I. Foundations & Logic
1. Mathematical Logic
- Propositional logic
- First-order logic
- Higher-order logic
- Modal logic
- Intuitionistic logic
- Non-classical logics
- Proof theory
- Model theory
- Recursion / computability theory
- Set theory
- Axiomatic set theory (ZFC and variants)
- Large cardinal theory
- Forcing & independence
- Descriptive set theory
- Reverse mathematics
2. Foundations of Mathematics
- Type theory
- Homotopy type theory
- Constructive mathematics
- Structuralism
- Category-theoretic foundations
- Philosophy of mathematics
II. Algebra (Structure Theory)
1. Group Theory
- Finite groups
- Infinite groups
- Abelian groups
- Group extensions
- Group cohomology
- Geometric group theory
- Lie groups
- Algebraic groups
2. Ring Theory
- Commutative algebra
- Noncommutative ring theory
- Ideal theory
- Noetherian & Artinian rings
- Homological dimensions
3. Field Theory
- Galois theory
- Valuation theory
- Local & global fields
- Class field theory
4. Linear & Multilinear Algebra
- Vector spaces
- Modules
- Tensor algebra
- Spectral theory
5. Universal Algebra
- Lattice theory
- Boolean algebras
- Algebraic systems
6. Representation Theory
- Finite group representations
- Lie algebra representations
- Algebraic group representations
- Modular representations
7. Homological Algebra
- Chain complexes
- Derived functors
- Ext & Tor
- Spectral sequences
8. Category Theory
- Functors
- Natural transformations
- Adjunctions
- Monoidal categories
- Topos theory
- Higher categories
- ∞-categories
III. Number Theory
- Elementary number theory
- Analytic number theory
- Algebraic number theory
- Arithmetic geometry
- Diophantine geometry
- Modular forms
- Automorphic forms
- Iwasawa theory
- p-adic number theory
- Langlands program
IV. Geometry
1. Classical Geometry
- Euclidean geometry
- Non-Euclidean geometry
- Projective geometry
- Affine geometry
2. Algebraic Geometry
- Varieties
- Schemes
- Sheaf theory
- Cohomology
- Moduli spaces
- Birational geometry
- Derived algebraic geometry
3. Differential Geometry
- Smooth manifolds
- Riemannian geometry
- Symplectic geometry
- Complex geometry
- Kähler geometry
- Global analysis
- Ricci flow
4. Discrete & Computational Geometry
V. Topology
1. General Topology
2. Algebraic Topology
- Homology
- Cohomology
- Homotopy theory
- Stable homotopy
3. Geometric Topology
- Low-dimensional topology
- Knot theory
- 3-manifolds
- 4-manifolds
4. Topological Groups
5. Topological Dynamics
VI. Analysis
1. Real Analysis
- Measure theory
- Integration theory
- Differentiation theory
2. Complex Analysis
- One complex variable
- Several complex variables
- Riemann surfaces
3. Functional Analysis
- Banach spaces
- Hilbert spaces
- Operator theory
- C*-algebras
- Banach algebras
4. Harmonic Analysis
- Fourier analysis
- Distribution theory
- Pseudodifferential operators
5. Partial Differential Equations
- Elliptic PDE
- Parabolic PDE
- Hyperbolic PDE
- Nonlinear PDE
6. Calculus of Variations
7. Dynamical Systems
- Chaos
- Bifurcation theory
- Ergodic theory
- Smooth dynamics
VII. Probability & Statistics
Probability Theory
- Measure-theoretic probability
- Stochastic processes
- Martingales
- Brownian motion
- Stochastic calculus
- Random matrices
- Large deviations
Statistics
- Parametric statistics
- Nonparametric statistics
- Bayesian statistics
- Asymptotic theory
- Multivariate statistics
- Statistical learning theory
VIII. Combinatorics & Discrete Mathematics
- Enumerative combinatorics
- Extremal combinatorics
- Probabilistic combinatorics
- Algebraic combinatorics
- Ramsey theory
- Design theory
- Graph theory
- Spectral graph theory
- Random graphs
IX. Applied Mathematics
- Numerical analysis
- Scientific computing
- Optimization
- Convex optimization
- Integer programming
- Operations research
- Control theory
- Game theory
- Mathematical physics
- Fluid mechanics
- Mathematical biology
- Financial mathematics
- Information theory
X. Theoretical Computer Science (Mathematical Core)
- Automata theory
- Formal languages
- Computational complexity
- Algorithmic information theory
- Cryptography
- Quantum computation theory
XI. Emerging & Hybrid Fields
- Tropical geometry
- Noncommutative geometry
- Free probability
- Information geometry
- Topological data analysis
- Algebraic statistics
- Network science
- Mathematical neuroscience
- Quantum topology