What Type Theory Is and Why It Matters · Core Syntax: Terms, Types, Contexts · Universe Levels · Simple Types and Functions · Basic Ground Types · Sum Types (Coproducts) · Dependent Function Types (Pi-Types) · Dependent Pair Types (Sigma-Types) · Inductive Types · Identity Types · Logic in Type Theory · Basic Mathematics in Type Theory · Preview of Later Bridges
I.11 Logic in Type Theory
The propositions-as-types framework · Encoding logical connectives using type formers — Conjunction · Disjunction · Implication · Negation · Universal quantification · Existential quantification · Constructive vs. classical logic · Role in the theory
First-Order Logic and the Membership Relation · The ZF Axioms · The Cumulative Hierarchy · Encoding Mathematical Objects as Sets · Natural Numbers and Peano Arithmetic · Ordinals and Transfinite Methods · Classical Number Systems · Cardinality and Cardinal Arithmetic · Common Mathematical Structures in Set Theory · Comparison with Type Theory · Role of ZF Going Forward
II.1 First-Order Logic and the Membership Relation
Syntax of first-order logic — Language · Formulas and terms · Free and bound variables · Structures, interpretations, and models · Membership as a primitive relation — Sets as the only kind of object
II.2 The ZF Axioms
Overview of the axiom system · Extensionality · Empty set · Pairing and union · Power set — Power set axiom · Infinity · Separation schema · Replacement schema · Foundation · Axiom of Choice
Basic Notions: Objects, Morphisms, and Composition · Functors · Natural Transformations · Universal Properties · Limits and Colimits · Adjunctions and Their Role · Monads and Algebraic Structure · Interactions with Other Foundations · Role of Category Theory Moving Forward
Foundations of Formal Theories · Gödel's Completeness Theorem · Proof Theory and Normalization · Model Theory · Arithmetization of Syntax · Gödel's Incompleteness Theorems · Tarski's Undefinability of Truth · Consistency, Models, and Relative Consistency · Undecidability and Computability · Decidable Theories · Metatheory in Broader Context · Role of Metatheory Going Forward
Motivation and Informal Overview · Elementary Toposes · Concrete Examples of Toposes · Logic Inside a Topos · Sheaves, Sites, and Grothendieck Toposes · Geometric Morphisms and Logical Structure · Classifying Toposes · Toposes and Set-Theoretic Ideas · Topos Theory in Relation to Other Foundations · Role of Topos Theory Going Forward
From Dependent Type Theory to Homotopy · Identity Types and Higher Groupoid Structure · The Univalence Axiom · Higher Inductive Types · Homotopy-Theoretic Constructions · Synthetic Homotopy Theory · Universes and Internal Structure · Relations to Other Foundations · Models of Homotopy Type Theory · HoTT as a Foundation for Mathematics · Role of HoTT Going Forward
Motivation and Historical Background · Turing Machines · Recursive and Recursively Enumerable Functions · The Halting Problem and Undecidability · Kolmogorov Complexity and Algorithmic Randomness · Degrees of Unsolvability · Recursion Theory on Structures · Lambda Calculus and Computation · Connections to Logic and Metatheory · Computability in Type Theory and Category Theory · Role of Computability Going Forward
What Algebra Is and Why It Matters · Universal Algebra: Operations, Relations, and Equations · Groups · Rings and Modules · Fields and Field Extensions · Linear Algebra as Algebra · Commutative Algebra · Categorical Algebra · Representation-Theoretic Motivation · Connections to Other Structural Domains · Role of Algebra Going Forward
VIII.2 Universal Algebra
Signatures of algebraic theories · Equational theories · Homomorphisms and subalgebras · Products, coproducts, and free algebras · Lattices and Boolean algebras — Partial orders and lattices · Distributive and modular lattices · Boolean algebras · Complete lattices and fixed points
VIII.3 Groups
Definition and basic properties — Axioms · Examples · Abelian vs non-abelian · Homomorphisms and isomorphisms · Isomorphism theorems · Subgroups, quotients, and normality · Direct and semidirect products · Classification of finitely generated abelian groups · Sylow theorems · Jordan–Hölder theorem · Simple groups and classification preview · Cayley's theorem · Free groups and presentations · Group actions · Examples and applications
