algebras, Induction, and Quantum-Serre Categorification	Examples & counterexamples	0
Ordinary Hecke Specht Polytabloids and Signed Murphy Comparison	Examples & counterexamples	0
Unequal-Parameter Hecke Cells and Representations	Examples & counterexamples	0
Cyclotomic KLR Algebras and Highest-Weight Categorification	Examples & counterexamples	1
Type-A Hecke Specht Modules and Cellular Bases	Examples & counterexamples	1
Cyclotomic Hecke and Quiver Hecke Algebras	Examples & counterexamples	2
Ariki Categorification and Graded Decomposition Numbers	Examples & counterexamples	3
The Degenerate Ariki Categorification Theorem	Examples & counterexamples	3

Homological Algebra

Reading order

68 pages

Page	Companion	Dependency level
BV, BRST, and Homological Physics Interfaces	Examples & counterexamples	0
Chain Complexes and Homology	Examples & counterexamples	0
Chain Homotopy and the Homotopy Category	Examples & counterexamples	1
Differential Graded Algebras and Homological Foundations	Examples & counterexamples	1
Spectral Sequences	Examples & counterexamples	1
A∞, L∞, and Deformation Complexes	Examples & counterexamples	2
Homological Gaussian Elimination	Examples & counterexamples	2
Mapping Cones Cylinders and Chain Triangles	Examples & counterexamples	2
Long Exact Sequences in Homology	Examples & counterexamples	3
Double Complexes Exact Couples and Convergence	Examples & counterexamples	4
Projective and Injective Resolutions	Examples & counterexamples	4
Triangulated Categories	Examples & counterexamples	4
Derived Functors	Examples & counterexamples	5
Delta Functors and Universality	Examples & counterexamples	6
Ext and Balanced Resolutions	Examples & counterexamples	7
Yoneda Extensions and Homological Dimension	Examples & counterexamples	8
Group Cohomology as a Derived Functor	Examples & counterexamples	9
Tor Flatness and Global Dimension	Examples & counterexamples	9
Derived Categories	Examples & counterexamples	10
Eilenberg–Watts Theorem and Natural Transformations	Examples & counterexamples	10
Graded Bimodules and Tensor Functors	Examples & counterexamples	10
Universal Coefficients and Kunneth Theorems	Examples & counterexamples	10
Bounded Bimodule Complexes and Derived Tensor	Examples & counterexamples	11
Grothendieck Groups and Graded Cartan Pairings	Examples & counterexamples	11
Grothendieck Spectral Sequences and Computations	Examples & counterexamples	11
Hochschild Homology and Diagonal Koszul Resolutions	Examples & counterexamples	11
Kunneth Exactness and Splittings over Principal Ideal Domains	Examples & counterexamples	11
Lefschetz Splitting In Bounded Triangulated Categories	Examples & counterexamples	11
Morita Bicategories and Projective Generators	Examples & counterexamples	11
Finite Abelian Categories and Eilenberg–Watts	Examples & counterexamples	12
Graded Eilenberg–Watts and Shift Coherence	Examples & counterexamples	12
Hochschild Hyperhomology and Cyclic Tensor Invariance	Examples & counterexamples	12
Perfect Complexes and Triangulated Grothendieck Groups	Examples & counterexamples	12
Deligne Products and Categorical Eilenberg–Watts	Examples & counterexamples	13

Hopf Algebras & Hecke Algebras

Reading order

42 pages

Page	Companion	Dependency level
Tensor Coherence and Algebraic Descent	Examples & counterexamples	0
Coalgebras, Counits, and the Fundamental Coalgebra Theorem	Examples & counterexamples	1
Double-Affine Hecke Algebras and Macdonald Theory	Examples & counterexamples	1
Generic Coxeter Hecke Algebras and the Standard Basis	Examples & counterexamples	1
Hopf, Hecke, Quantum-Group, and Braid Interfaces	Examples & counterexamples	1
Affine Hecke Algebras Beyond Type A	Examples & counterexamples	2
Bialgebras, Convolution, and Antipode Identities	Examples & counterexamples	2
Comodules, Matrix Coefficients, and Coalgebra Duality	Examples & counterexamples	2
Hecke Base Change, Semisimplicity, and Deformation	Examples & counterexamples	2
Hecke Parabolic Induction and Symmetrizing Traces	Examples & counterexamples	2
Hecke Symmetries and Generic Quantum Schur–Weyl Duality	Examples & counterexamples	2
Type A Affine Hecke Algebras and Bernstein PBW	Examples & counterexamples	2
Cyclotomic Hecke Quotients and Ariki–Koike PBW	Examples & counterexamples	3
Hopf Ideals, Finite Duals, and Basic Constructions	Examples & counterexamples	3
Hopf Module Tensor Products and Rigid Duality	Examples & counterexamples	3
Hopf Modules, Coinvariants, and the Fundamental Theorem	Examples & counterexamples	3
Quasi-Hopf, Weak Hopf, and Relative Tensor Products	Examples & counterexamples	3
Cyclotomic Murphy Bases and Cellular Module Constructions	Examples & counterexamples	4
Finite Hopf Integrals, Frobenius Duality, and Maschke	Examples & counterexamples	4
Quasitriangular Hopf Algebras and Braided Module Categories	Examples & counterexamples	4
Drinfeld Doubles and Yetter–Drinfeld Module Constructions	Examples & counterexamples	5

Information Theory

Reading order

48 pages

Page	Companion	Dependency level
F-Divergences, Rényi Information, and Hypothesis Testing	Examples & counterexamples	0
Finite Trellises, Viterbi, and Sequence Decoding	Examples & counterexamples	0
Finite-Alphabet Entropy and Divergence	Examples & counterexamples	0
Method of Types and Finite Large Deviations	Examples & counterexamples	0
Quantum Information Measures	Examples & counterexamples	0
Quantum Key Distribution and Composable Secrecy	Examples & counterexamples	0
Stationary Ergodic Sources and Universal Coding	Examples & counterexamples	0
Discrete Memoryless Channel Capacity	Examples & counterexamples	1
IID Lossless Source Coding and the AEP	Examples & counterexamples	1
Measure-Theoretic Information and Conditioning	Examples & counterexamples	1
Prefix Codes and Finite Lossless Coding	Examples & counterexamples	1
Quantum Relative Entropy, Data Processing, and Strong Subadditivity	Examples & counterexamples	1
Cost-Constrained, Gaussian, and Feedback Channels	Examples & counterexamples	2
Distributed Source Coding	Examples & counterexamples	2
Finite-Blocklength Bounds and Reliability	Examples & counterexamples	2
Quantum Source Compression and Classical Communication over Quantum Channels	Examples & counterexamples	2
Rate Distortion and Lossy Source Coding	Examples & counterexamples	2
Secrecy and Information Privacy	Examples & counterexamples	2
Finite Quantization, Lloyd Iteration, and Compact Companders	Examples & counterexamples	3
Lossy Source Coding with Side Information	Examples & counterexamples	3
Point-to-Point Source–Channel Separation	Examples & counterexamples	3
Quantum Channel Capacities and Entanglement Assistance	Examples & counterexamples	3
Multiuser Channel Networks	Examples & counterexamples	4
Relay and Interference Channel Bounds	Examples & counterexamples	5

Inverse Problems

Reading order

110 pages

Page	Companion	Dependency level
Active network tomography and identifiability by design	Examples & counterexamples	0
Analytic double fibrations and Bolker visibility	Examples & counterexamples	0
Credible sets and frequentist coverage	Examples & counterexamples	0
Dynamical observations and finite system realization	Examples & counterexamples	0
Euclidean Radon transform and filtered backprojection	Examples & counterexamples	0
Geodesic X-ray transforms on simple surfaces	Examples & counterexamples	0
Inverse maps, identifiability, and stability	Examples & counterexamples	0
Open frontiers in calibrated and robust learned inversion	Examples & counterexamples	0
Posterior computation and error accounting	Examples & counterexamples	0
Score and diffusion methods for inverse data	Examples & counterexamples	0
Sparse recovery and nullspace certificates	Examples & counterexamples	0
Spatial BV, finite perimeter, and TV conventions	Examples & counterexamples	0
Statistical inverse experiments and white noise	Examples & counterexamples	0
Compact linear inverse maps and the Picard criterion	Examples & counterexamples	1
Function-space priors and sample regularity	Examples & counterexamples	1
Hyperbolic inverse problems and finite-time visibility	Examples & counterexamples	1
Limited-angle visibility and instability	Examples & counterexamples	1
Linear feature identifiability and Gaussian MAP tests	Examples & counterexamples	1
Nonlinear maps and tangential cone control	Examples & counterexamples	1
Nonlinear Tikhonov existence and stability	Examples & counterexamples	1
Real phase retrieval and phaseless ambiguity	Examples & counterexamples	1
Scalar Calderón maps and conditional uniqueness	Examples & counterexamples	1
Stable compressed sensing with noise	Examples & counterexamples	1
Unknown operators, model mismatch, and robustness limits	Examples & counterexamples	1
Anisotropic gauge and unresolved uniqueness	Examples & counterexamples	2
Bregman distances and variational source rates	Examples & counterexamples	2
Data, operator, discretization, and solve errors	Examples & counterexamples	2
Deterministic source classes and optimal recovery	Examples & counterexamples	2
Fixed-frequency inverse scattering and Born linearization	Examples & counterexamples	2
Gaussian sequence inversion and exact risk	Examples & counterexamples	2
Likelihood potentials and posterior propriety	Examples & counterexamples	2
Nonlinear Landweber with discrepancy stopping	Examples & counterexamples	2
Parameter rules and the noise-free veto	Examples & counterexamples	2
PDE-constrained inversion and adjoint error control	Examples & counterexamples	2
Physical constraints and manifold-valued inversion	Examples & counterexamples	2
Regularized nonlinear Levenberg–Marquardt	Examples & counterexamples	2
Adaptive spectral regularization by SOLIT	Examples & counterexamples	3
Banach range invariance and EIT rates	Examples & counterexamples	3
Factorized and composite linear inversion	Examples & counterexamples	3
Function-space pCN and invariant posterior kernels	Examples & counterexamples	3
Hellinger stability and posterior discretization	Examples & counterexamples	3
Learned reconstruction targets and risk	Examples & counterexamples	3
Linear Gaussian inverse posteriors and Tikhonov	Examples & counterexamples	3
Minimax estimation over inverse source classes	Examples & counterexamples	3
Minimax signal detection and separation rates	Examples & counterexamples	3
Small-ball MAP and Cameron–Martin geometry	Examples & counterexamples	3
TV source certificates and level-set convergence	Examples & counterexamples	3
Bayesian measurement design and information gain	Examples & counterexamples	4
Benchmarks, prior shift, and solver admissibility	Examples & counterexamples	4
Conditional flow matching for inverse posteriors	Examples & counterexamples	4
Logarithmic EIT stability on explicit conductivity classes	Examples & counterexamples	4
Plug-and-play MMSE denoisers and stationarity	Examples & counterexamples	4
Posterior contraction in Gaussian sequence inversion	Examples & counterexamples	4
Unrolled reconstruction and finite-iteration limits	Examples & counterexamples	4
Noise-robust neural-operator approximation for EIT	Examples & counterexamples	5

