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Areas of mathematics

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The aim of this page is to list all areas of modern mathematics, with a brief explanation about their scope and links to other parts of this encyclopedia, set out in a systematic way.

The way research-level mathematics is internally organised is mostly determined by practitioners, and does change over time; this is in contrast with the apparently timeless syllabus divisions used in mathematics education, where calculus can seem to be much the same over a time scale of a century. Calculus itself does not appear as a major heading — most of the traditional material would be divided amongst topics under analysis. This illustrates, in part, the difficulty of communicating the principles of any large-scale organization. The research on most calculus topics was carried out in the eighteenth century, and has long been assimilated. The story of why fields exist as specialties involves, in most cases, quite a long intellectual history (and sometimes institutional history).

The American Mathematical Society's [Mathematics Subject Classification (2000 edition)] has been used as a starting point to ensure all areas are covered, and related areas are close together. However, the MSC aims to classify mathematical papers, not mathematics itself, so additional categories have been used. See also the list of mathematics lists.

Contents

Foundations / general

The study of structure starting with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fields, structures that generalize the properties possessed by everyday numbers. Long standing questions about compass and straightedge construction were finally settled by Galois theory. The physically important concept of vectors, generalized to vector spaces is studied in linear algebra.

Combinatorics (MSC 05)
Studies finite collections of objects that satisfy specified criteria. In particular, it is concerned with "counting" the objects in those collections (enumerative combinatorics) and with deciding whether certain "optimal" objects exist (extremal combinatorics). It includes graph theory, used to describe inter-connected objects (a graph in this sense is a collection of connected points). See also the list of combinatorics topics, list of graph theory topics and glossary of graph theory. While these are the classical definitions, a combinatorial flavour is present in many parts of problem-solving.
Order theory (MSC 06)
With any set of real numbers, it is possible to write them out in ascending order. Order Theory extends this idea to sets in general. It includes notions like lattices and ordered algebraic structures. See also the order theory glossary and the list of order topics.
General algebraic systems (MSC 08)
Given a set, ways of combining or relating members of that set can be defined. If these obey certain rules, then a particular algebraic structure is formed. Universal algebra is the more formal study of these structures and systems.
Number theory (MSC 11)
Number theory is traditionally concerned with the properties of integers. More recently, it has come to be concerned with wider classes of problems that have arisen naturally from the study of integers. It can be divided into elementary number theory (where the integers are studied without use of techniques from other mathematical fields); analytic number theory (where calculus and complex analysis are used as tools); algebraic number theory (which studies the algebraic numbers - the roots of polynomials with integer coefficients); Geometric number theory; combinatorial number theory and computational number theory. See also the list of number theory topics
Field theory and polynomials (MSC 12)
Field theory studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined. A polynomial is an expression in which constants and variables are combined using only addition, subtraction, and multiplication.
Commutative rings and algebras (MSC 13)
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, then a×b=b×a. Commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. It is foundational both for algebraic geometry and for algebraic number theory. The most prominent example for commutative rings are polynomial rings.
(Also transformation groups, abstract harmonic analysis)

Analysis

Analysis is primarily concerned with change. Rates of change, accumulated change, multiple things changing relative to (or independently of) one another, etc. (Also: probabilistic potential theory, numerical approximation, representation theory, analysis on manifolds)

Geometry

Geometry (MSC 51) deals with spatial relationships, using fundamental qualities or axioms. Such axioms can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions. See also List of geometry topics

Convex geometry and discrete geometry (MSC 52)
Includes the study of objects such as polytopes and polyhedra. See also List of convexity topics
Discrete or combinatorial geometry (MSC 52)
The study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation. It includes the study of shapes such as the Platonic solids and the notion of tessellation.
Differential geometry (MSC 53)
The study of geometry using calculus, and is very closely related to differential topology. Covers such areas as Riemannian geometry, curvature and differential geometry of curves. See also the glossary of differential geometry and topology.
Algebraic geometry (MSC 14)
Given a polynomial of two real variables, then the points on a plane where that function is zero will form a curve. An algebraic curve extends this notion to polynomials over a field in a given number of variables. Algebraic geometry may be viewed the study of these curves. See also the list of algebraic geometry topics and list of algebraic surfaces.
Topology
Deals with the properties of a figure that do not change when the figure is continuously deformed. The main areas are point set topology (or general topology), algebraic topology, and the topology of manifolds, defined below.
General topology (MSC 54)
Also called point set topology. Properties of topological spaces. Includes such notions as open and closed sets, compact spaces, continuous functions, convergence, separation axioms, metric spaces, dimension theory. See also the glossary of general topology and the list of general topology topics.
Algebraic topology (MSC 55)
Properties of algebraic objects associated with a topological space and how these algebraic objects capture properties of such spaces. Contains areas like homology theory, cohomology theory, homotopy theory, and homological algebra, some of them examples of functors. Homotopy deals with homotopy groups (including the fundamental group) as well as simplicial complexes and CW complexes (also called cell complexes). See also the list of algebraic topology topics.
Manifolds (MSC 57)
A manifold can be thought of as an n-dimensional generalization of a surface in the usual 3-dimensional Euclidean space. The study of manifolds includes differential topology, which looks at the properties of differentiable functions defined over a manifold. See also complex manifolds.

Probability and statistics

See also glossary of probability and statistics
Probability theory (MSC 60)
The study of how likely a given event is to occur. See also , and the list of probability topics.
Stochastic processes (MSC 60G/H)
Considers with aggregate effect of a random function, either over time (a time series) or physical space (a random field). See also List of stochastic processes topics, and .
Statistics (MSC 62)
Analysis of data, and how representative it is. See also the list of statistical topics.

Computational sciences

Numerical analysis, (MSC 65)
Many problems in mathematrics cannot in general be solved exactly (e.g. the quintic equation). Numerical analysis is the study of algorithms to provide an aproximate solution to problems to a given degree of accuracy. Includes numerical differentiation, numerical integration and numerical methods. See also List of numerical analysis topics

Physical sciences

Mechanics
Addresses what happens when a real physical object is subjected to forces. This divides naturally into the study of rigid solids, deformable solids, and fluids, detailed below.
Particle mechanics (MSC 70)
In mathematics, a particle is a point-like, perfectly rigid, solid object. Particle mechanics deals with the results of subjecting particles to forces. It includes celestial mechanics — the study of the motion of celestial objects.
Mechanics of deformable solids (MSC 74)
Most real-world objects are not point-like nor perfectly rigid. More importantly, objects change shape when subjected to forces. This subject has a very strong overlap with continuum mechanics, which is concerned with continuous matter. It deals with such notions as stress, strain and elasticity. See also continuum mechanics.
Fluid mechanics (MSC 76)
Fluids in this sense includes not just liquids, but flowing gases, and even solids under certain situations. (For example, dry sand can behave like a fluid). It includes such notions as viscosity, turbulent flow and laminar flow (its opposite). See also fluid dynamics.

Non-physical sciences

 

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