Mathematics
This section provides a collection of expository articles on mathematics, organized by subject area and progressing from introductory to advanced topics.
Foundations
Set Theory
Foundations of sets and mappings, axiomatic set theory, ZFC axioms, ordinals and cardinals.
Proof Techniques
Direct proof, proof by contradiction, mathematical induction, and other proof methods.
Functions
Basic concepts of functions, composition, inverse functions, and elementary functions.
Sequences
Arithmetic and geometric sequences, recurrence relations, convergence of series.
Algebra
Abstract Algebra
Groups, rings, and fields, Galois theory, homological algebra.
Linear Algebra
Vector spaces, determinants, eigenvalues, diagonalization, and function spaces.
Category Theory
Categories, functors, natural transformations, adjunctions, monads.
Lie Algebras
Fundamentals of Lie groups and Lie algebras, representation theory.
Number Theory & Discrete Mathematics
Geometry & Topology
Geometry
From trigonometry and coordinate geometry to projective and computational geometry.
Differential Geometry
Differential geometry of curves and surfaces, Riemannian geometry, connections and curvature.
Algebraic Geometry
Algebraic varieties, schemes, coherent sheaves.
Topology
Topological spaces, fundamental groups, homology, applied algebraic topology.
Analysis
Differentiation
Fundamentals and applications of differentiation: limits, derivatives, partial derivatives.
Matrix Calculus
Differentiation with respect to matrices and vectors — essential techniques for machine learning and optimization.
Integration
From indefinite and definite integrals to multiple integrals and line integrals.
Real Analysis
Lebesgue measure and integration, $L^p$ spaces, foundations of Fourier analysis.
Complex Analysis
Foundations of complex function theory, conformal mappings, and the residue theorem.
Functional Analysis
Banach spaces, Hilbert spaces, operator theory.
Differential Equations & Optimization
Applied Analysis & Integral Transforms
Fourier Analysis
Fourier series and the Fourier transform, from fundamentals to applications.
Laplace Transform
Definition of the Laplace transform, inverse transforms, and applications to differential equations.
Z-Transform
The Z-transform for discrete signals, inverse transforms, and applications to difference equations.
Radon Transform
Theory of the Radon transform and applications to CT image reconstruction.
Probability & Statistics
Computational Mathematics
Computer Algebra & Arbitrary-Precision Arithmetic
Symbolic computation (polynomial GCD, factorization, Gröbner bases, symbolic integration) and arbitrary-precision arithmetic (fast multiplication, floating-point, root extraction, constant computation) — 29 chapters in all.
Numerical Analysis
Interpolation and approximation, numerical integration, linear systems, eigenvalue problems, and numerical methods for ODEs/PDEs.
Verified Numerics & Interval Arithmetic
Interval arithmetic, verified numerical computation, random number generation.