What Topology Is and Why It Matters · Topological Spaces · Separation and Countability Axioms · Connectedness and Compactness · Metric Spaces · Constructions of New Spaces · Homotopy and Basic Invariants · Topological Groups · Stone Duality and Boolean Algebras · Categorical Topology · Connections to Other Structural Domains · Role of Topology Going Forward
IX.2 Topological Spaces
Open sets and the topology axioms — Definition · Examples · Closed sets · Continuous maps · Bases and subbases · Subspaces, products, and quotients · Embedding theorems · Proper maps
What Analysis Is and Why It Matters · Real Numbers and Basic Structures · Metric and Normed Spaces · Topological Foundations of Analysis · Differentiation · Integration · Function Spaces · Fourier Analysis · Differential Equations · Probability Theory (Measure-Theoretic Core) · Harmonic and Functional Analysis · Categorical and Structural Perspectives · Connections to Other Structural Domains · Role of Analysis Going Forward
X.6 Integration
Riemann integration · Lebesgue integration — Measure theory foundation · Convergence theorems · Fubini's theorem · Radon–Nikodym theorem · Change of variables and differentiation · Egorov's and Lusin's theorems
What These Fields Are and Why They Matter · Homotopy Theory: The Fundamental Group · Covering Spaces · Homology and Cohomology · CW Complexes and Cellular Techniques · Higher Homotopy Groups · Categories, Functors, and Natural Transformations in Topology · Spectral Sequences · Algebraic Geometry: Varieties and Morphisms · Sheaves and Local Structures · Schemes · Intersection Theory · Connections Between Algebraic Topology and Algebraic Geometry · Role of These Fields Going Forward
Why Modern Structures Matter · Sheaves and Presheaves · Sites and Grothendieck Topologies · Sheaf Cohomology · Derived Categories · Model Categories and Homotopical Algebra · Higher Categories and Infinity-Categories · Higher Toposes · Higher Algebra · Derived Algebraic Geometry · Connections to Homotopy Type Theory · Role of Modern Structures Going Forward
What Harmonic Analysis Is and Why It Matters · Fourier Series on the Circle · The Fourier Transform on R · Distribution Theory and Generalized Functions · Convolution and Filtering · L^p Theory and Interpolation · Harmonic Analysis on Groups · Fourier Analysis on Manifolds · Littlewood–Paley Theory · Hardy Spaces and Singular Integrals · Connections to PDE · Connections to Probability · Connections to Representation Theory · Role of Harmonic Analysis Going Forward
What Probability Is and Why It Matters · Probability Spaces · Expectation and Integration · Modes of Convergence · Laws of Large Numbers and the Central Limit Theorem · Random Series and Martingales · Markov Chains · Stochastic Processes · Gaussian Processes · Brownian Motion · Stochastic Calculus · Connections to Harmonic Analysis and PDE · Connections to Geometry and Topology · Role of Probability and Stochastic Processes Going Forward
XIV.2 Probability Spaces
Sigma-algebras and measurable sets — Definition · Examples · Generated sigma-algebras · Probability measures — Definition · Examples · Properties · Product measures and Fubini — Product spaces · Fubini's theorem · Measurable functions as random variables — Random variables · Distribution and density · Common distributions · Change of variables · Joint distributions and marginalization — Joint distributions · Marginal distributions · The multivariate Gaussian · Conditional probability and Bayes' rule — Conditional probability · Chain rule of probability · Bayes' rule · Law of total probability · Independence — Independent events · Independent random variables · Conditional independence · Product spaces and independence · Borel–Cantelli lemmas · 0-1 laws
XIV.3 Expectation and Integration
Expectation as integral · Moments and distributions · Conditional expectation · Radon-Nikodym and densities · Inequalities — Markov's inequality · Chebyshev's inequality · Jensen's inequality · Hölder and Cauchy-Schwarz
XIV.5 Laws of Large Numbers and the Central Limit Theorem
