# Lists of mathematics topics

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This article itemizes the various **lists** of **mathematics topics**. 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.

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

#### 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
- List of abstract algebra topics
- List of algebraic structures
- List of Boolean algebra topics
- List of category theory topics
- 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
- Glossary of field theory
- Glossary of group theory
- Glossary of linear algebra
- Glossary of ring theory
- List of cohomology theories

#### Calculus and analysis

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.

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

#### Geometry and topology

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.

- List of geometry topics
- List of geometric shapes
- List of curves topics
- List of triangle topics
- List of circle topics
- List of general topology topics
- List of differential geometry topics
- List of algebraic geometry topics
- List of algebraic surfaces

- List of algebraic topology topics
- List of cohomology theories

- List of geometric topology topics
- List of knot theory topics
- List of Lie group topics
- Glossary of differential geometry and topology
- Glossary of general topology
- List of mathematical properties of points
- Glossary of Riemannian and metric geometry
- Glossary of scheme theory

#### 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*).

#### Logic

Logic is the foundation which 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
- Glossary of order theory
- List of set theory topics

#### Number theory

Number theory 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

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 partial differential equation topics
- List of nonlinear partial differential equations

#### 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}

#### Computation

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 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 electrical engineering 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.

#### Probability and statistics

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).

#### 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, evolutionary biology, 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 performance of real-world systems.

## Methodology

## 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.

## 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 numeral system topics
- List of order topics
- List of partition topics
- List of polynomial topics
- List of properties of sets of reals
- List of transforms
- List of permutation topics

## 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 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 mathematical functions
- List of mathematical knots and links
- List of manifolds
- List of mathematical shapes
- 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 algebraic surfaces
- List of surfaces
- Table of Lie groups

## Equations named after people

## About mathematics

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

### 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).

### Work of particular mathematicians

- List of things named after Niels Henrik Abel
- List of things named after Archimedes
- List of things named after Emil Artin
- List of things named after Thomas Bayes
- 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 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 Fibonacci
- 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 Jacques Hadamard
- 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 W. V. D. Hodge
- List of things named after Carl Gustav Jacob Jacobi
- List of things named after Felix Klein
- List of things named after Joseph-Louis Lagrange
- 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 John Milnor
- List of things named after Isaac Newton
- 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 Jean-Pierre Serre
- List of things named after James Joseph Sylvester
- List of things named after Alfred Tarski
- 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 Ernst Witt
- List of things named after John von Neumann

## Reference tables

### 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

## Others

## Notes

**^Note 1**: Definition from the*Journal of Mathematical Physics*.

## External links and references

- 2000 Mathematics Subject Classification from the American Mathematical Society, a scheme authors find many mathematics research journals asking them to use to classify their submissions; those published then include these classifications.
- The Mathematical Atlas