Knot Theory

Reading order

108 pages

Page	Companion	Dependency level
Crossing signs, writhe, and linking number	Examples & counterexamples	0
Knot exteriors and peripheral systems	Examples & counterexamples	0
Satellite knots and JSJ tori	Examples & counterexamples	0
Wirtinger presentations and knot groups	Examples & counterexamples	0
Dehn surgery on knots and homology	Examples & counterexamples	1
Fox colorings and quandle invariants	Examples & counterexamples	1
Hyperbolic knot complements and volume	Examples & counterexamples	1
Kauffman bracket state sum and the Jones polynomial	Examples & counterexamples	1
Multivariable Alexander modules and chain models	Examples & counterexamples	1
Satellite operators and concordance filtrations	Examples & counterexamples	1
Seifert surfaces and knot genus	Examples & counterexamples	1
Vassiliev invariants and chord diagrams	Examples & counterexamples	1
Alternating diagrams and crossing-number minimality	Examples & counterexamples	2
Hyperbolic Dehn filling and exceptional slopes	Examples & counterexamples	2
Kauffman and Dubrovnik polynomials from BMW algebras	Examples & counterexamples	2
Khovanov homology and Jones categorification	Examples & counterexamples	2
Milnor invariants and link peripheral nilpotent quotients	Examples & counterexamples	2
Planar-diagram codes and knot-table enumeration	Examples & counterexamples	2
Ribbon categories and Reshetikhin–Turaev knot invariants	Examples & counterexamples	2
Seifert matrices and the Alexander module	Examples & counterexamples	2
Skein modules for links in 3-manifolds	Examples & counterexamples	2
The Kontsevich integral and universality	Examples & counterexamples	2
Three-manifold surgery presentations and Kirby moves	Examples & counterexamples	2
Twisted Alexander polynomials and Reidemeister torsion	Examples & counterexamples	2
Alexander polynomials from Seifert matrices	Examples & counterexamples	3
Colored Jones and the volume conjecture	Examples & counterexamples	3
Normal-surface unknot recognition	Examples & counterexamples	3
Root-of-unity skein TQFT for 3-manifolds	Examples & counterexamples	3
Smooth and topological knot concordance	Examples & counterexamples	3
Terminating computations for classical knot invariants	Examples & counterexamples	3
The knot-complement theorem	Examples & counterexamples	3
Connected sums and prime decomposition	Examples & counterexamples	4
Cyclic branched covers and knot determinants	Examples & counterexamples	4
Freedman’s theorem and topological sliceness	Examples & counterexamples	4
Levine–Tristram signatures and the Arf invariant	Examples & counterexamples	4
Ribbon knots and the slice-ribbon problem	Examples & counterexamples	4
Torus knots and Seifert-fibered exteriors	Examples & counterexamples	4
Algebraic concordance and metabolic forms	Examples & counterexamples	5
Casson–Gordon invariants and sliceness obstructions	Examples & counterexamples	5
Character varieties and the A-polynomial	Examples & counterexamples	5
Fox–Milnor factorization and slice obstructions	Examples & counterexamples	5
Grid homology and combinatorial knot Floer theory	Examples & counterexamples	6
Open questions in knot polynomials and surgery	Examples & counterexamples	6
Smooth sliceness and four-ball genus obstructions	Examples & counterexamples	6
Branched-cover correction terms and sliceness	Examples & counterexamples	7
Khovanov homology detects the unknot	Examples & counterexamples	7
Knot Floer genus and fibering detection	Examples & counterexamples	7
Lee deformation and Rasmussen’s s-invariant	Examples & counterexamples	7
The Khovanov to Floer spectral sequence	Examples & counterexamples	7
Knot cobordism maps and functoriality	Examples & counterexamples	8
Zero traces and the Conway nonsliceness theorem	Examples & counterexamples	8
Bridge presentations and unknotting number	Examples & counterexamples	9
The tau invariant and four-ball genus	Examples & counterexamples	9
Epsilon and Upsilon concordance invariants	Examples & counterexamples	10

Lie Theory

Reading order

56 pages

Page	Companion	Dependency level
Lie Algebra Representations, Enveloping Algebras, and PBW	Examples & counterexamples	0
Lie Groups, Invariant Fields, and the Exponential Map	Examples & counterexamples	0
Lie Subgroups, Actions, and Homogeneous Spaces	Examples & counterexamples	1
Solvable and Nilpotent Lie Algebras	Examples & counterexamples	1
Gauge Groups and Representation-Theoretic Interfaces	Examples & counterexamples	2
Semisimple Lie Algebras, Cohomology, and Levi Theory	Examples & counterexamples	2
Cartan Subalgebras and Root Space Decompositions	Examples & counterexamples	3
Finite Weyl Invariants, Bruhat Order, and Kostant Harmonics	Examples & counterexamples	4
Root Systems, Dynkin Diagrams, and the Cartan-Killing Classification	Examples & counterexamples	4
Harish Chandra Isomorphism Casimir and Central Characters	Examples & counterexamples	5
Highest Weight Theory for Complex Semisimple Lie Algebras	Examples & counterexamples	5
Lie Superalgebras and Super-Representations	Examples & counterexamples	5
Compact Lie Groups, Maximal Tori, and Peter–Weyl Theory	Examples & counterexamples	6
Kac Moody Algebras from Generalized Cartan Matrices	Examples & counterexamples	6
Verma Modules and Shapovalov Forms	Examples & counterexamples	6
Affine Lie Algebras and Loop Central Extensions	Examples & counterexamples	7
Homomorphisms Between Verma Modules and Linkage	Examples & counterexamples	7
Integrable Highest Weight Kac Moody Modules	Examples & counterexamples	7
Primitive Ideals and Duflo Theorem	Examples & counterexamples	7
Real Forms and Real Semisimple Lie Algebras	Examples & counterexamples	7
Category O Finiteness Duality and Blocks	Examples & counterexamples	8
The Weyl Kac Character Formula	Examples & counterexamples	8
Projectives Standard Filtrations and Bgg Reciprocity	Examples & counterexamples	9
The BGG Resolution	Examples & counterexamples	9
Lie Algebra Cohomology and Kostants Nilradical Theorem	Examples & counterexamples	10
Weyl Character and Multiplicity Formulas	Examples & counterexamples	10
Borel Weil and Borel Weil Bott	Examples & counterexamples	11
Tensor Product Multiplicities and Littlewood Richardson	Examples & counterexamples	11

Linear Algebra

Reading order

44 pages

Page	Companion	Dependency level
Vector Spaces, Linear Subspaces, Span and Direct Sums	Examples & counterexamples	0
Linear Independence, Bases and Dimension	Examples & counterexamples	1
Linear Transformations, Rank-Nullity and Quotient Spaces	Examples & counterexamples	2
Matrices, the Matrix of a Linear Map, and Change of Basis	Examples & counterexamples	3
Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form	Examples & counterexamples	4
Determinants of Matrices over a Commutative Ring	Examples & counterexamples	5
Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia	Examples & counterexamples	6
The Determinant of a Linear Operator, Cofactors and Cramer's Rule	Examples & counterexamples	6
Eigenvalues, Eigenvectors and the Characteristic Polynomial	Examples & counterexamples	7
Inner Product Spaces, Gram-Schmidt, Projections and Adjoints	Examples & counterexamples	7
Diagonalisation and the Minimal Polynomial	Examples & counterexamples	8
Complexification, Realification and Real Structures	Examples & counterexamples	9
Triangularisation, Generalised Eigenspaces and Jordan Canonical Form	Examples & counterexamples	9
Exterior Powers, Orientation and Hodge Duality	Examples & counterexamples	10
The Spectral Theorem, Positive Operators and Singular Value Decomposition	Examples & counterexamples	10
Matrix Norms, Condition Numbers and Numerical Stability	Examples & counterexamples	11
Direct Matrix Factorisations: LU, Cholesky and QR	Examples & counterexamples	12
Matrix Differentiation and First-order Spectral Perturbation	Examples & counterexamples	12
Eigenvalue Iterations and the QR Algorithm	Examples & counterexamples	13
The Moore--Penrose Pseudoinverse and Regularised Least Squares	Examples & counterexamples	13
Krylov Subspaces, Arnoldi and GMRES	Examples & counterexamples	14
Conjugate Gradients, MINRES and Preconditioning	Examples & counterexamples	15