Weak and strong laws — Weak law of large numbers · Strong law of large numbers · Interpretation · Central limit theorem — Classical CLT · Universality of the Gaussian · Berry-Esseen theorem · Lindeberg-Feller CLT · Characteristic functions — Definition · Fourier proof of CLT · Stable distributions — Definition · Examples · Applications — Statistical inference · Physics · Monte Carlo methods
XIV.6 Random Series and Martingales
Martingales — Definition · Examples · Submartingales and supermartingales · Stopping times · Optional stopping theorem · Doob's maximal inequalities · Convergence theorems · Azuma–Hoeffding inequality
XIV.7 Markov Chains
Discrete-time Markov processes — Markov property · Transition matrix · Examples · Classification of states · Stationary distributions — Definition · Convergence to stationarity · Ergodic theorem for Markov chains · Coupling and total variation — Coupling · Total variation distance · Mixing time · Applications — Markov chain Monte Carlo · Random walks on graphs · Probabilistic methods in combinatorics
XIV.8 Stochastic Processes
Random functions — Definition · Sample paths · Finite-dimensional distributions — Definition · Consistency · Kolmogorov extension theorem — Statement · Limitations · Poisson process
XIV.9 Gaussian Processes
Definition and characterization — Definition via finite-dimensional distributions · Gaussian measure on function spaces · Characterization by mean and covariance · Positive-definite kernels — Definition and basic properties · Mercer's theorem · Kernel algebra · Common kernel families · Stationarity and spectral representation — Stationary and isotropic kernels · Bochner's theorem · Spectral density and frequency content · Reproducing kernel Hilbert spaces — Definition and reproducing property · RKHS norm as smoothness measure · Cameron–Martin space · Path regularity — Mean-square continuity · Mean-square differentiability · Sample path smoothness from kernel smoothness · Conditioning and posterior processes — Gaussian conditioning in infinite dimensions · Posterior mean and covariance · Predictive distributions · Marginal likelihood · Connections and applications — Brownian motion as a Gaussian process · Spatial statistics and kriging · Function-space priors in inference
XIV.10 Brownian Motion
Definition and basic properties — Standard Brownian motion · Wiener measure · Multidimensional Brownian motion · Path properties — Nowhere differentiability · Hölder continuity · Quadratic variation · Martingale properties — Brownian motion as martingale · Reflection principle and maximal distribution · Symmetries and invariance — Scaling property · Time inversion · Donsker's invariance principle
XIV.11 Stochastic Calculus
Itô integrals — Motivation · Construction · Properties · Itô's lemma — Statement · Heuristic derivation · Applications · Stratonovich integral · Stochastic differential equations · Girsanov theorem · Feynman-Kac formula · Malliavin calculus (mention) · Applications
What Differential Geometry Is and Why It Matters · Smooth Manifolds · Tangent and Cotangent Spaces · Tensor Fields · Vector Fields and Flows · Differential Forms · Integration on Manifolds · Stokes' Theorem · de Rham Cohomology · Riemannian Geometry · Lie Groups and Lie Algebras · Bundles and Connections · Geometric Operators · Global Differential Geometry · Symplectic Geometry · Connections to Other Areas · Role of Differential Geometry Going Forward
What PDEs Are and Why They Matter · Classical PDEs: First Definitions · First-Order PDEs and Characteristics · Second-Order Linear PDEs · Elliptic PDE · Parabolic PDE · Hyperbolic PDE · Weak Solutions and Sobolev Spaces · Well-Posedness Theory · Fourier Methods for PDE · Nonlinear PDE · Stochastic PDE · PDE on Manifolds · Numerical Methods · Connections to Other Areas · Role of PDE Going Forward
What Representation Theory Is and Why It Matters · Group Representations · Modules and Linear Actions · Characters and Finite Groups · Symmetric Groups and Young Tableaux · Compact Lie Groups · Lie Algebras and Representations · Associative Algebras and Modules · Induced Representations · Tensor Products and Schur–Weyl Duality · Geometric Representation Theory · Connections to Number Theory · Categorical and Higher Representation Theory · Computational Methods and Applications · Role of Representation Theory Going Forward
