**Foundations of Mathematics** 1. Logic - Propositional logic - First-order logic - Higher-order logic - Modal logic - Fuzzy logic 2. Set Theory - Naive set theory - Zermelo-Fraenkel set theory (ZFC) - Axiom of choice - Continuum hypothesis 3. Category Theory - Objects and morphisms - Functors - Natural transformations - Adjoint functors - Topos theory 4. Proof Theory - Natural deduction - Sequent calculus - Hilbert-style deduction - Automated theorem proving **Number Theory** 1. Elementary Number Theory - Divisibility and primes - Congruences - Diophantine equations - Quadratic reciprocity 2. Analytic Number Theory - Prime number theorem - Riemann zeta function - L-functions - Modular forms 3. Algebraic Number Theory - Number fields - Rings of integers - Ideal class groups - Valuations 4. Computational Number Theory - Primality testing - Integer factorization - Elliptic curve cryptography - Lattice-based cryptography **Algebra** 1. Group Theory - Cyclic groups - Symmetric groups - Abelian groups - Solvable groups - Representation theory 2. Ring Theory - Commutative rings - Integral domains - Euclidean domains - Principal ideal domains - Polynomial rings 3. Field Theory - Finite fields - Galois theory - Algebraic geometry - Transcendental numbers 4. Linear Algebra - Vector spaces - Matrices - Determinants - Eigenvalues and eigenvectors - Inner product spaces **Geometry and Topology** 1. Euclidean Geometry - Planes and angles - Triangles and polygons - Circles and spheres - Congruence and similarity 2. Non-Euclidean Geometry - Hyperbolic geometry - Spherical geometry - Projective geometry - Affine geometry 3. Differential Geometry - Manifolds - Riemannian geometry - Curvature - Geodesics - Lie groups 4. Topology - Metric spaces - Topological spaces - Continuity and homeomorphisms - Homotopy theory - Knot theory **Analysis** 1. Real Analysis - Sequences and series - Limits and continuity - Differentiation - Integration - Measure theory 2. Complex Analysis - Complex numbers - Analytic functions - Cauchy's integral formula - Residue theorem - Conformal mappings 3. Functional Analysis - Banach spaces - Hilbert spaces - Operator theory - Spectral theory - C*-algebras 4. Harmonic Analysis - Fourier series - Fourier transforms - Wavelets - Sobolev spaces - Partial differential equations **Discrete Mathematics** 1. Combinatorics - Permutations and combinations - Generating functions - Recurrence relations - Polya's enumeration theorem - Ramsey theory 2. Graph Theory - Connectivity - Planarity - Coloring - Matching - Network flows 3. Coding Theory - Error-correcting codes - Linear codes - Cyclic codes - Reed-Solomon codes - Convolutional codes 4. Game Theory - Zero-sum games - Nash equilibrium - Cooperative games - Shapley value - Mechanism design **Probability and Statistics** 1. Probability Theory - Random variables - Probability distributions - Expectation and variance - Conditional probability - Stochastic processes 2. Mathematical Statistics - Estimation theory - Hypothesis testing - Regression analysis - Analysis of variance (ANOVA) - Bayesian inference 3. Stochastic Processes - Markov chains - Poisson processes - Brownian motion - Martingales - Stochastic calculus 4. Machine Learning - Supervised learning - Unsupervised learning - Reinforcement learning - Neural networks - Support vector machines **Applied Mathematics** 1. Numerical Analysis - Interpolation and approximation - Numerical integration - Numerical linear algebra - Optimization - Finite element methods 2. Dynamical Systems - Ordinary differential equations (ODEs) - Stability theory - Bifurcation theory - Chaos theory - Hamiltonian systems 3. Fluid Dynamics - Navier-Stokes equations - Turbulence - Boundary layers - Compressible flow - Computational fluid dynamics (CFD) 4. Control Theory - Linear systems - Stability and feedback - Optimal control - Robust control - Nonlinear control 5. Mathematical Biology - Population dynamics - Epidemiology - Neuroscience - Systems biology - Bioinformatics 6. Mathematical Finance - Options pricing - Portfolio optimization - Risk management - Stochastic volatility models - Algorithmic trading 7. Cryptography - Public-key cryptography - Digital signatures - Hash functions - Zero-knowledge proofs - Quantum cryptography 8. Operations Research - Linear programming - Integer programming - Network optimization - Queuing theory - Inventory management 9. Computer Science - Algorithms and complexity - Data structures - Computational geometry - Parallel and distributed computing - Quantum computing 10. Mathematical Physics - Classical mechanics - Quantum mechanics - Relativity - Statistical mechanics - Quantum field theory This map covers a wide range of mathematical fields and their applications, but it is by no means exhaustive. Mathematics is a vast and ever-expanding discipline, with new areas of research and applications constantly emerging. **Foundations of Mathematics** 5. Model Theory - First-order models - Compactness theorem - Löwenheim-Skolem theorems - Ultraproducts - Stability theory 6. Recursion Theory - Computable functions - Turing machines - Recursively enumerable sets - Degrees of