Mathematical Statistics

Reading order

74 pages

Page	Companion	Dependency level
Concentration	Examples & counterexamples	0
Empirical-Process Complexity	Examples & counterexamples	0
Hilbert-Valued CLT for Functional Time Series	Examples & counterexamples	0
Statistical Experiments, Models and Identifiability	Examples & counterexamples	0
Consistency and Limiting Distributions	Examples & counterexamples	1
Domination and Likelihood Ratios	Examples & counterexamples	1
Estimators and Randomized Procedures	Examples & counterexamples	1
Chaining and Uniform Limits	Examples & counterexamples	2
Conformal Prediction and Exchangeable Ranks	Examples & counterexamples	2
Density Estimation	Examples & counterexamples	2
Gaussian Orthogonal Decomposition and Quadratic Forms	Examples & counterexamples	2
Loss, Risk and Decision Theory	Examples & counterexamples	2
M- and Z-Estimators	Examples & counterexamples	2
Semiparametric Models and Tangent Spaces	Examples & counterexamples	2
Sufficiency, Completeness and Exponential Families	Examples & counterexamples	2
Contamination and Robust Risk	Examples & counterexamples	3
Delta Method and Asymptotic Normality	Examples & counterexamples	3
Finite-Sample Point Estimation	Examples & counterexamples	3
Minimax Lower Bounds	Examples & counterexamples	3
Regression Estimation	Examples & counterexamples	3
Tests, Neyman–Pearson and Valid P-values	Examples & counterexamples	3
Conditional and Finite-Group Exact Inference	Examples & counterexamples	4
Confidence Sets and Inversion	Examples & counterexamples	4
Decision-Theoretic Upper Bounds	Examples & counterexamples	4
Robust Means under Heavy Tails	Examples & counterexamples	4
Scores and Information Bounds	Examples & counterexamples	4
Sequential Validity and Ville's Inequality	Examples & counterexamples	4
Sparse Estimation and Covariance	Examples & counterexamples	4
UMP Tests	Examples & counterexamples	4
Influence Functions and Efficient Scores	Examples & counterexamples	5
Likelihood Theory and Classical Calibration	Examples & counterexamples	5
Robust Covariance and Sparse Robust Estimation	Examples & counterexamples	5
Splines and Series Methods	Examples & counterexamples	5
Influence and Breakdown	Examples & counterexamples	6
LAN, Contiguity and Efficiency	Examples & counterexamples	6
Nonregular and Misspecified Limits	Examples & counterexamples	6
Semiparametric Efficiency	Examples & counterexamples	7

Measure Theory

Reading order

56 pages

Page	Companion	Dependency level
Pointwise Extended-Real Function Products	Examples & counterexamples	0
Sigma Algebras and Borel Sets	Examples & counterexamples	0
Measures and Their Basic Properties	Examples & counterexamples	1
Outer Measure and the Caratheodory Extension Theorem	Examples & counterexamples	2
Lebesgue Measure on Euclidean Space	Examples & counterexamples	3
Lebesgue-Stieltjes Measures and Distribution Functions	Examples & counterexamples	4
Non Measurable Sets and the Cost of Choice	Examples & counterexamples	4
Measurable Functions and Simple Approximation	Examples & counterexamples	5
Hausdorff Measure and Hausdorff Dimension	Examples & counterexamples	6
The Lebesgue Integral and the Convergence Theorems	Examples & counterexamples	6
Modes of Convergence Egorov and Lusin	Examples & counterexamples	7
Product Measures and the Fubini Tonelli Theorems	Examples & counterexamples	7
Signed and Complex Measures Hahn and Jordan	Examples & counterexamples	7
The Lebesgue and Riemann Integrals Compared	Examples & counterexamples	7
The Lᵖ Spaces Holder Minkowski and Riesz Fischer	Examples & counterexamples	8
The Radon Nikodym Theorem and Lebesgue Decomposition	Examples & counterexamples	8
Density Separability and Convolution in Lᵖ	Examples & counterexamples	9
The Duality of Lᵖ and L^q	Examples & counterexamples	9
Complex Lp Spaces and Test-Function Conventions	Examples & counterexamples	10
Radon Measures and the Riesz Markov Kakutani Theorem	Examples & counterexamples	10
The Maximal Function and Lebesgue Differentiation	Examples & counterexamples	10
Complex Riesz–Thorin Endpoint Interpolation	Examples & counterexamples	11
Differentiation of Monotone Functions and the Vitali Covering Theorem	Examples & counterexamples	11
Measure-Preserving Systems and Mixing Criteria	Examples & counterexamples	11
Absolute Continuity and the Sharp Fundamental Theorem of Calculus	Examples & counterexamples	12
Measure Preserving Transformations and Poincare Recurrence	Examples & counterexamples	12
Weak Mixing and the Chacon Transformation	Examples & counterexamples	12
The Ergodic Theorems of von Neumann and Birkhoff	Examples & counterexamples	13

Neural Networks and Learning Dynamics

Reading order

56 pages

Page	Companion	Dependency level
Adaptive gradient algorithms and regret limits	Examples & counterexamples	0
Algorithmic stability of stochastic training	Examples & counterexamples	0
Automatic differentiation and backpropagation	Examples & counterexamples	0
Compression generalization for trained networks	Examples & counterexamples	0
Convolution and finite-group equivariance	Examples & counterexamples	0
Feedforward networks and single-hidden-layer universality	Examples & counterexamples	0
Marchenko–Pastur law and sample-covariance edge bounds	Examples & counterexamples	0
Message-passing GNNs and the 1-WL ceiling	Examples & counterexamples	0
Momentum and quadratic acceleration	Examples & counterexamples	0
Nonconvex gradient descent and stationarity	Examples & counterexamples	0
Norm and margin generalization for neural networks	Examples & counterexamples	0
Recurrent, LSTM, and structured state-space models	Examples & counterexamples	0
Stochastic gradient descent and random-iterate guarantees	Examples & counterexamples	0
Wasserstein metric and continuity-equation foundations	Examples & counterexamples	0
Attention, transformer universality, and sequence limits	Examples & counterexamples	1
Bilevel learning, hypergradients, and implicit layers	Examples & counterexamples	1
Existence, uniqueness, and stability of Wasserstein gradient flows	Examples & counterexamples	1
Implicit bias of gradient-based learning	Examples & counterexamples	1
Initialization and signal propagation	Examples & counterexamples	1
Low-rank adaptation and the geometry of parameter updates	Examples & counterexamples	1
Neural operators and PINN consistency	Examples & counterexamples	1
Neural tangent kernels and lazy training	Examples & counterexamples	1
Overparameterized networks and training to interpolation	Examples & counterexamples	1
ReLU approximation rates, depth separation, and linear regions	Examples & counterexamples	1
Sample-covariance resolvent deterministic equivalents	Examples & counterexamples	1
Exact tiled attention and memory-aware differentiation	Examples & counterexamples	2
Mean-field training and feature learning	Examples & counterexamples	2
Minimum-norm interpolation and double descent	Examples & counterexamples	2

Number Theory

Reading order

54 pages

Page	Companion	Dependency level
Divisibility, Greatest Common Divisors and Bézout's Identity	Examples & counterexamples	0
Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic	Examples & counterexamples	1
Regular Continued Fractions and Diophantine Approximation	Examples & counterexamples	1
Congruences, the Integers Modulo n and the Chinese Remainder Theorem	Examples & counterexamples	2
Pell Equations and Generalized Pell Orbits	Examples & counterexamples	2
Absolute Values Completions and P Adic Numbers	Examples & counterexamples	3
Arithmetic Functions and Dirichlet Convolution	Examples & counterexamples	3
Number Fields Rings of Integers and Discriminants	Examples & counterexamples	3
Positive Definite Binary Quadratic Forms and Reduction	Examples & counterexamples	3
Primitive Roots and Unit Groups Modulo N	Examples & counterexamples	3
Dirichlet Series and Euler Products	Examples & counterexamples	4
Prime Ideal Decomposition Ramification and the Different	Examples & counterexamples	4
Quadratic Residues and the Legendre Symbol	Examples & counterexamples	4
Decomposition Inertia and Frobenius	Examples & counterexamples	5
Minkowski Theory and Number Field Class Groups	Examples & counterexamples	5
Quadratic Reciprocity and the Jacobi Symbol	Examples & counterexamples	5
Sums of Two Squares	Examples & counterexamples	5
Average Orders Divisor Sums and Representation Counts	Examples & counterexamples	6
Cyclotomic Arithmetic and Reciprocity via Frobenius	Examples & counterexamples	6
Dirichlets Unit Theorem Regulators and S Units	Examples & counterexamples	6
Lagrange Four Square Theorem	Examples & counterexamples	6
Chebyshev Bounds and Mertens Theorems	Examples & counterexamples	7
Dirichlet Characters L Functions and Primes in Progressions	Examples & counterexamples	8
Perron Inversion and the Explicit Formula	Examples & counterexamples	8
Classical Zero Free Region and the Prime Number Theorem	Examples & counterexamples	9
Hilbert Symbols and the Quadratic Local Global Principle	Examples & counterexamples	9
Primitive Dirichlet L Functions and Functional Equations	Examples & counterexamples	9