What Number Theory Is and Why It Matters · Divisibility and the Euclidean Algorithm · Congruences and Modular Arithmetic · Fermat's Little Theorem and Euler's Theorem · Linear Congruences and Chinese Remainder Theorem · Primitive Roots and Order of Elements · Quadratic Residues · Multiplicative Functions · Prime Numbers · Continued Fractions · Diophantine Equations · Sum of Squares · Quadratic Forms · Gaussian Integers · Unique Factorization Domains · Geometry of Numbers · Elementary Analytic Methods · Partition Function · Perfect Numbers and Mersenne Primes · Fibonacci Numbers and Divisibility · Computational Aspects · Group-Theoretic Perspectives · Connections to Other Areas · Role of Elementary Number Theory Going Forward
What Analytic Number Theory Is and Why It Matters · Dirichlet Series and Euler Products · The Riemann Zeta Function · Prime Number Theorem · Dirichlet Characters and L-Functions · Zero-Free Regions · Analytic Techniques · Class Number Formula · Multiplicative Functions and Mean Values · Sieve Methods · Explicit Formulas · Exponential Sums · Modular Forms · Probabilistic Number Theory · Additive Combinatorics · Circle Method · Equidistribution · Connections to Other Areas · Role of Analytic Number Theory Going Forward
What Algebraic Number Theory Is · Number Fields and Rings of Integers · Ideals and Unique Factorization · Units and Dirichlet's Unit Theorem · Class Group and Class Number · Valuations and Completions · Local Fields · Ramification Theory · Dedekind Zeta Functions · Artin L-functions · Class Field Theory · Adeles and Ideles · Galois Representations · Iwasawa Theory · Elliptic Curves over Number Fields · Mordell–Weil Theorem · Reciprocity Laws · Complex Multiplication · Selmer Groups and Shafarevich–Tate Group · Birch and Swinnerton-Dyer Conjecture · Modular Forms and Connections · Computational Aspects · Diophantine Equations and Algebraic Methods · Connections to Other Areas · Role of Algebraic Number Theory Going Forward · Open Problems
What Dynamical Systems Are and Why They Matter · Foundations of Dynamical Systems · Phase Space and State Spaces · Orbits, Fixed Points, and Periodic Points · Linear Dynamics · Topological Dynamics · Recurrence and Poincaré's Theorem · Chaos and Sensitivity · Symbolic Dynamics · Topological Conjugacy and Equivalence · Invariant Measures · Ergodic Theory Fundamentals · The Ergodic Theorems · Mixing and Recurrence Properties · Entropy and Information · Hyperbolic Dynamics · Attractors and Basins of Attraction · Lyapunov Exponents · Conservative vs. Dissipative Systems · Hamiltonian Systems and KAM Theory · Geodesic Flows and Geometry · Random Dynamical Systems · Smooth Ergodic Theory · Rotation Numbers and Circle Maps · Thermodynamic Formalism · Dimension Theory and Fractals · Furstenberg Correspondence and Combinatorics · Computational and Applied Aspects · Connections to Other Domains · Role of Dynamical Systems Going Forward
What Complex Analysis Is and Why It Matters · Foundations of Complex Analysis · Power Series and Analytic Continuation · Cauchy's Theorem and Integral Formula · Singularities and Residues · Argument Principle and Rouché's Theorem · Maximum Modulus and Mapping Principles · Harmonic Functions and Complex Methods · Conformal Mappings · Riemann Mapping Theorem · Weierstrass Factorization and Mittag-Leffler Theorems · Infinite Products · Normal Families and Montel's Theorem · Picard Theorems · Elliptic Functions · Riemann Surfaces · Meromorphic Functions and Divisors · Differentials and Holomorphic Forms · Topology of Riemann Surfaces · Uniformization Theorem · Riemann–Roch Theorem and Complex Geometry · Covering Spaces and Monodromy · Moduli Spaces and Teichmüller Theory · Connections to Other Areas · Advanced Topics and Generalizations · Role of Complex Analysis and Riemann Surfaces Going Forward
