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 below 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.
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

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.
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.
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.
concerns the study of discrete objects. Aspects include "counting" the objects satisfying certain criteria, deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria, finding "largest", "smallest", or "optimal" objects, and finding algebraic structures these objects may have.
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.
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.

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