Numerical Differential Equations and Stochastic Dynamics

Reading order

78 pages

Page	Companion	Dependency level
Euler–Maruyama Scheme and Coefficient Conditions	Examples & counterexamples	0
Finite bit random streams and variate generation	Examples & counterexamples	0
Finite generator master equations and uniformization	Examples & counterexamples	0
Fractional evolution and convolution quadrature	Examples & counterexamples	0
Geometric numerical integration	Examples & counterexamples	0
Langevin splittings and invariant measure error	Examples & counterexamples	0
Linear multistep methods and Dahlquist equivalence	Examples & counterexamples	0
Linear numerical boundary-value problems	Examples & counterexamples	0
Milstein stochastic taylor and iterated integrals	Examples & counterexamples	0
Nonlinear Boundary-Value Solvers	Examples & counterexamples	0
One-step ODE Consistency and Local Error	Examples & counterexamples	0
Pseudo arclength continuation and fold certificates	Examples & counterexamples	0
Stochastic Reaction Jump Simulation and Coupling	Examples & counterexamples	0
Adaptive stochastic paths and brownian bridge coupling	Examples & counterexamples	1
Certified stationary distribution and hitting statistic numerics	Examples & counterexamples	1
Euler–Maruyama Strong and Weak Convergence	Examples & counterexamples	1
Global ODE and Log-density Integration Error	Examples & counterexamples	1
Local bifurcation normal forms and numerical certificates	Examples & counterexamples	1
Tau Leaping and Reaction Diffusion Model Error	Examples & counterexamples	1
Coagulation Fragmentation and Sectional Population Balance Numerics	Examples & counterexamples	2
Delay differential equations and history discretization	Examples & counterexamples	2
Exponential and integrating factor integrators	Examples & counterexamples	2
Implicit stiff time stepping and algebraic error	Examples & counterexamples	2
Multirate and partitioned time integration	Examples & counterexamples	2
Periodic orbit shooting collocation and Floquet error	Examples & counterexamples	2
Positive Conservative Production Destruction Integrators	Examples & counterexamples	2
Renewal age and size structured numerical methods	Examples & counterexamples	2
Runge–Kutta Methods and Order	Examples & counterexamples	2
Adaptive ODE solvers and error estimation	Examples & counterexamples	3
Constrained dynamics shake and rattle	Examples & counterexamples	3
Deferred correction and collocation iterations	Examples & counterexamples	3
Differential algebraic equations and consistent initialization	Examples & counterexamples	3
Imex and operator splitting methods	Examples & counterexamples	3
Parallel in time parareal and mgrit	Examples & counterexamples	3
Rosenbrock and linearly implicit methods	Examples & counterexamples	3
Stability regions and stiff equations	Examples & counterexamples	3
Stable sde schemes for superlinear drift	Examples & counterexamples	3
Event location and hybrid trajectory error	Examples & counterexamples	4
Nonsmooth contact time stepping and projections	Examples & counterexamples	5

Numerical Linear Algebra and Approximation

Reading order

102 pages

Page	Companion	Dependency level
Composite quadrature rules	Examples & counterexamples	0
Fast multipole expansions and error	Examples & counterexamples	0
Finite-precision direct factorizations and least squares	Examples & counterexamples	0
Matrix exponential and phi function actions	Examples & counterexamples	0
Multigrid methods and complexity	Examples & counterexamples	0
Optimal star-discrepancy order in dimension at least three	Examples & counterexamples	0
Outward rounded interval and ball arithmetic	Examples & counterexamples	0
Polynomial eigenproblems and linearization	Examples & counterexamples	0
Polynomial interpolation and remainder	Examples & counterexamples	0
Radial basis and kernel interpolation	Examples & counterexamples	0
Randomized low-rank approximation	Examples & counterexamples	0
Sparse matrix models and kernel costs	Examples & counterexamples	0
Stationary iterations and relaxation	Examples & counterexamples	0
Tensor train compression and truncation error	Examples & counterexamples	0
Adaptive quadrature and error indicators	Examples & counterexamples	1
Alternating tensor optimization and effective eigenproblems	Examples & counterexamples	1
Certified geometric predicates	Examples & counterexamples	1
Computed fast transforms and convolution	Examples & counterexamples	1
Computed singular values and rank revealing factorizations	Examples & counterexamples	1
Hierarchical low rank matrices and error propagation	Examples & counterexamples	1
Interval newton krawczyk and polynomial root isolation	Examples & counterexamples	1
Krylov methods and Lanczos convergence	Examples & counterexamples	1
Oscillatory integrals and filon levin methods	Examples & counterexamples	1
Polynomial approximation and Chebyshev interpolation	Examples & counterexamples	1
Preconditioner algorithms and residual metrics	Examples & counterexamples	1
Shape preserving piecewise polynomial interpolation	Examples & counterexamples	1
Sparse direct elimination and ordering	Examples & counterexamples	1
Sparse grid cubature and mixed regularity	Examples & counterexamples	1
Spline evaluation and interpolation algorithms	Examples & counterexamples	1
Square root covariance update error and stability	Examples & counterexamples	1
Stable special function recurrences and certified evaluation	Examples & counterexamples	1
Wavelet transforms and nonlinear approximation	Examples & counterexamples	1
Barycentric rational and aaa approximation	Examples & counterexamples	2
Clenshaw–Curtis and extrapolation quadrature	Examples & counterexamples	2
Computed Schur decompositions and nonnormal spectra	Examples & counterexamples	2
Delaunay triangulations and voronoi computation	Examples & counterexamples	2
Ewald periodic potentials and particle mesh error	Examples & counterexamples	2
Gaussian basis integral recurrences and boys error	Examples & counterexamples	2
Interpolative cur and nystrom decompositions	Examples & counterexamples	2
Minimax approximation and Remez algorithms	Examples & counterexamples	2
Nonuniform fast fourier transforms	Examples & counterexamples	2
Sketched least squares and subspace embeddings	Examples & counterexamples	2
Toeplitz circulant and displacement structured solvers	Examples & counterexamples	2
Generalized eigenvalue pencils and QZ methods	Examples & counterexamples	3
Sylvester and lyapunov matrix equations	Examples & counterexamples	3
Verified linear systems and spectral enclosures	Examples & counterexamples	3
Matrix Functions and Scaling Squaring	Examples & counterexamples	4
Numerical algebraic Riccati equations and stabilizing branches	Examples & counterexamples	4
Restarted and block eigensolvers	Examples & counterexamples	4
Gaussian quadrature	Examples & counterexamples	5
Singular and infinite interval quadrature	Examples & counterexamples	6

Numerical Optimization Inverse Problems and Uncertainty Quantification

Reading order

70 pages

Page	Companion	Dependency level
Automatic differentiation and computational graphs	Examples & counterexamples	0
Finite basis projector energies and variational electronic structure	Examples & counterexamples	0
Global sensitivity anova and sobol estimation	Examples & counterexamples	0
Inequality KKT duality and optimality certificates	Examples & counterexamples	0
Nonlinear equations and Newton methods	Examples & counterexamples	0
Numerical differentiation and derivative error control	Examples & counterexamples	0
Numerical Inverse Problems and Regularization Error	Examples & counterexamples	0
Numerical optimization globalization and inexact certificates	Examples & counterexamples	0
Polynomial chaos stochastic collocation and Galerkin UQ	Examples & counterexamples	0
Rare event importance sampling and multilevel splitting	Examples & counterexamples	0
Trust region and cubic regularization methods	Examples & counterexamples	0
Barrier and primal dual interior point methods	Examples & counterexamples	1
Discrete continuous and pde adjoint sensitivities	Examples & counterexamples	1
Gauss-Newton and Levenberg-Marquardt methods	Examples & counterexamples	1
Global optimization and certified branch and bound	Examples & counterexamples	1
Minimum energy paths string neb and saddle certificates	Examples & counterexamples	1
Nonconvex line search and inexact newton methods	Examples & counterexamples	1
Proximal gradient and accelerated composite optimization	Examples & counterexamples	1
Revised simplex linear programming and certificates	Examples & counterexamples	1
Riemannian optimization on stiefel and grassmann manifolds	Examples & counterexamples	1
Self consistent projector iterations and certified mixing	Examples & counterexamples	1
Spectral regularization and parameter choice	Examples & counterexamples	1
Bfgs and limited memory quasi newton methods	Examples & counterexamples	2
Coordinate and block descent methods	Examples & counterexamples	2
Iterative regularization and early stopping	Examples & counterexamples	2
Nonlinear conjugate gradient methods	Examples & counterexamples	2
Numerical identifiability and optimal experiment design	Examples & counterexamples	2
Operator splitting ADMM and primal dual methods	Examples & counterexamples	2
Stochastic approximation and stochastic gradient methods	Examples & counterexamples	2
Subgradient bundle and derivative free optimization	Examples & counterexamples	2
Simulation gradient estimation and simultaneous perturbation	Examples & counterexamples	3
Sqp active set and augmented lagrangian methods	Examples & counterexamples	3
Variance reduced finite sum and adaptive gradient optimization	Examples & counterexamples	3
Direct transcription and numerical ode optimal control	Examples & counterexamples	4
Finite element pde constrained optimization	Examples & counterexamples	4

