Lists of mathematics topics

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 extremely long list of mathematics articles contains all mathematical articles in alphabetical order. This article brings together the same content organised in a manner better suited to 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.

Areas of basic mathematics

These lists include topics typically taught in secondary education or in the first year of university.

*List of basic discrete mathematics topics
*List of calculus topics
*List of geometry topics
*List of topics in logic
*List of trigonometry topics
**List of basic trigonometry topics
*List of trigonometric identities

Areas of advanced mathematics

See also 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".

*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
*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 studies the same subjects, but on a more rigorous level, and also topics that evolved from calculus.

*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 multivariable calculus topics
*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 curve topics
*List of triangle topics
*List of circle topics
**List of topics related to π
*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
*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").

*List of combinatorics topics
*List of graph theory topics
*Glossary of graph theory

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 different "kinds" of prime numbers (each with an accompanying table)

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.

*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
*List of wave topics

Computing

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 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.

*List of information theory topics
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*List of algebraic coding theory topics
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Probability and statistics

Probability theory is the formalization and study of the mathematics of uncertain events or knowledge. The related field of statistics concerns the mathematics and science of collecting and analyzing data.

*List of probability distributions
*List of probability topics
*List of statistics topics
*List of stochastic processes topics

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.

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*Glossary of 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.

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*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
**Erdős conjecture — a list of conjectures by Paul Erdős
*Combinatorial principles
*List of equations
*List of formulae involving π
*List of fundamental theorems
*List of mathematical identities
*List of inequalities
*List of lemmas
*List of mathematical proofs
*List of NP-complete problems
*List of statements undecidable in ZFC
*List of undecidable problems
*List of theorems

General concepts

*List of convexity topics
*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

*Scientific 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).

*List of geometers
*List of logicians
*List of mathematicians
*List of probabilists
*List of statisticians

Work of particular mathematicians

*List of things named after Paul Erdős
*List of topics named after Leonhard Euler
*List of topics named after Carl Friedrich Gauss
*List of topics named after Srinivasa Ramanujan

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 arc hyperbolic functions
*List of integrals of exponential functions
*List of integrals of 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

Meta-lists

*List of important publications in mathematics
*List of important publications in statistics
*List of mathematical theories
*List of mathematics articles
*List of mathematics categories

External links and references

* [http://www.ams.org/msc 2000 Mathematics Subject Classification] from the American Mathematical Society. Many research journals in mathematics ask authors to classify their papers according to this scheme; the classifications appear in the published work.
* [http://www.math-atlas.org The Mathematical Atlas]

Notes

*fnb|1 Definition from the "Journal of Mathematical Physics" [http://jmp.aip.org/jmp/staff.jsp] .