Editing Lists of mathematics topics (section)

==Areas of advanced mathematics==
{{See also|Areas of mathematics|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===
{{Main|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]]
* [[Glossary of field theory]]
* [[Glossary of group theory]]
* [[Glossary of linear algebra]]
* [[Glossary of ring theory]]
* [[List of abstract algebra topics]]
* [[List of algebraic structures]]
* [[List of Boolean algebra topics]]
* [[List of category theory topics]]
* [[List of cohomology theories]]
* [[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]]

====Calculus and analysis====
[[File:Gibbs phenomenon 10.png|thumb|200px|right|[[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. [[Mathematical analysis|Analysis]] evolved from calculus.
* [[Glossary of tensor theory]]
* [[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 mathematical series]]
* [[List of multivariable calculus topics]]
* [[List of q-analogs]]
* [[List of real analysis topics]]
* [[List of variational topics]]
* See also [[#Dynamical systems and differential equations|Dynamical systems and differential equations]] section below.

====Geometry and topology====
[[File:Ford circles.svg|200px|right|thumb|[[Ford circle]]s—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.
* [[Glossary of differential geometry and topology]]
* [[Glossary of general topology]]
* [[Glossary of Riemannian and metric geometry]]
* [[Glossary of scheme theory]]
* [[List of algebraic geometry topics]]
** [[List of algebraic surfaces]]
* [[List of algebraic topology topics]]
** [[List of cohomology theories]]
* [[List of circle topics]]
** [[List of topics related to pi]]
* [[List of curves topics]]
* [[List of differential geometry topics]]
* [[List of general topology topics]]
* [[List of geometric shapes]]
* [[List of geometric topology topics]]
* [[List of geometry topics]]
* [[List of knot theory topics]]
* [[List of Lie group topics]]
* [[List of mathematical properties of points]]
* [[List of topology topics]]
** [[List of topologies]]
** [[Topological property]]
* [[List of triangle topics]]

====Combinatorics====
[[Combinatorics]] concerns the study of [[Countable set|discrete]] (and usually [[Finite set|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 design]]s and [[matroid]] theory''), finding "largest", "smallest", or "optimal" objects (''[[extremal combinatorics]]'' and ''[[combinatorial optimization]]''), and finding [[algebra]]ic structures these objects may have (''[[algebraic combinatorics]]'').
* [[Outline of combinatorics]]
* [[Glossary of graph theory]]
* [[List of graph theory topics]]

====Logic====
[[File:Venn A intersect B.svg|thumb|200px|right|[[Venn diagram]]s 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.
* [[List of Boolean algebra topics]]
* [[List of first-order theories]]
* [[List of large cardinal properties]]
* [[List of mathematical logic topics]]
* [[List of set theory topics]]
* [[Glossary of order theory]]

====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.
* [[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]]&mdash;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====
[[File:Limitcycle.svg|right|200px|thumb|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.
* [[List of dynamical systems and differential equations topics]]
* [[List of nonlinear partial differential equations]]
* [[List of partial differential equation topics]]

====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".{{ref|1|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]]
* [[Index of wave articles]]

====Theory of computation====
[[File:Raytraced image jawray.jpg|thumb|200px|right|[[Ray tracing (graphics)|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.
* [[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 [[social science]] involving the quantification of [[information]]. Historically, information theory was developed to find fundamental limits on compressing and reliably [[communication|communicating]] data.

[[Signal processing]] is the analysis, interpretation, and manipulation of [[Signal (electrical engineering)|signal]]s. Signals of interest include [[Audio signal processing|sound]], [[image processing|images]], biological signals such as [[ECG]], [[radar]] signals, and many others. Processing of such signals includes [[Filter (signal processing)|filtering]], storage and reconstruction, separation of information from [[noise]], [[Data compression|compression]], and [[feature extraction]].
* [[List of algebraic coding theory topics]]
* [[List of information theory topics]]
* [[Wikipedia:WikiProject Cryptography/List of cryptography topics|List of cryptography topics]]

====Probability and statistics====
[[File:Normal approximation to binomial.svg|200px|right|thumb|The "bell curve"—the probability density function of the [[normal distribution]].]]
{{Main|Lists of statistics topics}}
[[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]]).
* [[Catalog of articles in probability theory]]
* [[List of probability topics]]
* [[List of stochastic processes topics]]
* [[List of probability distributions]]
* [[List of statistics topics]]
* [[Outline of regression analysis]]

====Game theory====
[[Game theory]] is a branch of [[mathematics]] that uses [[model (abstract)|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]].
* [[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 the performance of real-world systems.
* [[List of knapsack problems]]
* [[List of network theory topics]]