Numerical Pdes and Spatial Discretization

Reading order

110 pages

Page	Companion	Dependency level
ALE, interfaces and partitioned multiphysics coupling	Examples & counterexamples	0
Conservative finite-volume weak limits	Examples & counterexamples	0
Elliptic finite-difference methods	Examples & counterexamples	0
Finite-element meshes and interpolation	Examples & counterexamples	0
High-resolution TVD, ENO and WENO reconstruction	Examples & counterexamples	0
Interacting particle mean field and vortex blob error	Examples & counterexamples	0
Linear scheme stability and Lax equivalence	Examples & counterexamples	0
Meshfree kernel approximation and conservative sph	Examples & counterexamples	0
Nonlinear full-approximation multigrid	Examples & counterexamples	0
Nystrom product integration and fredholm error	Examples & counterexamples	0
Riemann solvers for hyperbolic systems	Examples & counterexamples	0
Spectral collocation and Galerkin methods	Examples & counterexamples	0
Céa and Aubin–Nitsche error estimates	Examples & counterexamples	1
Mixed finite elements and Stokes stability	Examples & counterexamples	1
Monotone finite-volume entropy convergence	Examples & counterexamples	1
Parabolic finite-difference methods	Examples & counterexamples	1
Particle in cell and particle mesh kinetic methods	Examples & counterexamples	1
Positivity, invariant domains and entropy-stable systems fluxes	Examples & counterexamples	1
Substructuring, FETI-DP and BDDC	Examples & counterexamples	1
Symmetric interior-penalty discontinuous Galerkin	Examples & counterexamples	1
Volume integral scattering and partial wave error	Examples & counterexamples	1
Adaptive finite-element methods	Examples & counterexamples	2
Finite-element eigenvalue approximation	Examples & counterexamples	2
Hyperbolic DG and summation-by-parts methods	Examples & counterexamples	2
Hyperbolic finite-difference methods	Examples & counterexamples	2
Linear elasticity, nearly incompressible limits and locking	Examples & counterexamples	2
Mixed H(div) fluxes, Darcy flow and local conservation	Examples & counterexamples	2
Nitsche methods for weak boundary enforcement	Examples & counterexamples	2
Overlapping Schwarz and stable subspace decompositions	Examples & counterexamples	2
Parabolic finite-element methods	Examples & counterexamples	2
Projection, pressure correction and saddle-point solvers	Examples & counterexamples	2
Variational crimes and Strang lemmas	Examples & counterexamples	2
Well-balanced balance laws and wet/dry fronts	Examples & counterexamples	2
Crouzeix–Raviart finite elements	Examples & counterexamples	3
Curved geometry, mesh generation and conservative transfer	Examples & counterexamples	3
Discrete de Rham complexes and edge-element foundations	Examples & counterexamples	3
First-order least-squares FEM and practical DPG	Examples & counterexamples	3
Graph diffusion and metric network pde numerics	Examples & counterexamples	3
hp finite elements and spectral elements	Examples & counterexamples	3
Nonlinear parabolic compactness and monotone evolution	Examples & counterexamples	3
SUPG, residual stabilization and variational multiscale methods	Examples & counterexamples	3
Variational inequalities and spatial contact discretization	Examples & counterexamples	3
Evolutionary incompressible flow discretization	Examples & counterexamples	4
Finite-volume diffusion on admissible meshes	Examples & counterexamples	4
Isogeometric Galerkin discretization	Examples & counterexamples	4
Maxwell eigenproblems and compatible vector wave approximation	Examples & counterexamples	4
Semi-Lagrangian characteristic transport	Examples & counterexamples	4
Yee FDTD and transient Maxwell finite elements	Examples & counterexamples	4
Compressible viscous flow and entropy-stable viscous fluxes	Examples & counterexamples	5
Homogenization, multiscale FEM and heterogeneous media	Examples & counterexamples	5
Monotone Hamilton–Jacobi–Bellman schemes and policy iteration	Examples & counterexamples	5
Phase-field and dissipative gradient-flow computation	Examples & counterexamples	5
Reaction-diffusion systems, invariants and pattern computation	Examples & counterexamples	5
Boundary-integral equations and Galerkin BEM	Examples & counterexamples	6
Exterior waves, absorbing boundaries and PML	Examples & counterexamples	7

Online and Reinforcement Learning

Reading order

42 pages

Page	Companion	Dependency level
Empirical-Bernstein confidence sequences under adaptive sampling	Examples & counterexamples	0
Finite MDPs and Bellman operators	Examples & counterexamples	0
Online protocols and regret benchmarks	Examples & counterexamples	0
Preference learning and direct preference optimization	Examples & counterexamples	0
Average-reward regret and UCRL2	Examples & counterexamples	1
Function approximation, off-policy instability, and deep-RL limits	Examples & counterexamples	1
Hedge and online gradient descent	Examples & counterexamples	1
Imitation learning and DAgger	Examples & counterexamples	1
Linear contextual bandits and LinUCB	Examples & counterexamples	1
Mirror descent and adaptive regularization	Examples & counterexamples	1
POMDPs and belief-state planning	Examples & counterexamples	1
Stochastic finite-armed bandits	Examples & counterexamples	1
Tabular prediction and control by interaction	Examples & counterexamples	1
Adversarial bandits and importance-weighted feedback	Examples & counterexamples	2
Model-based planning, Dyna, and model exploitation	Examples & counterexamples	2
Offline reinforcement learning and off-policy evaluation	Examples & counterexamples	2
Optimistic exploration and PAC-MDP sample complexity	Examples & counterexamples	2
Policy gradients, REINFORCE, and actor–critic	Examples & counterexamples	2
Constrained MDPs and policy safety	Examples & counterexamples	3
Contextual expert bandits	Examples & counterexamples	3
Verifiable generation, group-relative updates, and test-time sampling	Examples & counterexamples	3

Partial Differential Equations

Reading order

58 pages

Page	Companion	Dependency level
Bessel-Potential Completions and Real-Order Sobolev Spaces	Examples & counterexamples	0
Euclidean Surface Measure, Divergence, and Green Identities	Examples & counterexamples	0
Hamilton Jacobi Equations and Viscosity Solutions	Examples & counterexamples	0
Partial Differential Equations and Characteristics	Examples & counterexamples	0
Strongly Continuous Semigroups and Hille Yosida	Examples & counterexamples	0
Weak Derivatives and Sobolev Spaces	Examples & counterexamples	0
Analytic Majorants and the Cauchy–Kovalevskaya Theorem	Examples & counterexamples	1
Harmonic Functions and Mean Values in Rn	Examples & counterexamples	1
Smooth Approximation and Sobolev Extension	Examples & counterexamples	1
Maximum Principles Harnack and Liouville in Rn	Examples & counterexamples	2
Quasilinear Characteristics and Cauchy Kovalevskaya	Examples & counterexamples	2
Sobolev Traces and Zero Boundary Values	Examples & counterexamples	2
Fundamental Solutions Newtonian Potentials and Green Functions	Examples & counterexamples	3
Poisson Problems and Interior Harmonic Estimates	Examples & counterexamples	4
Sobolev Poincare and Morrey Inequalities	Examples & counterexamples	5
The Heat Kernel and the Cauchy Problem	Examples & counterexamples	5
Wave Equation Representation Formulas	Examples & counterexamples	5
Heat Equation Maximum Principles Duhamel and Smoothing	Examples & counterexamples	6
Rellich Kondrachov and Sobolev Compactness	Examples & counterexamples	6
Wave Energy, Finite Propagation and Huygens' Principle	Examples & counterexamples	6
Lax--Milgram and Weak Elliptic Solutions	Examples & counterexamples	7
Scalar Conservation Laws and Entropy Solutions	Examples & counterexamples	7
Fredholm Elliptic Problems and the Elliptic Spectrum	Examples & counterexamples	8
Schauder and Lᵖ Elliptic Estimates	Examples & counterexamples	8
Interior and Boundary Sobolev Elliptic Regularity	Examples & counterexamples	9
Analytic Semigroups and Linear Evolution Equations	Examples & counterexamples	10
The Direct Method and Euler--Lagrange Equations	Examples & counterexamples	10
Weak Elliptic Maximum Principles and Holder Regularity	Examples & counterexamples	10
Constrained Variational Problems and Variational Inequalities	Examples & counterexamples	11

Probability

Reading order

50 pages

Page	Companion	Dependency level
Probability Spaces Random Variables and Expectation	Examples & counterexamples	0
Conditional Expectation	Examples & counterexamples	1
Independence Borel Cantelli and Zero One Laws	Examples & counterexamples	1
Discrete Time Martingales	Examples & counterexamples	2
Infinite Product Measures and Kolmogorov Extension	Examples & counterexamples	2
Modes of Convergence for Random Variables	Examples & counterexamples	2
Standard-Borel Real Codings and Determining Classes	Examples & counterexamples	3
Weak Laws and Series of Independent Random Variables	Examples & counterexamples	3
Conditional Distributions and Regular Conditional Probability	Examples & counterexamples	4
Strong Laws of Large Numbers	Examples & counterexamples	4
Weak Convergence Tightness and Representation	Examples & counterexamples	5
Characteristic Functions Inversion and Continuity	Examples & counterexamples	6
Central Limit Theorems	Examples & counterexamples	7
Brownian Motion Construction and Continuity	Examples & counterexamples	8
Martingale Inequalities and Convergence	Examples & counterexamples	8
Stopping Times and Optional Stopping	Examples & counterexamples	9
Brownian Motion, Markov Properties and Hitting Times	Examples & counterexamples	10
Markov Kernels and Markov Chains	Examples & counterexamples	10
Brownian Path Properties	Examples & counterexamples	11
Recurrence Transience and Hitting Times for Markov Chains	Examples & counterexamples	11
Stationary Markov Chains and Ergodic Limits	Examples & counterexamples	12
The Ito Integral with Respect to Brownian Motion	Examples & counterexamples	12
Itos Formula and Brownian Martingales	Examples & counterexamples	13
Forward Itô SDE Well-Posedness	Examples & counterexamples	14
Gaussian smoothing densities, scores, and endpoint limits	Examples & counterexamples	15

Quantum Algebra and Representation Theory

Reading order

28 pages

Page	Companion	Dependency level
Affine Current Algebras, Virasoro, and Sugawara	Examples & counterexamples	0
Brauer, BMW, and Partition Diagram Algebras	Examples & counterexamples	0
Temperley–Lieb Diagram Algebra and Catalan Basis	Examples & counterexamples	0
Universal Clifford Algebras and Their PBW Basis	Examples & counterexamples	0
CAR Algebras, Majorana Generators, and Clifford Comparison	Examples & counterexamples	1
Clifford Classification, Modules, and Graded Tensor Products	Examples & counterexamples	1
Temperley–Lieb Link-State Modules and Cellularity	Examples & counterexamples	1
Vertex Algebra Axioms and State–Field Constructions	Examples & counterexamples	1
Pin and Spin Groups and Their Representations	Examples & counterexamples	2
Temperley–Lieb Semisimplicity and Generic Structure	Examples & counterexamples	2
Vertex Modules, Rationality, and Logarithmic Regimes	Examples & counterexamples	2
Temperley–Lieb Root-of-Unity and Modular Representation Theory	Examples & counterexamples	3
Temperley–Lieb Jones–Wenzl Projectors, Traces, and Quotients	Examples & counterexamples	4
Temperley–Lieb Towers, Spin Chains, and Scaling Interfaces	Examples & counterexamples	4

