Encyclopedia Reading Routes

Suggested connections between published concepts. Choose a branch, revisit a prerequisite when needed, or start further along if you already know the material.

Links open encyclopedia explanations unless marked as short terminology definitions.

Sets, proof and mathematical language

Foundations & Logic

  1. Set
  2. function between sets Terminology: short definition
  3. Equivalence Relation
  4. Cardinality of a Set
  5. Countable Set
  6. Universal Quantifier
  7. Completeness of First-Order Logic

Start further along: If functions, equivalence relations and elementary proof methods are familiar, start with cardinality or first-order logic.

Algebra toward geometry and homological methods

Algebra & Algebraic Geometry

  1. Group
  2. Commutative Ring
  3. Ideal of a Commutative Ring
  4. Noetherian Commutative Ring
  5. Localization of a Commutative Ring
  6. Prime Spectrum of a Commutative Ring
  7. Chain Complex of Modules
  8. Ext Group of Modules

Start further along: With a solid first algebra course, start at localization or chain complexes. The spectrum and homological branches need not be read in a fixed order.

Integration, Fourier methods and weak solutions

Analysis, Measure & PDE

  1. Lebesgue Measure
  2. Lebesgue Integral
  3. Dominated Convergence Theorem
  4. Lebesgue Lp Space
  5. Fourier Transform
  6. Distribution in Analysis
  7. Weak Derivative
  8. Sobolev Space
  9. Weak Solution of a PDE

Start further along: If measure theory and Lp spaces are secure, start with distributions or Fourier transforms; return to convergence theorems when a proof needs them.

From rings to sheaves and schemes

Algebra & Algebraic Geometry

  1. Commutative Ring
  2. Ideal of a Commutative Ring
  3. Localization of a Commutative Ring
  4. Prime Spectrum of a Commutative Ring
  5. Sheaf
  6. Scheme

Start further along: If ideals and localization are familiar, start with the prime spectrum. Study sheaf gluing before the scheme definition; homological algebra is a separate next branch.

Manifolds, curvature and symplectic geometry

Geometry & Topology

  1. Topological Space
  2. Smooth Manifold
  3. Tangent Space
  4. Smooth Vector Bundle
  5. Differential Form
  6. Riemannian Metric
  7. Levi–Civita Connection
  8. Symplectic Manifold
  9. Hamiltonian Floer Homology

Start further along: If smooth manifolds and tangent spaces are familiar, choose the differential-forms or Riemannian branch directly.

Topology toward homology and homotopy

Geometry & Topology

  1. Topological Space
  2. Connected Space
  3. Topological Compactness
  4. Quotient Topology
  5. Fundamental Group
  6. Covering Space
  7. Homotopy of Maps
  8. Singular Homology
  9. Stable Homotopy Group

Start further along: Readers comfortable with point-set topology can start at the fundamental group. Homology and homotopy are parallel tools, not interchangeable invariants.

Arithmetic toward algebraic and analytic number theory

Number Theory

  1. Integer Divisibility
  2. Euclidean Algorithm for Integers
  3. Integer Congruence
  4. Number Field
  5. ring of integers of a number field Terminology: short definition
  6. Fractional Ideal of a Number Field
  7. Ideal Class Group
  8. Dirichlet Character
  9. Riemann Zeta Function
  10. Dirichlet L-Function

Start further along: After elementary congruences, choose algebraic number fields or the analytic branch through Dirichlet characters; neither branch requires finishing the other first.

Probability toward martingales and stochastic processes

Probability

  1. Probability Space
  2. Random Variable
  3. expectation Terminology: short definition
  4. Conditional Expectation
  5. Convergence in Distribution
  6. Law of Large Numbers
  7. Central Limit Theorem
  8. Markov Chain
  9. Martingale
  10. Stopping Time
  11. Brownian Motion
  12. Stochastic Differential Equation

Start further along: If measure-theoretic probability is familiar, start with conditional expectation or martingales. Limit theorems and Markov processes form additional branches.

Counting, graph theory and additive methods

Combinatorics

  1. Inclusion–Exclusion Principle
  2. Ordinary Generating Function
  3. Finite Simple Graph
  4. path in a graph Terminology: short definition
  5. Tree in Graph Theory
  6. Hall’s Marriage Theorem
  7. Probabilistic Method
  8. Extremal Graph Theory
  9. Additive Combinatorics

Start further along: Choose counting or graph theory as an entry point. Familiarity with elementary counting lets you start at generating functions or the probabilistic method.

Orbits, recurrence and ergodic ideas

Dynamical Systems

  1. Orbit of a Dynamical System
  2. Dynamical Flow
  3. Topological Transitivity
  4. Invariant Measure
  5. Poincaré Recurrence Theorem
  6. Ergodic Transformation
  7. Strong Mixing in Measure-Preserving Dynamics

Start further along: After orbits and invariant measures, choose the topological or measure-preserving branch. Mixing is stronger than ergodicity, not its definition.

Categorical language toward higher structures

Category Theory

  1. Category
  2. Functor
  3. Natural Transformation
  4. Categorical Product
  5. Universal Property
  6. Categorical Pullback
  7. Representable Functor
  8. Yoneda Lemma
  9. Equivalence of Categories
  10. Adjunction
  11. Higher Categories and the Topos Bridge

Start further along: If functors and natural transformations are already comfortable, start at representability or adjunctions; products and pullbacks offer concrete refresher examples.

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