unsolvability - Hyperarithmetical hierarchy 7. Constructive Mathematics - Intuitionistic logic - Constructive analysis - Realizability theory - Martin-Löf type theory - Homotopy type theory **Number Theory** 5. Diophantine Geometry - Rational points on curves and surfaces - Heights and the Mordell-Weil theorem - Fermat's last theorem - ABC conjecture - Elliptic curves over number fields 6. Transcendental Number Theory - Liouville numbers - Algebraic independence - Schanuel's conjecture - Baker's theorem - Periods and the Grothendieck period conjecture 7. Analytic and Probabilistic Number Theory - Vaughan's identity - Hardy-Littlewood circle method - Goldbach's conjecture - Twin prime conjecture - Erdős-Kac theorem **Algebra** 5. Homological Algebra - Exact sequences - Homology and cohomology - Derived functors - Spectral sequences - Sheaf cohomology 6. Representation Theory - Lie algebras - Representation of finite groups - Character theory - Induced representations - Langlands program 7. Commutative Algebra - Localization - Completion - Dimension theory - Homological dimensions - Cohen-Macaulay rings 8. Non-Commutative Algebra - Free algebras - Hopf algebras - Quantum groups - Noncommutative geometry - Operator algebras **Geometry and Topology** 5. Algebraic Topology - Fundamental group - Homology and cohomology - Homotopy groups - Spectral sequences - K-theory 6. Geometric Topology - Manifolds - Triangulations - Cobordism theory - Surgery theory - Floer homology 7. Symplectic Geometry - Symplectic manifolds - Hamiltonian mechanics - Symplectic reduction - Moment maps - Mirror symmetry 8. Complex Geometry - Complex manifolds - Kähler geometry - Hodge theory - Calabi-Yau manifolds - Geometric invariant theory **Analysis** 5. Nonlinear Analysis - Calculus of variations - Critical point theory - Degree theory - Fixed point theorems - Monotone operators 6. Dynamical Systems and Ergodic Theory - Topological dynamics - Ergodic measures - Lyapunov exponents - Entropy - Symbolic dynamics 7. Partial Differential Equations - Elliptic equations - Parabolic equations - Hyperbolic equations - Sobolev spaces - Regularity theory 8. Geometric Analysis - Minimal surfaces - Harmonic maps - Yang-Mills theory - Ricci flow - Geometric evolution equations **Discrete Mathematics** 5. Matroid Theory - Independence axioms - Greedy algorithm - Duality - Minors - Representability 6. Algebraic Combinatorics - Symmetric functions - Young tableaux - Schubert calculus - Cluster algebras - Quiver representations 7. Extremal Combinatorics - Ramsey numbers - Turán's theorem - Szemerédi's theorem - Hypergraph theory - Additive combinatorics 8. Enumerative Combinatorics - Catalan numbers - Stirling numbers - Partitions - Lattice paths - Bijective combinatorics **Probability and Statistics** 5. Survival Analysis - Kaplan-Meier estimator - Cox proportional hazards model - Accelerated failure time model - Competing risks - Frailty models 6. Multivariate Analysis - Principal component analysis (PCA) - Factor analysis - Canonical correlation analysis - Discriminant analysis - Cluster analysis 7. Nonparametric Statistics - Kernel density estimation - Spline regression - Locally weighted regression (LOESS) - Permutation tests - Bootstrap methods 8. Bayesian Statistics - Prior and posterior distributions - Conjugate priors - Markov chain Monte Carlo (MCMC) - Gibbs sampling - Variational inference **Applied Mathematics** 11. Mathematical Economics - General equilibrium theory - Game theory - Social choice theory - Mechanism design - Econometrics 12. Mathematical Epidemiology - SIR model - SEIR model - Contact networks - Vaccination strategies - Parameter estimation 13. Computational Biology - Sequence alignment - Phylogenetics - Gene expression analysis - Protein structure prediction - Systems biology 14. Computational Geometry - Convex hull algorithms - Voronoi diagrams - Delaunay triangulations - Polygon triangulation - Mesh generation 15. Numerical Optimization - Linear programming - Nonlinear programming - Convex optimization - Stochastic optimization - Semidefinite programming 16. Approximation Theory - Polynomial approximation - Spline approximation - Rational approximation - Padé approximation - Wavelets and multiresolution analysis 17. Information Theory - Entropy and mutual information - Channel capacity - Source coding - Error-correcting codes - Cryptography 18. Quantum Information Theory - Quantum circuits - Quantum algorithms - Quantum error correction - Quantum key distribution - Quantum machine learning 19. Mathematical Geosciences - Geostatistics - Inverse problems in geophysics - Seismic wave propagation - Atmospheric and ocean modeling - Climate modeling 20. Industrial Mathematics - Process modeling and optimization - Supply chain management - Scheduling and logistics - Quality control and reliability - Financial mathematics This expanded map covers even more areas of mathematics and their applications, showcasing the incredible breadth and depth of the field. However, it is still not a complete list, as mathematics continues to grow and evolve, with new connections and applications being discovered all the time.