Quantum Algorithms

Reading order

74 pages

Page	Companion	Dependency level
Algorithm Composition, Precision Budgets, and Dequantization	Examples & counterexamples	0
Block Encodings	Examples & counterexamples	0
Computational Problems, Promises, and Cost Models	Examples & counterexamples	0
Continuous-Time Quantum Walks	Examples & counterexamples	0
Discrete Quantum Walks and Szegedy Theory	Examples & counterexamples	0
Finite Abelian Fourier Analysis	Examples & counterexamples	0
QMA Verification and Local Hamiltonian	Examples & counterexamples	0
Quantum Communication and Fingerprinting	Examples & counterexamples	0
Quantum Complexity Classes and Oracle Separations	Examples & counterexamples	0
Quantum Fourier-Transform Circuits	Examples & counterexamples	0
Quantum Signal Processing	Examples & counterexamples	0
Amplitude Amplification	Examples & counterexamples	1
Finite Abelian Hidden-Subgroup Algorithms	Examples & counterexamples	1
Hamiltonian Simulation and Product Formulas	Examples & counterexamples	1
Quantum Phase Estimation	Examples & counterexamples	1
Quantum Singular-Value Transformation	Examples & counterexamples	1
Reversible Arithmetic and Oracle Conversion	Examples & counterexamples	1
Adiabatic Computation and Circuit Equivalence	Examples & counterexamples	2
Amplitude Estimation and Counting	Examples & counterexamples	2
Exact Oracle Algorithms	Examples & counterexamples	2
Grover Search and Invariant-Plane Rotations	Examples & counterexamples	2
Linear Combinations of Unitaries and Taylor Simulation	Examples & counterexamples	2
Polynomial Method and Approximate Degree	Examples & counterexamples	2
Quantum Order Finding and Rational Reconstruction	Examples & counterexamples	2
Quantum Spectral Estimation and Matrix Functions	Examples & counterexamples	2
Quantum-Walk Search and Element Distinctness	Examples & counterexamples	2
Adversary Methods and Quantum Query Complexity	Examples & counterexamples	3
Period, Hidden-Shift, and Number-Theoretic Algorithms	Examples & counterexamples	3
Quantum Query Lower Bounds and Search Optimality	Examples & counterexamples	3
Quantum Sampling, Expectation, and Monte Carlo	Examples & counterexamples	3
Qubitization and Near-Optimal Hamiltonian Simulation	Examples & counterexamples	3
Shor’s Factoring Algorithm	Examples & counterexamples	3
Simon’s Algorithm and Finite-Field Recovery	Examples & counterexamples	3
Topological Quantum Algorithms and Braid Compilation	Examples & counterexamples	3
Phase-Estimation Linear-System Algorithms	Examples & counterexamples	4
Block-Encoding Linear Systems and Variable-Time Amplification	Examples & counterexamples	5
Quantum Optimization, Semidefinite Programs, and Variational Algorithms	Examples & counterexamples	6

Quantum Computing

Reading order

54 pages

Page	Companion	Dependency level
Finite-Dimensional Complex Inner-Product Spaces and Matrix Interfaces	Examples & counterexamples	0
Registers, Tensor Products, and Partial Trace	Examples & counterexamples	0
Unitary Fusion Spaces, F/R Symbols, and Anyon Models	Examples & counterexamples	0
Classical and Quantum Error-Correcting Codes	Examples & counterexamples	1
Pure and Mixed States, Ensembles, and Schmidt Decomposition	Examples & counterexamples	1
Measurements, POVMs, and Instruments	Examples & counterexamples	2
Qubits, Bloch Sphere, Pauli Algebra, and One-Qubit Gates	Examples & counterexamples	2
Channels, Kraus, Choi, and Stinespring Representations	Examples & counterexamples	3
Controlled Gates, Phase, and Entanglement	Examples & counterexamples	3
Circuit Syntax, Denotational Semantics, and Measurement	Examples & counterexamples	4
Knill–Laflamme Recovery and Subsystem Codes	Examples & counterexamples	4
Pauli and Clifford Groups, Binary Symplectic Tableaux	Examples & counterexamples	4
Trace Distance, Fidelity, and Information Limits	Examples & counterexamples	4
Bell States, Teleportation, and Dense Coding	Examples & counterexamples	5
Reversible Computation, Ancillas, and Coherent Oracles	Examples & counterexamples	5
Stabilizer Circuits and Gottesman–Knill Simulation	Examples & counterexamples	5
Continuous-Gate Universality and Exact Synthesis	Examples & counterexamples	6
Stabilizer and CSS Codes, Syndromes, and Procedures	Examples & counterexamples	6
Approximate Universality and Solovay–Kitaev	Examples & counterexamples	7
Graph States and Measurement-Based Computation	Examples & counterexamples	7
Toric Code Anyons, Logical Operators, and Protection	Examples & counterexamples	7
Clifford+T Exact and Approximate Synthesis	Examples & counterexamples	8
Noise, Fault Tolerance, Concatenation, and Thresholds	Examples & counterexamples	8
State Preparation, Isometry Synthesis, and Limits	Examples & counterexamples	8
Uniform Circuit Families, Promise Problems, and Complexity	Examples & counterexamples	9
Circuit Optimization, Resource Tradeoffs, and Lower Bounds	Examples & counterexamples	10
Topological Braid Gates, Encodings, and Universality	Examples & counterexamples	10

Quantum Groups and Crystal Bases

Reading order

18 pages

Page	Companion	Dependency level
Quantized Enveloping Algebras and Quantum Serre Relations	Examples & counterexamples	0
Yangians, Quantum Affine Algebras, and Transfer Matrices	Examples & counterexamples	0
Divided Powers, Integral Forms, and Quantum Highest Weights	Examples & counterexamples	1
Kashiwara Operators and Crystal Lattices	Examples & counterexamples	2
Tensor-Product Crystals and Highest-Weight Crystal Uniqueness	Examples & counterexamples	3
Balanced Triples and Global Crystal Bases	Examples & counterexamples	4
Quantum Affine Type A and Level-One Fock Space	Examples & counterexamples	5
Quantum Groups at Roots of Unity and Tilting Quotients	Examples & counterexamples	5
Level-One Fock Space and Residue Crystals	Examples & counterexamples	6

Real Analysis

Reading order

112 pages

Page	Companion	Dependency level
Construction of the Real Numbers via Cauchy Sequences		0
Construction of the Real Numbers via Dedekind Cuts		0
Foundations of the Real Numbers for Analysis		1
Suprema and Infima	Examples & counterexamples	2
Countability and Uncountability	Examples & counterexamples	3
Roots, Rational Powers, and Classical Inequalities	Examples & counterexamples	3
Sequences and Limits	Examples & counterexamples	4
Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness	Examples & counterexamples	5
limsup, liminf, and Subsequential Limits	Examples & counterexamples	6
The Formal Laurent Series Field ℝ((t⁻¹)): Cauchy Complete, Non-Archimedean, Not Complete		6
Topology of ℝ	Examples & counterexamples	6
Equivalent Forms of Completeness	Examples & counterexamples	7
Limits of Real Functions	Examples & counterexamples	7
Series: Convergence and the Nonnegative Tests	Examples & counterexamples	7
Absolute and Conditional Convergence; Rearrangement; Products	Examples & counterexamples	8
Continuity, IVT, EVT, and Uniform Continuity	Examples & counterexamples	8
The Cantor Set, Baire Category, and Measure Zero in ℝ	Examples & counterexamples	8
Monotone Functions, Discontinuities, and Continuity Sets	Examples & counterexamples	9
The Derivative and the Mean Value Theorems	Examples & counterexamples	10
The Riemann Integral: Definition and Integrability	Examples & counterexamples	10
Darboux, L'Hôpital, and Taylor's Theorem	Examples & counterexamples	11
Properties of the Integral and the Working FTC	Examples & counterexamples	11
The Gauge Integral and Cousin's Lemma	Examples & counterexamples	11
Bounded Variation and the Riemann–Stieltjes Integral	Examples & counterexamples	12
Convexity	Examples & counterexamples	12
Improper Integrals	Examples & counterexamples	12
Rⁿ as a Normed Space; Vector-Valued Functions	Examples & counterexamples	12
Sequences and Series of Functions; Uniform Convergence	Examples & counterexamples	12
Approximation and Compactness in C(K)	Examples & counterexamples	13
Arc Length and Rectifiable Curves	Examples & counterexamples	13
Power Series and Real-Analytic Functions	Examples & counterexamples	13
The Fundamental Theorems of Calculus	Examples & counterexamples	13
The Riemann Integral in Rᵐ and Jordan Content	Examples & counterexamples	13
The Total Derivative in ℝᵐ → ℝⁿ	Examples & counterexamples	13
Mixed Partials, Taylor Formulae, and Extrema	Examples & counterexamples	14
Sine, Cosine, and the Definition of Pi	Examples & counterexamples	14
The Exponential Function	Examples & counterexamples	14
The Inverse and Implicit Function Theorems	Examples & counterexamples	14
Convex and Semicontinuous Functions on Rⁿ	Examples & counterexamples	15
Fubini and Change of Variables	Examples & counterexamples	15
Fundamental Trigonometric Identities	Examples & counterexamples	15
pi: the Equivalent Characterizations	Examples & counterexamples	15
Picard-Lindelöf and First-Order Ordinary Differential Equations	Examples & counterexamples	15
The Logarithm and General Powers	Examples & counterexamples	15
Areas of Elementary Plane Figures	Examples & counterexamples	16
Further Trigonometric Identities and Inverse Functions	Examples & counterexamples	16
Improper and Parameter-Dependent Multiple Integrals	Examples & counterexamples	16
Line Integrals and the Gradient Theorem	Examples & counterexamples	16
The Complex Exponential and Euler's Formula	Examples & counterexamples	16
The Integral Logarithm and the Equivalence of Its Characterisations	Examples & counterexamples	16
The Inverse Function Theorem Completed	Examples & counterexamples	16
Trigonometric and Oscillatory Examples in Several Variables	Examples & counterexamples	16
Constant Rank, Submersions, Immersions and Regular Level Sets	Examples & counterexamples	17
Trigonometric and Oscillatory Examples in One Variable	Examples & counterexamples	17
Volumes of Elementary Solids and Solids of Revolution	Examples & counterexamples	17
Regular Surfaces and Surface Integrals	Examples & counterexamples	18
The Real Gamma and Beta Functions	Examples & counterexamples	18
The Divergence Theorem and Classical Stokes	Examples & counterexamples	19

