Lists of mathematics topics

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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.

Contents

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

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.

Calculus and analysis

Fourier series approximation of square wave in five steps. Gibbs phenomenon 10.png
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.

Geometry and topology

Ford circles--A circle rests upon each fraction in lowest terms. Each touches its neighbors without crossing. Ford circles.svg
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.

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

Venn diagrams are illustrations of set theoretical, mathematical or logical relationships. Venn A intersect B.svg
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.

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.

Applied mathematics

Dynamical systems and differential equations

Phase portrait of a continuous-time dynamical system, the Van der Pol oscillator. Limitcycle.svg
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.

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

Theory of computation

Ray tracing is a process based on computational mathematics. Raytraced image jawray.jpg
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.

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.

Probability and statistics

The "bell curve"--the probability density function of the normal distribution. Normal approximation to binomial.svg
The "bell curve"—the probability density function of the normal distribution.

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, anthropology, political science, social psychology and military strategy.

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.

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

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.

Equations named after people

About 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

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.

Journals

Meta-lists

See also

Others

Notes

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This article includes a mathematics-related list of lists.

Related Research Articles

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.

<span class="mw-page-title-main">Discrete mathematics</span> Study of discrete mathematical structures

Discrete mathematics is the study of mathematical structures that can be considered "discrete" rather than "continuous". Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete mathematics".

<span class="mw-page-title-main">Differential geometry</span> Branch of mathematics dealing with functions and geometric structures on differentiable manifolds

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries.

In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign =. The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation.

<span class="mw-page-title-main">History of mathematics</span>

The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars.

<span class="mw-page-title-main">Mathematical analysis</span> Branch of mathematics

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

<span class="mw-page-title-main">Discrete geometry</span> Branch of geometry that studies combinatorial properties and constructive methods

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme that has collaboratively been produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH. The MSC is used by many mathematics journals, which ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The current version is MSC2020.

<span class="mw-page-title-main">Outline of trigonometry</span> Overview of and topical guide to trigonometry

The following outline is provided as an overview of and topical guide to trigonometry:

Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.

This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.

<span class="mw-page-title-main">Mathematics education in the United States</span> Overview of mathematics education in the United States

Mathematics education in the United States varies considerably from one state to the next, and even within a single state. However, with the adoption of the Common Core Standards in most states and the District of Columbia beginning in 2010, mathematics content across the country has moved into closer agreement for each grade level. The SAT, a standardized university entrance exam, has been reformed to better reflect the contents of the Common Core. However, many students take alternatives to the traditional pathways, including accelerated tracks. As of 2023, twenty-seven states require students to pass three math courses before graduation from high school, and seventeen states and the District of Columbia require four.

Mathematics is a field of study that investigates topics such as number, space, structure, and change.

Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers.

The Colloquium Lecture of the American Mathematical Society is a special annual session of lectures.

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