Lists of mathematics topics

Lists of mathematics topics cover a variety of topics related to mathematics. Some of these lists link to hundreds of articles; some link only to a few. The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. Lists cover aspects of basic and advanced mathematics, methodology, mathematical statements, integrals, general concepts, mathematical objects, and reference tables. They also cover equations named after people, societies, mathematicians, journals, and meta-lists.

The purpose of this list is not similar to that of the Mathematics Subject Classification formulated by the American Mathematical Society. Many mathematics journals ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The subject codes so listed are used by the two major reviewing databases, Mathematical Reviews and Zentralblatt MATH. This list has some items that would not fit in such a classification, such as list of exponential topics and list of factorial and binomial topics, which may surprise the reader with the diversity of their coverage.
Basic mathematics

This branch is typically taught in secondary education or in the first year of university.
Areas of advanced mathematics
See also: Areas of mathematics and Glossary of areas of mathematics

As a rough guide, this list is divided into pure and applied sections although in reality, these branches are overlapping and intertwined.
Pure mathematics
Main article: Pure mathematics

Algebra

Algebra includes the study of algebraic structures, which are sets and operations defined on these sets satisfying certain axioms. The field of algebra is further divided according to which structure is studied; for instance, group theory concerns an algebraic structure called group.

Outline of algebra
Glossary of field theory
Glossary of group theory
Glossary of linear algebra
Glossary of ring theory
List of abstract algebra topics
List of algebraic structures
List of Boolean algebra topics
List of category theory topics
List of cohomology theories
List of commutative algebra topics
List of homological algebra topics
List of group theory topics
List of representation theory topics
List of linear algebra topics
List of reciprocity laws

Calculus and analysis
Fourier series approximation of square wave in five steps.

Calculus studies the computation of limits, derivatives, and integrals of functions of real numbers, and in particular studies instantaneous rates of change. Analysis evolved from calculus.

Glossary of tensor theory
List of complex analysis topics
List of functional analysis topics
List of vector spaces in mathematics
List of integration and measure theory topics
List of harmonic analysis topics
List of Fourier analysis topics
List of mathematical series
List of multivariable calculus topics
List of q-analogs
List of real analysis topics
List of variational topics
See also Dynamical systems and differential equations section below.

Geometry and topology
Ford circles—A circle rests upon each fraction in lowest terms. Each touches its neighbors without crossing.

Geometry is initially the study of spatial figures like circles and cubes, though it has been generalized considerably. Topology developed from geometry; it looks at those properties that do not change even when the figures are deformed by stretching and bending, like dimension.

Glossary of differential geometry and topology
Glossary of general topology
Glossary of Riemannian and metric geometry
Glossary of scheme theory
List of algebraic geometry topics
List of algebraic surfaces
List of algebraic topology topics
List of cohomology theories
List of circle topics
List of topics related to pi
List of curves topics
List of differential geometry topics
List of general topology topics
List of geometric shapes
List of geometric topology topics
List of geometry topics
List of knot theory topics
List of Lie group topics
List of mathematical properties of points
List of topology topics
List of topologies
Topological property
List of triangle topics

Combinatorics

Combinatorics concerns the study of discrete (and usually finite) objects. Aspects include "counting" the objects satisfying certain criteria (enumerative combinatorics), deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and finding algebraic structures these objects may have (algebraic combinatorics).

Outline of combinatorics
Glossary of graph theory
List of graph theory topics

Logic
Venn diagrams are illustrations of set theoretical, mathematical or logical relationships.

Logic is the foundation that underlies mathematical logic and the rest of mathematics. It tries to formalize valid reasoning. In particular, it attempts to define what constitutes a proof.

List of Boolean algebra topics
List of first-order theories
List of large cardinal properties
List of mathematical logic topics
List of set theory topics
Glossary of order theory

Number theory

The branch of mathematics deals with the properties and relationships of numbers, especially positive integers. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theory also studies the natural, or whole, numbers. One of the central concepts in number theory is that of the prime number, and there are many questions about primes that appear simple but whose resolution continues to elude mathematicians.

List of algebraic number theory topics
List of number theory topics
List of recreational number theory topics
Glossary of arithmetic and Diophantine geometry
List of prime numbers—not just a table, but a list of various kinds of prime numbers (each with an accompanying table)
List of zeta functions

Applied mathematics
Dynamical systems and differential equations
Phase portrait of a continuous-time dynamical system, the Van der Pol oscillator.

