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Areas of mathematics
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 Wikipedia, 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 doesn't 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 organisation. 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 maths itself, so additional categories have been used. See also list of mathematical topics.
Foundations / general
Algebra
Algebra is primarily concerned with the notions of symmetry, discrete sets and their manipulations and rules for working with operators. It is an extension of the algebra topics encountered at school. See also Category:Algebra, Category:Abstract 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 ). See also list of combinatorics topics.
(Also transformation groups, abstract harmonic analysis)
Analysis
- 26: Real functions, including fractional derivatives and integrals
- 28: Measure and integration
- 30: Complex functions, including all approximation theory in the complex domain
- 31: Potential theory
- 32: Several complex variables and analytic spaces
- 33: Special functions
- 34: Ordinary differential equations
- 35: Partial differential equations
- 37: Dynamical systems and ergodic theory
- 39: Difference equations and functional equations
- 40: Sequences, series, summability
- 41: Approximations and expansions
- 42: Fourier analysis, including Fourier transforms, trigonometric approximation and trigonometric interpolation, and orthogonal functions
- 43: Abstract harmonic analysis
- 44: Integral transforms, operational calculus
- 45: Integral equations
- 46: Functional analysis, including infinite-dimensional holomorphy, integral transforms in distribution spaces
- 47: Operator theory
- 49: Calculus of variations and optimal control; optimization (including geometric integration theory )
- 58: Global analysis , analysis on manifolds (including infinite-dimensional holomorphy)
(Also: probabilistic potential theory , numerical approximation, representation theory, analysis on manifolds)
Geometry
Geometry 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 Category:Geometry.
- Convex geometry (MSC 52)
- (Main article )
- Discrete or combinatorial geometry (MSC 52)
- may be loosely defined as 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 tesselation. See also the Category:Discrete geometry category, and the Discrete geometry main article.
- Differential geometry (MSC 53)
- is 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 Glossary of differential geometry and topology, Main article.
- Topology
- Deals with the properties of a figure that do not change when the figure is continuously deformed. (Main article). The two main areas are point set topology (or general topology) and algebraic topology, defined below.
- Point set or general topology (MSC 54)
- Properties of topological spaces (sets with some subsets defined as open). Includes such notions as open and closed sets, compact spaces, continuous functions, convergence, separation axioms, metric spaces, dimension theory. See also glossary of general topology for detailed definitions, the list of general topology topics and Main article.
- Algebraic topology (MSC 55)
- Properties of algebraic objects within a topological space. Includes areas like homotopy groups (including the fundamental group), and homological algebra. See also list of algebraic topology topics, Main article.
- Manifolds (MSC 57)
- A manifold can be thought of as an N-dimensional generalization of a surface in normal 3D Euclidean space. The study of manifolds covers differential topology, which looks at the properties of differentiable functions defined over a manifold. See also complex manifolds, Main article.
- Cell complexes
- (Main article)
(Also algebraic geometry.)
Applied mathematics
Probability and statistics
- Probability theory (MSC 60)
- the study of how likely a given event is to occur. See also probability
- Statistics (MSC 62)
- Analysis of data, and how representative it is. See also List of statistical topics
Computational sciences
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. See also category:mechanics, main article.
- Particle mechanics (MSC 70)
- In maths, 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. (Main Article)
- 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. It has a very strong overlap with continuum mechanics, which deals with continuous matter. It deals with such notions as stress, strain and elasticity. See also Category:Continuum mechanics, Main Article .
- 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 Category:fluid mechanics, Category:fluid dynamics, fluid dynamics, main article
Non-physical sciences
Last updated: 01-04-2007 01:18:57
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