List of mathematics lists
Encyclopedia

This article itemizes the various lists of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 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
Mathematics Subject Classification
The Mathematics Subject Classification is an alphanumerical classification scheme collaboratively produced by staff of and based on the coverage of the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH...

 formulated by the American Mathematical Society
American Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...

. 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
Mathematical Reviews
Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses of many articles in mathematics, statistics and theoretical computer science.- Reviews :...

and Zentralblatt MATH
Zentralblatt MATH
Zentralblatt MATH is a service providing reviews and abstracts for articles in pure and applied mathematics, published by Springer Science+Business Media. It is a major international reviewing service which covers the entire field of mathematics...

. 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.

Areas of advanced mathematics

See also Areas of mathematics
Areas of mathematics
Mathematics has become a vastly diverse subject over history, and there is a corresponding need to categorize the different areas of mathematics. A number of different classification schemes have arisen, and though they share some similarities, there are differences due in part to the different...

.

As a rough guide this list is divided into pure and applied sections although in reality these branches are overlapping and intertwined.

Algebra

Algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

 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

Calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

 studies the computation of limits, derivatives, and integrals of functions of real numbers, and in particular studies instantaneous rates of change. Analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

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

Geometry and topology

Geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

 is initially the study of spatial figures like circles and cubes, though it has been generalized considerably. Topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

 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
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

 concerns the study of discrete
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...

 (and usually finite) objects. Aspects include "counting" the objects satisfying certain criteria (enumerative combinatorics
Enumerative combinatorics
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations...

), deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial design
Combinatorial design
Combinatorial design theory is the part of combinatorial mathematics that deals with the existence and construction of systems of finite sets whose intersections have specified numerical properties....

s and matroid
Matroid
In combinatorics, a branch of mathematics, a matroid or independence structure is a structure that captures the essence of a notion of "independence" that generalizes linear independence in vector spaces....

 theory
), finding "largest", "smallest", or "optimal" objects (extremal combinatorics
Extremal combinatorics
Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics. Extremal combinatorics studies how large or how small a collection of finite objects can be, if it has to satisfy certain restrictions.For example, how many people can we invite to a party where among each...

and combinatorial optimization
Combinatorial optimization
In applied mathematics and theoretical computer science, combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects. In many such problems, exhaustive search is not feasible...

), and finding algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

ic structures these objects may have (algebraic combinatorics
Algebraic combinatorics
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra....

).
  • Outline of combinatorics 
  • List of graph theory topics 
  • Glossary of graph theory
    Glossary of graph theory
    Graph theory is a growing area in mathematical research, and has a large specialized vocabulary. Some authors use the same word with different meanings. Some authors use different words to mean the same thing. This page attempts to keep up with current usage....

     

Logic

Logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

 is the foundation which underlies mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

 and the rest of mathematics. It tries to formalize valid reasoning. In particular, it attempts to define what constitutes a proof.

Number theory

Number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

 studies the natural, or whole, numbers. One of the central concepts in number theory is that of the prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

, 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
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

 is an equation involving an unknown function and its derivatives.

In a dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

, 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
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

 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".

Computing

The fields of mathematics and computing intersect both in computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, 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
Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...

 is a branch of applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

 and electrical engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

 involving the quantification of information
Information
Information in its most restricted technical sense is a message or collection of messages that consists of an ordered sequence of symbols, or it is the meaning that can be interpreted from such a message or collection of messages. Information can be recorded or transmitted. It can be recorded as...

. Historically, information theory was developed to find fundamental limits on compressing and reliably communicating
Communication
Communication is the activity of conveying meaningful information. Communication requires a sender, a message, and an intended recipient, although the receiver need not be present or aware of the sender's intent to communicate at the time of communication; thus communication can occur across vast...

 data.

Signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

 is the analysis, interpretation, and manipulation of signal
Signal (electrical engineering)
In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying or spatial-varying quantity....

s. Signals of interest include sound
Audio signal processing
Audio signal processing, sometimes referred to as audio processing, is the intentional alteration of auditory signals, or sound. As audio signals may be electronically represented in either digital or analog format, signal processing may occur in either domain...

, images
Image processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...

, biological signals such as ECG, radar
Radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...

 signals, and many others. Processing of such signals includes filtering
Filter (signal processing)
In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...

, storage and reconstruction, separation of information from noise
Noise
In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise is random unwanted perturbation to a wanted signal; it is called noise as a generalisation of the acoustic noise heard when listening to a weak radio transmission with significant electrical noise...

, compression
Data compression
In computer science and information theory, data compression, source coding or bit-rate reduction is the process of encoding information using fewer bits than the original representation would use....

, and feature extraction
Feature extraction
In pattern recognition and in image processing, feature extraction is a special form of dimensionality reduction.When the input data to an algorithm is too large to be processed and it is suspected to be notoriously redundant then the input data will be transformed into a reduced representation...

.

Probability and statistics


Probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 is the formalization and study of the mathematics of uncertain events or knowledge. The related field of mathematical statistics
Mathematical statistics
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis...

 develops statistical theory
Statistical theory
The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics. The theory covers approaches to statistical-decision problems and to statistical inference, and the actions and deductions that...

 with mathematics. Statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the science concerned with collecting and analyzing data, is an autonomous discipline (and not a subdiscipline of applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...

).

Game theory

Game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

 is a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 that uses models to study interactions with formalized incentive structures ("games"). It has applications in a variety of fields, including economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...

, evolutionary biology, political science
Political science
Political Science is a social science discipline concerned with the study of the state, government and politics. Aristotle defined it as the study of the state. It deals extensively with the theory and practice of politics, and the analysis of political systems and political behavior...

, social psychology
Social psychology
Social psychology is the scientific study of how people's thoughts, feelings, and behaviors are influenced by the actual, imagined, or implied presence of others. By this definition, scientific refers to the empirical method of investigation. The terms thoughts, feelings, and behaviors include all...

 and military strategy
Military strategy
Military strategy is a set of ideas implemented by military organizations to pursue desired strategic goals. Derived from the Greek strategos, strategy when it appeared in use during the 18th century, was seen in its narrow sense as the "art of the general", 'the art of arrangement' of troops...

.

Operations research

Operations research
Operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

 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


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

  • Scientific equations named after people
    Scientific equations named after people
    This is a list of scientific equations named after people .-See also:* Eponym* List of eponymous laws* List of laws in science* List of equations* Scientific constants named after people* Scientific phenomena named after people...

     

About mathematics


Mathematicians

Mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

s 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
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

 and theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...

).

Work of particular mathematicians


Reference tables


Integrals

In calculus, the integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 of a function is a generalization of area, mass, volume, sum, and total. The following pages list the integrals of many different 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 
  • Table of mathematical symbols
    Table of mathematical symbols
    This is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts being installed, and in , as an image.-Symbols:-Variations:...

  • Table of logic symbols
    Table of logic symbols
    In logic, a set of symbols is commonly used to express logical representation. As logicians are familiar with these symbols, they are not explained each time they are used. So, for students of logic, the following table lists many common symbols together with their name, pronunciation and related...


External links and references

  • 2000 Mathematics Subject Classification from the American Mathematical Society
    American Mathematical Society
    The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...

    . Many research journals in mathematics ask authors to classify their papers according to this scheme; the classifications appear in the published work.
  • The Mathematical Atlas
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