Formal

Logic

• Propositional (sentential) logic
• Predicate (first-order) logic
• Higher-order logic
• Classical logic
• Free logic
• Modal logic
• Temporal logic
• Deontic logic
• Epistemic logic
• Doxastic logic
• Provability logic
• Dynamic logic
• Intuitionistic logic
• Fuzzy logic
• Many-valued logic
• Paraconsistent logic
• Substructural logic
• Relevance logic
• Linear logic
• Quantum logic
• Combinatory logic
• Non-monotonic logic
• Description logic
• Mathematical logic
• Proof theory
• Model theory
• Set theory
• Recursion / computability theory
• Category-theoretic logic
• Philosophical logic
• Informal logic / argumentation theory

Mathematics

Foundations

• Set theory
• Category theory
• Type theory
• Model theory
• Proof theory
• Mathematical logic
• Reverse mathematics

Pure

• Number theory
• Abstract algebra
• Linear algebra
• Commutative algebra
• Homological algebra
• Universal algebra
• Galois theory
• Geometry
• Topology
• Real analysis
• Complex analysis
• Functional analysis
• Harmonic analysis
• Measure theory
• Differential equations
• Dynamical systems
• Combinatorics
• Graph theory
• Order theory
• Lattice theory
• Representation theory
• K-theory

Applied

• Numerical analysis
• Optimization theory
• Mathematical physics
• Approximation theory
• Discrete mathematics
• Computational mathematics

Probability & Statistics

• Probability theory
• Stochastic processes
• Measure-theoretic probability
• Mathematical statistics
• Descriptive statistics
• Inferential statistics
• Bayesian statistics
• Frequentist statistics
• Nonparametric statistics
• Multivariate statistics
• Time-series analysis
• Experimental design
• Statistical learning theory
• Actuarial science

Theoretical Computer Science

• Theory of computation
• Computability theory
• Computational complexity theory
• Automata theory
• Formal language theory
• Algorithm theory / analysis of algorithms
• Data structures (theoretical)
• Type theory
• Programming language theory
• Formal methods / formal verification
• Lambda calculus
• Domain theory
• Cryptography (theoretical)
• Quantum computation theory
• Distributed computation theory
• Computational geometry
• Computational number theory

Information Theory

• Coding theory
• Error-correcting codes
• Data compression theory
• Algorithmic information theory (Kolmogorov complexity)
• Quantum information theory
• Signal processing (theoretical)

Systems & Cybernetics

• Systems theory / general systems theory
• Systems science
• Cybernetics
• Control theory
• Dynamical systems theory
• Chaos theory
• Complexity theory (complex systems)
• Network theory
• Automata & self-organization

Decision, Strategy & Optimization

• Decision theory
• Game theory
• Social choice theory
• Mechanism design
• Utility theory
• Operations research
• Optimization
• Queueing theory
• Mathematical programming

Bridging Fields

• Theoretical / formal linguistics
• Mathematical economics
• Econometrics (theoretical)
• Mathematical finance / quantitative finance
• Formal epistemology
• Formal logic in philosophy
• Mereology
• Formal ontology
• Mathematical biology (theoretical portions)
• Theoretical / mathematical physics
• Biostatistics
• Psychometrics (formal measurement theory)
• Measurement theory