A differential equation is an equation involving an unknown function and its derivatives.

In a dynamical system, a fixed rule describes the time dependence of a point in a geometrical space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems.

List of dynamical systems and differential equations topics
List of nonlinear partial differential equations
List of partial differential equation topics

Mathematical physics

Mathematical physics is concerned with "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories".1

List of mathematical topics in classical mechanics
List of mathematical topics in quantum theory
List of mathematical topics in relativity
List of string theory topics
Index of wave articles

Theory of computation
Ray tracing is a process based on computational mathematics.

The fields of mathematics and computing intersect both in computer science, the study of algorithms and data structures, and in scientific computing, the study of algorithmic methods for solving problems in mathematics, science, and engineering.

List of algorithm general topics
List of computability and complexity topics
Lists for computational topics in geometry and graphics
List of combinatorial computational geometry topics
List of computer graphics and descriptive geometry topics
List of numerical computational geometry topics
List of computer vision topics
List of formal language and literal string topics
List of numerical analysis topics
List of terms relating to algorithms and data structures

Information theory and signal processing

Information theory is a branch of applied mathematics and Social science involving the quantification of information. Historically, information theory was developed to find fundamental limits on compressing and reliably communicating data.

Signal processing is the analysis, interpretation, and manipulation of signals. Signals of interest include sound, images, biological signals such as ECG, radar signals, and many others. Processing of such signals includes filtering, storage and reconstruction, separation of information from noise, compression, and feature extraction.

List of algebraic coding theory topics
List of information theory topics
List of cryptography topics

Probability and statistics
The "bell curve"—the probability density function of the normal distribution.
Main article: Lists of statistics topics

Probability theory is the formalization and study of the mathematics of uncertain events or knowledge. The related field of mathematical statistics develops statistical theory with mathematics. Statistics, the science concerned with collecting and analyzing data, is an autonomous discipline (and not a subdiscipline of applied mathematics).

Catalog of articles in probability theory
List of probability topics
List of stochastic processes topics
List of probability distributions
List of statistics topics
Outline of regression analysis

Game theory

Game theory is a branch of mathematics that uses models to study interactions with formalized incentive structures ("games"). It has applications in a variety of fields, including economics, anthropology , political science, social psychology and military strategy.

Glossary of game theory
List of games in game theory

Operations research

Operations research is the study and use of mathematical models, statistics, and algorithms to aid in decision-making, typically with the goal of improving or optimizing the performance of real-world systems.

List of knapsack problems
List of network theory topics

Methodology

List of graphical methods
List of mathematics-based methods
List of rules of inference

Mathematical statements

A mathematical statement amounts to a proposition or assertion of some mathematical fact, formula, or construction. Such statements include axioms and the theorems that may be proved from them, conjectures that may be unproven or even unprovable, and also algorithms for computing the answers to questions that can be expressed mathematically.

List of algorithms
List of axioms
List of conjectures
List of conjectures by Paul Erdős
Combinatorial principles
List of equations
List of formulae involving pi
List of representations of e
List of inequalities
List of lemmas
List of mathematical identities
List of mathematical proofs
List of theorems

General concepts

List of convexity topics
List of dualities
List of exceptional set concepts
List of exponential topics
List of factorial and binomial topics
List of fractal topics
List of logarithm topics
List of mathematical properties of points
List of numeral system topics
List of order topics
List of partition topics
List of permutation topics
List of polynomial topics
List of properties of sets of reals
List of transforms

Mathematical objects

Among mathematical objects are numbers, functions, sets, a great variety of things called "spaces" of one kind or another, algebraic structures such as rings, groups, or fields, and many other things.