What Symplectic Geometry Is and Why It Matters · Symplectic Linear Algebra · Symplectic Manifolds and Forms · Cotangent Bundles · Darboux and Moser Theorems · Hamiltonian Vector Fields and Flows · Symplectomorphisms · Lagrangian Submanifolds · Generating Functions · Symplectic Capacities and Non-Squeezing · Moment Maps and Symplectic Reduction · Coadjoint Orbits and Representation Theory · Action-Angle Coordinates · Toric Manifolds · Geometric Quantization · J-Holomorphic Curves · Floer Homology · Fukaya Categories · Symplectic Cobordism · Contact Geometry · Deformation Quantization · KAM Theory · Arnold Conjectures · Connections to Other Areas · Computational Applications · Open Problems · Role Going Forward
What Lie Theory Is and Why It Matters · Classical Matrix Groups · Lie Groups: Basic Concepts · Lie Algebras · The Exponential Map · One-Parameter Subgroups and Flows · Lie's Three Theorems · Homomorphisms and Representations · Adjoint Representation · Universal Enveloping Algebra and PBW Theorem · Structure Theory of Lie Algebras · Nilpotent and Solvable Lie Groups · Root Systems and Highest-Weight Theory · Dynkin Diagrams · Classification of Simple Lie Algebras · Real Forms and Complexification · Lie Groups as Geometric Objects · Coadjoint Orbits · Haar Measure and Integration on Lie Groups · Compact Lie Groups · Peter–Weyl Theorem · Weyl Character Formula and Casimir Operators · Borel–Weil–Bott Theorem · Parabolic and Borel Subgroups · Non-Compact Lie Groups · Lattices and Discrete Subgroups · Central Extensions and Cohomology · Infinite-Dimensional Lie Groups and Algebras · Lie Theory and Physics · Connections to Other Areas · Computational Aspects · Role of Lie Groups and Lie Algebras Going Forward
What Higher Algebra Is and Why It Matters · Constructions of Spectra · Stable Homotopy Groups · Suspension and Loop Space Functors · Stable Infinity-Categories · Model Categories for Spectra · Spanier–Whitehead Duality · Brown Representability Theorem · Ring Spectra and Module Spectra · Classical Examples of Ring Spectra · Generalized Cohomology Theories · Thom Spectra and Bordism · Topological K-Theory · Complex Cobordism MU · Algebraic K-Theory Spectrum · Topological Modular Forms · Smash Products and Monoidal Structures · Steenrod Algebra and Cohomology Operations · Adams Spectral Sequence · Atiyah–Hirzebruch Spectral Sequence · Postnikov and Whitehead Towers in Stable Setting · Bousfield Localization and Completion · Formal Group Laws and Complex-Oriented Cohomology · Chromatic Homotopy Theory · Adams–Novikov Spectral Sequence · Equivariant Stable Homotopy Theory · Motivic Stable Homotopy Theory · Brave New Algebra and Structured Ring Spectra · Derived and Spectral Algebraic Geometry · Computational Aspects · Connections to Other Areas · Role of Higher Algebra Going Forward
What Spectral Sequences Are and Why They Matter · Basic Homological Algebra · Filtered Complexes · Definition of a Spectral Sequence · Exact Couples · Convergence Conditions · Edge Homomorphisms and Five-Term Exact Sequences · Transgression and Connecting Homomorphisms · The Serre Spectral Sequence · The Grothendieck Spectral Sequence · The Leray Spectral Sequence · Eilenberg–Moore Spectral Sequence · Bockstein Spectral Sequence · Lyndon–Hochschild–Serre Spectral Sequence · Adams Spectral Sequence · Atiyah–Hirzebruch Spectral Sequence · Multiplicative Structure on Spectral Sequences · Spectral Sequences from Double Complexes · Künneth Spectral Sequence · Universal Coefficient Spectral Sequences · Change of Rings Spectral Sequence · Hochschild and Cyclic Homology Spectral Sequences · Hodge–de Rham Spectral Sequence · Frölicher Spectral Sequence · Bar and Cobar Spectral Sequences · Equivariant Spectral Sequences · Motivic Spectral Sequences · Algebraic K-Theory Spectral Sequences · Computational Techniques · Extension Problems · Worked Examples · Connections to Other Areas · Role of Spectral Sequences Going Forward
What Model Categories Are and Why They Matter · Homotopy Theory Inside Categories · Definitions and Axioms of a Model Category · Weak Equivalences · Factorization Systems and the Small Object Argument · Cofibrant Generation · Proper Model Categories · Combinatorial Model Categories · Homotopy Categories · Cofibrant and Fibrant Replacements · Homotopies and Cylinder/Path Objects · Quillen Functors and Quillen Equivalences · Transferred Model Structures · Examples of Model Categories · Simplicial Enrichment · Homotopy Colimits · Homotopy Limits · Reedy Model Structures · Derived Functors · Ken Brown's Lemma · Acyclic Models · Monoidal Model Categories · Model Structures on Categories of Algebras · Bousfield Localization · Applications to Geometry and Topology · Simplicial Localization and Infinity-Categories · Computational Techniques · Connections to Other Areas · Historical Development · Role of Model Categories Going Forward