Representation Learning and Generative Models

Reading order

34 pages

Page	Companion	Dependency level
Anderson reverse-time diffusion	Examples & counterexamples	0
Discrete diffusion chains and DDPM objectives	Examples & counterexamples	0
Discrete masked diffusion and finite-state reverse generation	Examples & counterexamples	0
Energy models, score matching, and contrastive divergence	Examples & counterexamples	0
Exact-density normalizing flows	Examples & counterexamples	0
Finite-metric transport duality	Examples & counterexamples	0
Foundation models, transfer, and scaling evidence	Examples & counterexamples	0
GAN objectives and the ideal discriminator	Examples & counterexamples	0
Multimodal contrastive alignment and transfer	Examples & counterexamples	0
Neural ODEs and continuous normalizing flows	Examples & counterexamples	0
Reconstruction and masked autoencoders	Examples & counterexamples	0
Self-supervised and contrastive representations	Examples & counterexamples	0
Variational autoencoders and pathwise training	Examples & counterexamples	0
Conditional flow matching and transport paths	Examples & counterexamples	1
Kantorovich–Rubinstein duality on compact metric spaces	Examples & counterexamples	1
Score SDEs and reverse-time generation	Examples & counterexamples	1
Integral probability metrics and Wasserstein GANs	Examples & counterexamples	2

Representation Theory of Groups

Reading order

64 pages

Page	Companion	Dependency level
Artin Induction and Rational Characters	Examples & counterexamples	0
Blocks Defect Groups and the Brauer Homomorphism	Examples & counterexamples	0
Brauer Induction and Elementary Subgroups	Examples & counterexamples	0
Bruhat Decomposition and Flags over Finite Fields	Examples & counterexamples	0
Clifford Theory over Normal Subgroups	Examples & counterexamples	0
Equivariant Constructible Perverse Foundations	Examples & counterexamples	0
Frobenius Groups and the Normal Complement Theorem	Examples & counterexamples	0
Galois Orbits and Descent of Simple Finite-Group Modules	Examples & counterexamples	0
Haar Measure Existence and Uniqueness	Examples & counterexamples	0
Unitary Representations, Positive Type and GNS	Examples & counterexamples	0
Monomial Characters and M Groups	Examples & counterexamples	1
Principal Series Representations of GL N over a Finite Field	Examples & counterexamples	1
Projective Extensions and the Little Group Method	Examples & counterexamples	1
Schur Indices and Fields of Definition	Examples & counterexamples	1
Signed Quiver Flag Convolution And Klr Ext Realization	Examples & counterexamples	1
The Modular Function and L1 Group Algebras	Examples & counterexamples	1
Vertices Sources and the Green Correspondence	Examples & counterexamples	1
Brauers First Main Theorem	Examples & counterexamples	2
Complete Reducibility for Compact Groups	Examples & counterexamples	2
Induced Unitary Representations of Locally Compact Groups	Examples & counterexamples	2
Quiver Full Flag Projective Semisimplicity	Examples & counterexamples	2
Brauers Second Main Theorem	Examples & counterexamples	3
Mackeys Imprimitivity Theorem	Examples & counterexamples	3
Peter Weyl Theory for General Compact Groups	Examples & counterexamples	3
Exponentiated Weyl Relations and Regular Representations	Examples & counterexamples	4
Group C Star Algebras and the Fell Unitary Dual	Examples & counterexamples	4
Unitary Poincaré Representations and Little Groups	Examples & counterexamples	4
Amenability Reiter Nets and Folner Conditions	Examples & counterexamples	5
Direct Integral Decomposition and Type I Groups	Examples & counterexamples	5
Sl2 R Principal and Complementary Series	Examples & counterexamples	5
Kazhdan's Property T and Spectral Gap	Examples & counterexamples	6
SL2(R): Discrete Series and the Unitary Dual	Examples & counterexamples	6

Scheme Theory

Reading order

54 pages

Page	Companion	Dependency level
Presheaves Sheaves Stalks and Sheafification	Examples & counterexamples	0
Sheaf Operations Exactness Ringed Spaces and Module Pullback	Examples & counterexamples	1
Affine Schemes and the Structure Sheaf	Examples & counterexamples	2
Sheaf Cohomology Cech Cohomology and Comparison	Examples & counterexamples	2
Schemes Subschemes and Morphisms Locally of Finite Type	Examples & counterexamples	3
Fibre Products Base Change and Scheme Theoretic Fibres	Examples & counterexamples	4
Diagonals Separated Morphisms and Valuative Uniqueness	Examples & counterexamples	5
Group Schemes of Finite Type over a Field	Examples & counterexamples	5
Kahler Differentials Conormal Sequences and Infinitesimal Lifting	Examples & counterexamples	5
Affine Group Schemes, Hopf Algebras, and Rational Representations	Examples & counterexamples	6
Finite Proper and Projective Morphisms	Examples & counterexamples	6
Flat Smooth and Etale Morphisms	Examples & counterexamples	7
Quasi Coherent and Coherent Sheaves and Vector Bundles	Examples & counterexamples	7
Cartier and Weil Divisors Line Bundles and Picard Groups	Examples & counterexamples	8
Lie Algebras and Infinitesimal Group Schemes	Examples & counterexamples	8
Proj Projective Schemes Twisting Sheaves and Ampleness	Examples & counterexamples	8
Algebraic Group Actions, Orbits, Stabilizers, and Controlled Quotients	Examples & counterexamples	9
Cohomology of Quasi Coherent Sheaves on Affine and Projective Schemes	Examples & counterexamples	9
Algebraic Spaces, Stacks, and Derived Algebraic Geometry Foundations	Examples & counterexamples	10
Compact Support Sheaf Calculus	Examples & counterexamples	10
Hilbert Functors and Projective Hilbert Schemes	Examples & counterexamples	10
Smooth Proper Curves Divisors Genus and Ramification	Examples & counterexamples	10
Deformation Theory of Schemes and Obstruction Spaces	Examples & counterexamples	11
Riemann Roch for Curves via Euler Characteristics	Examples & counterexamples	11
Blowups, Exceptional Divisors, and Strict Transforms	Examples & counterexamples	12
Residues Serre Duality for Curves and the Full Riemann Roch Theorem	Examples & counterexamples	12
Analytic Derived Six Operations And Duality	Examples & counterexamples	13

Set Theory

Reading order

81 pages

Page	Companion	Dependency level
The ZFC Axioms and the Basic Set Constructions	Examples & counterexamples	0
Relations, Functions, and Quotients	Examples & counterexamples	1
Construction of the Natural Numbers		2
Dependent Choice and the Complete-Metric Baire Theorem	Examples & counterexamples	3
Formal Set-Theoretic Syntax, Structures, and Satisfaction	Examples & counterexamples	3
Order, Zorn's Lemma, and the Axiom of Choice	Examples & counterexamples	3
Filters and Ultrafilters	Examples & counterexamples	4
Ordinals, Cardinals, and Transfinite Recursion	Examples & counterexamples	4
Ordinal Arithmetic and the First Uncountable Ordinal	Examples & counterexamples	5
Well-Founded Relations, Rank, and the Cumulative Hierarchy	Examples & counterexamples	5
Cardinal Arithmetic, Cofinality and the Alephs	Examples & counterexamples	6
Borel and Analytic Sets, Perfect Sets, and Determinacy	Examples & counterexamples	7
Club, Stationary Sets, and Pressing Down	Examples & counterexamples	7
Deduction, Soundness, Completeness, and Compactness	Examples & counterexamples	7
Weak Choice Principles and Sierpiński's Theorem	Examples & counterexamples	7
Arithmetization, Incompleteness, and Relative Consistency	Examples & counterexamples	8
Boolean Algebras, Stone Duality, and the Prime Ideal Theorem	Examples & counterexamples	8
Reflection, Absoluteness, and Elementary Submodels	Examples & counterexamples	8
Set-Theoretic Trees, Delta Systems, and Diamond	Examples & counterexamples	8
Forcing Orders, Names, and Generic Extensions	Examples & counterexamples	9
PCF Scales and ZFC Dowker Spaces	Examples & counterexamples	9
The Constructible Hierarchy and Inner Models	Examples & counterexamples	9
Condensation, GCH, and Diamond in L	Examples & counterexamples	10
Large Cardinals, Measures, and Elementary Embeddings	Examples & counterexamples	10
The Forcing Theorem and Formal Consistency Transfer	Examples & counterexamples	10
Permutation Models and Transfer to ZF	Examples & counterexamples	11
Preservation, Cohen Forcing, and the Continuum	Examples & counterexamples	11
Finite-Support Iterations and Martin's Axiom	Examples & counterexamples	12
Solovay's Model and Regularity of All Sets of Reals	Examples & counterexamples	12
Symmetric Extensions and Basic Choice-Failure Models	Examples & counterexamples	12
Boolean Prime Ideal Theorem in the Basic Cohen Model	Examples & counterexamples	13
Choice Strength in Baire, Urysohn, Stone, and Tychonoff	Examples & counterexamples	13
Easton's Theorem and Cardinal Invariants of the Continuum	Examples & counterexamples	13
Normal Moore Spaces, PMEA, and Consistency Strength	Examples & counterexamples	13
Prikry Forcing and Gitik's Singular-Cardinal Model	Examples & counterexamples	13
Suslin Trees, Lines, Algebras, and Independence	Examples & counterexamples	13
Symmetric Collapse and Ultrafilter-Free Models	Examples & counterexamples	13
Halpern–Läuchli and BPI Without Choice	Examples & counterexamples	14
Proper Forcing, Countable-Support Iterations, and PFA	Examples & counterexamples	14
Shelah's Baire-Property Model and Inner-Model Lower Bounds	Examples & counterexamples	14
Minimal Walks, Oscillation, and L- and S-Spaces	Examples & counterexamples	15