List of mathematical examples
List of algebraic surfaces
List of curves
List of complex reflection groups
List of complexity classes
List of examples in general topology
List of finite simple groups
List of Fourier-related transforms
List of manifolds
List of mathematical constants
List of mathematical functions
List of mathematical knots and links
List of mathematical shapes
List of mathematical spaces
List of matrices
List of numbers
List of polygons, polyhedra and polytopes
List of regular polytopes
List of simple Lie groups
List of small groups
List of special functions and eponyms
List of surfaces
Table of Lie groups

Equations named after people

Scientific equations named after people

About mathematics

List of letters used in mathematics and science
List of mathematical societies
List of mathematics competitions
List of mathematics history topics
List of publications in mathematics
List of mathematics journals

Mathematicians
Main article: List of mathematicians

Mathematicians study and research in all the different areas of mathematics. The publication of new discoveries in mathematics continues at an immense rate in hundreds of scientific journals, many of them devoted to mathematics and many devoted to subjects to which mathematics is applied (such as theoretical computer science and theoretical physics).

List of films about mathematicians
List of game theorists
List of geometers
List of logicians
List of mathematicians
List of mathematical probabilists
List of statisticians

Work of particular mathematicians
See also: Category:Lists of things named after mathematicians

List of things named after Niels Henrik Abel
List of things named after George Airy
List of things named after Jean d'Alembert
List of things named after Archimedes
List of things named after Vladimir Arnold
List of things named after Emil Artin
List of things named after Stefan Banach
List of things named after Thomas Bayes
List of things named after members of the Bernoulli family
List of things named after Jakob Bernoulli
List of things named after Friedrich Bessel
List of things named after Élie Cartan
List of things named after Augustin-Louis Cauchy
List of things named after Arthur Cayley
List of things named after Pafnuty Chebyshev
List of things named after John Horton Conway
List of things named after Richard Dedekind
List of things named after Pierre Deligne
List of things named after Peter Gustav Lejeune Dirichlet
List of things named after Albert Einstein
List of things named after Euclid
List of things named after Leonhard Euler
List of things named after Paul Erdős
List of things named after Pierre de Fermat
List of things named after Fibonacci
List of things named after Joseph Fourier
List of things named after Erik Fredholm
List of things named after Ferdinand Georg Frobenius
List of things named after Carl Friedrich Gauss
List of things named after Évariste Galois
List of things named after Hermann Grassmann
List of things named after Alexander Grothendieck
List of things named after Jacques Hadamard
List of things named after William Rowan Hamilton
List of things named after Erich Hecke
List of things named after Eduard Heine
List of things named after Charles Hermite
List of things named after David Hilbert
List of things named after W. V. D. Hodge
List of things named after Carl Gustav Jacob Jacobi
List of things named after Johannes Kepler
List of things named after Felix Klein
List of things named after Joseph-Louis Lagrange
List of things named after Johann Lambert
List of things named after Pierre-Simon Laplace
List of things named after Adrien-Marie Legendre
List of things named after Gottfried Leibniz
List of things named after Sophus Lie
List of things named after Joseph Liouville
List of things named after Andrey Markov
List of things named after John Milnor
List of things named after Hermann Minkowski
List of things named after John von Neumann
List of things named after Isaac Newton
List of things named after Emmy Noether
List of things named after Henri Poincaré
List of things named after Siméon Denis Poisson
List of things named after Pythagoras
List of things named after Srinivasa Ramanujan
List of things named after Bernhard Riemann
List of things named after Issai Schur
List of things named after Anatoliy Skorokhod
List of things named after George Gabriel Stokes
List of things named after Jean-Pierre Serre
List of things named after James Joseph Sylvester
List of things named after Alfred Tarski
List of things named after Alan Turing
List of things named after Stanislaw Ulam
List of things named after Karl Weierstrass
List of things named after André Weil
List of things named after Hermann Weyl
List of things named after Norbert Wiener
List of things named after Ernst Witt

Reference tables

List of mathematical reference tables
List of moments of inertia
Table of derivatives

Integrals

In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. The following pages list the integrals of many different functions.

Lists of integrals
List of integrals of exponential functions
List of integrals of hyperbolic functions
List of integrals of inverse hyperbolic functions
List of integrals of inverse trigonometric functions
List of integrals of irrational functions
List of integrals of logarithmic functions
List of integrals of rational functions
List of integrals of trigonometric functions

Journals

List of mathematics journals
List of mathematics education journals
Category:History of science journals
Category:Philosophy of science literature

Meta-lists

Glossary of mathematical symbols
List of important publications in mathematics
List of important publications in statistics
List of mathematical theories
List of mathematics categories
List of mathematical symbols by subject
Table of logic symbols
Table of mathematical symbols