What Schemes Are and Why They Matter · Affine Schemes · Morphisms of Schemes · Functor of Points · Properties of Morphisms · Gluing and General Schemes · Noetherian Schemes · Sheaves and Sheaf Cohomology · Sheaf Operations · Higher Direct Images and Base Change · Cohomological Methods in Geometry · Projective Schemes · Ampleness and Line Bundles · Cohomology on Projective Schemes · Explicit Cohomology Computations · Grothendieck–Riemann–Roch · Flatness, Smoothness, and Dimension · Kähler Differentials and Differential Forms · Étale Morphisms and Étale Cohomology · Grothendieck Topologies and Descent · Group Schemes · Formal Schemes · Derived Geometry · Schemes and Arithmetic · Connections to Other Areas · Role of Schemes Going Forward
What Arithmetic Geometry Is and Why It Matters · Arithmetic Schemes · Local and Global Fields · p-Adic Numbers and Analysis · Points on Varieties · Diophantine Equations · Rational Points on Curves by Genus · Class Field Theory · Elliptic Curves and Abelian Varieties · Birch and Swinnerton-Dyer Conjecture · Selmer Groups and Tate–Shafarevich Group · Faltings' Theorem and Finiteness Results · Heights and Diophantine Geometry · Tate Modules and l-adic Representations · Good and Bad Reduction · Galois Representations and Arithmetic · l-adic Cohomology · Weil Conjectures · L-functions and Automorphic Forms · Functional Equations and Special Values · Modularity Theorems · Modular Curves and Modular Forms · Shimura Varieties · Langlands Program and Reciprocity · Iwasawa Theory · Arakelov Geometry · Arithmetic Surfaces · Anabelian Geometry · The abc Conjecture · Duality Theorems in Arithmetic · Computational Aspects · Perfectoid Spaces · Connections to Other Areas · Recent Developments and Open Problems · Role of Arithmetic Geometry Going Forward
What Moduli Theory Is and Why It Matters · Moduli Problems as Functors · Fine vs Coarse Moduli Spaces · Deformation Theory Foundations · Tangent-Obstruction Complexes · Grothendieck's Hilbert and Quot Schemes · Geometric Invariant Theory · Moduli of Curves · Compactifications · Moduli of Vector Bundles · GIT Stability · Slope Stability · Harder–Narasimhan Filtration · Bridgeland Stability · Moduli of Sheaves on Surfaces · Why Stacks Are Necessary · Fibered Categories and Descent · Algebraic Stacks · Morphisms of Stacks · Quotient Stacks · Picard Stacks and Line Bundles on Stacks · Inertia Stacks and Orbifold Cohomology · Cohomology of Stacks · Derived Stacks · Virtual Fundamental Classes · Moduli of Elliptic Curves · Moduli of Abelian Varieties · Moduli of Principal G-Bundles · Moduli of Stable Maps · Character Varieties and Higgs Bundles · Teichmüller Theory and Mapping Class Groups · Arithmetic Applications · Topological Applications · Physics Connections · Role of Moduli Spaces and Stacks Going Forward
What TQFT Is and Why It Matters · The Atiyah–Segal Axioms · Cobordism Categories · Functorial vs Topological Field Theory · 1D and 2D TQFT Examples · Classification of 2D TQFTs · Cobordism Hypothesis · Fully Extended TQFTs · Oriented, Framed, and Structured Bordisms · Frobenius Algebras · Modular Tensor Categories · Fusion Categories · Quantum Groups and Braiding · Ribbon Categories and Tangles · Chern–Simons Theory · Witten–Reshetikhin–Turaev Invariants · State-Sum Models · Khovanov Homology · 3-Manifold Invariants and Volume Conjecture · Infinity-Categories for TQFT · Dualizability and Adjoints · Factorization Algebras · Defects, Boundaries, and Domain Walls · Anomalies and Invertible Field Theories · Conformal Field Theory · Gromov–Witten Theory as TQFT · Homological Mirror Symmetry · Topological Modular Forms and Elliptic Cohomology · Symplectic Field Theory and Fukaya Categories · Topological Phases of Matter · Higher Chern–Simons and BF Theory · Categorical Symmetries and Gauging · Representation Theory Connections · Role of TQFT and Higher Geometry Going Forward