Statistical Learning and Classical Models

Reading order

68 pages

Page	Companion	Dependency level
Additive-noise causal discovery and identifiability	Examples & counterexamples	0
Adversarial robustness and robust risk	Examples & counterexamples	0
CART trees and pruning	Examples & counterexamples	0
Differential privacy mechanisms and composition	Examples & counterexamples	0
Federated averaging, local SGD, and SCAFFOLD	Examples & counterexamples	0
Gaussian-process regression and prediction	Examples & counterexamples	0
K-means, Gaussian mixtures, and soft assignments	Examples & counterexamples	0
Latent factors, ICA, NMF, and representation ambiguity	Examples & counterexamples	0
Linear classifiers, discriminants, and logistic models	Examples & counterexamples	0
Linear prediction, shrinkage, and sparse regression	Examples & counterexamples	0
Monotone risk calibration and finite-sample deployment guarantees	Examples & counterexamples	0
Nearest neighbors and learned metrics	Examples & counterexamples	0
PC algorithm and Markov-equivalence structure learning	Examples & counterexamples	0
PCA, kernel PCA, MDS, and manifold embeddings	Examples & counterexamples	0
Prediction problems, risks, and empirical minimization	Examples & counterexamples	0
RKHS regularization, kernels, and support-vector machines	Examples & counterexamples	0
Undirected graphical models and sparse precision estimation	Examples & counterexamples	0
AdaBoost and gradient boosting	Examples & counterexamples	1
Algorithmic stability and generalization	Examples & counterexamples	1
Bagging and random forests	Examples & counterexamples	1
Calibration, proper losses, and surrogate classification	Examples & counterexamples	1
Distribution shift and domain adaptation	Examples & counterexamples	1
Finite-class PAC learning and ERM guarantees	Examples & counterexamples	1
Minimax learning lower bounds and optimal rates	Examples & counterexamples	1
Rademacher complexity and uniform generalization	Examples & counterexamples	1
Sample compression and PAC-Bayes bounds	Examples & counterexamples	1
Sketched kernel learning and certified approximate solves	Examples & counterexamples	1
Spectral clustering and graph cuts	Examples & counterexamples	1
Active querying, version spaces, and abstention	Examples & counterexamples	2
Fairness criteria, impossibility boundaries, and limits	Examples & counterexamples	2
Predictive calibration and uncertainty under shift	Examples & counterexamples	2
Private ERM and private online learning	Examples & counterexamples	2
Semi-supervised graph propagation and pseudo-labeling limits	Examples & counterexamples	2
VC dimension, growth functions, and classification sample complexity	Examples & counterexamples	2

Statistical Models and Experiments

Reading order

124 pages

Page	Companion	Dependency level
BH independent FDR	Examples & counterexamples	0
Counting-process compensators, integrals, brackets	Examples & counterexamples	0
Detection and quantification limits	Examples & counterexamples	0
Discrete test supermartingales and Ville	Examples & counterexamples	0
Full factorials	Examples & counterexamples	0
Group-sequential alpha spending	Examples & counterexamples	0
Independent-subject terminal influence limits	Examples & counterexamples	0
IV and LATE	Examples & counterexamples	0
Measurement error and misclassification	Examples & counterexamples	0
Method agreement	Examples & counterexamples	0
Missingness and nonidentification	Examples & counterexamples	0
Poisson and Cox point processes	Examples & counterexamples	0
Polyhedral selective inference	Examples & counterexamples	0
Potential outcomes and identification	Examples & counterexamples	0
Power and sample size	Examples & counterexamples	0
Reproducibility and transparent reporting	Examples & counterexamples	0
Scientific model contracts and applicability	Examples & counterexamples	0
Sequential likelihood-ratio tests	Examples & counterexamples	0
Stationary means and sample autocovariance	Examples & counterexamples	0
Two-stage conditional-error adaptation	Examples & counterexamples	0
Valid p-values, Bonferroni, simultaneous coverage	Examples & counterexamples	0
Assignment randomization and sharp-null tests	Examples & counterexamples	1
BY arbitrary dependence and BH PRDS	Examples & counterexamples	1
Calibration and measurement uncertainty	Examples & counterexamples	1
Event observation, censoring, truncation	Examples & counterexamples	1
Finite-history linear prediction	Examples & counterexamples	1
Finite-window spatial likelihood	Examples & counterexamples	1
Fixed-design regression, estimability, and Gauss–Markov	Examples & counterexamples	1
Fractional factorial aliases and foldover	Examples & counterexamples	1
Generalized linear models and Fisher scoring	Examples & counterexamples	1
Holm Step-Down Familywise Error Control	Examples & counterexamples	1
Missingness weights and sensitivity	Examples & counterexamples	1
Multivariate normal, Wishart, and Hotelling T²	Examples & counterexamples	1
PCA and eigengap stability	Examples & counterexamples	1
Random fields and variograms	Examples & counterexamples	1
SCM, DAGs, adjustment	Examples & counterexamples	1
Spatial summaries and Palm laws	Examples & counterexamples	1
Spectral measures and periodogram bias	Examples & counterexamples	1
Standardization and IPW identification	Examples & counterexamples	1
State-space model laws	Examples & counterexamples	1
Sub-Gaussian mixture confidence sequences	Examples & counterexamples	1
Temporal intensity and likelihood	Examples & counterexamples	1
ARMA roots, spectra and inference	Examples & counterexamples	2
Blocking, pairs, and Latin squares	Examples & counterexamples	2
Cox PH and partial likelihood	Examples & counterexamples	2
Exact Gaussian regression inference and prediction	Examples & counterexamples	2
Finite populations and design probability	Examples & counterexamples	2
Interference and exposure mappings	Examples & counterexamples	2
Kriging	Examples & counterexamples	2
Logrank and weighted logrank	Examples & counterexamples	2
Mediation and cross-world assumptions	Examples & counterexamples	2
Mixed effects and longitudinal models	Examples & counterexamples	2
Nelson–Aalen, Kaplan–Meier, Greenwood	Examples & counterexamples	2
Parametric survival and reliability	Examples & counterexamples	2
Response surfaces and optimal design	Examples & counterexamples	2
Spatial summary estimators	Examples & counterexamples	2
Batch and measurement-process variation	Examples & counterexamples	3
Biological and technical replication	Examples & counterexamples	3
Cluster and two-stage sampling	Examples & counterexamples	3
Competing risks and cumulative incidence	Examples & counterexamples	3
HT variance and SRSWOR	Examples & counterexamples	3
Stratification and allocation	Examples & counterexamples	4

Symmetric Groups and Symmetric Functions

Reading order

32 pages

Page	Companion	Dependency level
Young Diagrams Tableaux and Permutation Modules	Examples & counterexamples	0
Specht Modules and the Irreducibles of the Symmetric Group	Examples & counterexamples	1
Symmetric Functions, the Hall Inner Product, and Schur Bases	Examples & counterexamples	1
Integral Specht Modules and Modular Simple Modules	Examples & counterexamples	2
The Branching Rule and the Young Graph	Examples & counterexamples	2
The Hook Length Formula and Rsk Correspondence	Examples & counterexamples	2
Frobenius Characteristic and the Symmetric-Group Character Dictionary	Examples & counterexamples	3
Jucys–Murphy Elements and Seminormal Forms	Examples & counterexamples	3
Modular Branching and the Good-Node Crystal	Examples & counterexamples	4
Outer Products, Skew Specht Modules, and Littlewood–Richardson Coefficients	Examples & counterexamples	4
Plancherel Measure and Asymptotic Young Diagrams	Examples & counterexamples	4
Kronecker Coefficients and Internal Products	Examples & counterexamples	5
Rim Hooks and the Murnaghan–Nakayama Rule	Examples & counterexamples	5
Character Polynomials, FI-Modules, and Representation Stability	Examples & counterexamples	6
Cores, Quotients, and Blocks of Symmetric Groups	Examples & counterexamples	6
Murnaghan Stability via FI-Modules	Examples & counterexamples	7

Topology

Reading order

64 pages

Page	Companion	Dependency level
Metric Spaces	Examples & counterexamples	0
Completeness, Completion, and Uniform Continuity	Examples & counterexamples	1
Topological Spaces and Continuity	Examples & counterexamples	1
Compactness in Metric Spaces	Examples & counterexamples	2
Subspaces, Products, and Quotients	Examples & counterexamples	2
Connectedness	Examples & counterexamples	3
Compactness	Examples & counterexamples	4
Convergence: Nets and Filters	Examples & counterexamples	5
Countability Axioms and Cardinal Functions	Examples & counterexamples	5
Function Space Topologies and the Exponential Law	Examples & counterexamples	5
Separation Axioms: the Hierarchy	Examples & counterexamples	5
The Topology of Euclidean Space	Examples & counterexamples	5
Hausdorff via the Diagonal	Examples & counterexamples	6
Homotopy and Homotopy Equivalence	Examples & counterexamples	6
Stone–Weierstrass in General	Examples & counterexamples	6
The Ascoli–Arzelà Theorem	Examples & counterexamples	6
Hereditary and Productive Behaviour of the Separation Axioms	Examples & counterexamples	7
The Fundamental Group	Examples & counterexamples	7
Uniform Spaces: the Three Definitions	Examples & counterexamples	7
Urysohn's Lemma and the Tietze Extension Theorem	Examples & counterexamples	7
Covering Spaces and Lifting	Examples & counterexamples	8
Partitions of Unity and Paracompactness	Examples & counterexamples	8
The Tychonoff Embedding and the Stone–Čech Compactification	Examples & counterexamples	8
Complete Metrizability, Čech-Completeness, and Baire Category	Examples & counterexamples	9
Discrete bases in metric spaces	Examples & counterexamples	9
The Fundamental Group of the Circle	Examples & counterexamples	9
Uniform Completeness, Completion, and the Samuel Compactification	Examples & counterexamples	9
Classification of Covering Spaces	Examples & counterexamples	10
Metrization: Urysohn, Nagata–Smirnov, Bing, Smirnov	Examples & counterexamples	10
The Seifert–van Kampen Theorem	Examples & counterexamples	10
Applications of the Fundamental Group	Examples & counterexamples	11
Classification of Compact Connected Surfaces	Examples & counterexamples	11