Mathematics
This section provides a collection of expository articles on mathematics, organized by subject area and progressing from introductory to advanced topics.
If you are starting from scratch
As one possible route, read Set Theory → Proofs → Functions → Linear Algebra → Differentiation and Integration. This is only a suggestion — feel free to start from whichever subject interests you.
Foundations
Set Theory
Foundations of sets and mappings, axiomatic set theory, ZFC axioms, ordinals and cardinals.
Proof Techniques
How to write proofs: direct proof, proof by contradiction, mathematical induction, and other proof methods.
Mathematical Logic
What a proof is, formally: formal systems, proof theory, type theory, the Curry–Howard correspondence, incompleteness theorems, and the foundations of proof assistants.
Functions
Basic concepts of functions, composition, inverse functions, and elementary functions.
Sequences and Series
Arithmetic and geometric sequences, recurrence relations, convergence of series.
Theory of Computation
Turing machines, automata, computability, undecidability, computational complexity, formal languages.
Algebra
Algebra
From high-school equations to groups, rings and fields, Galois theory and 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
From polynomials and shapes to algebraic varieties and schemes.
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 — techniques widely used in machine learning and optimization.
Integration
From indefinite and definite integrals to multiple integrals and line integrals.
Real Analysis
Lebesgue measure and integration, Lp 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 & Transform Methods
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.
Numerical Analysis: Special Topics
Interval arithmetic, verified numerical computation, random number generation.
Frequently Asked Questions
What topics are covered in these mathematics notes?
The notes cover a broad range of mathematical topics: foundations (set theory, proofs, functions, sequences), algebra (algebra, linear algebra, category theory, Lie algebras), number theory and discrete mathematics, geometry and topology, analysis (calculus, real analysis, complex analysis, functional analysis), differential equations, applied analysis and transform methods (Fourier, Laplace, Z- and Radon transforms), probability and statistics, and computational mathematics (computer algebra, numerical analysis).
Are these notes suitable for beginners?
Each subject is split into introductory, basic, intermediate and advanced levels, so a beginner can start from the introductory articles. Some subjects go as far as graduate-level material (algebraic geometry, functional analysis), so how far you go depends on the background you bring.