This is a glossary of terms that are or have been considered areas of study in mathematics.
- Absolute differential calculus: the original name for tensor calculus developed around 1890.
- Absolute geometry: an extension of ordered geometry that is sometimes referred to as neutral geometry because its axiom system is neutral to the parallel postulate.
- Abstract algebra: the study of algebraic structures and their properties. Originally it was known as modern algebra.
- Abstract analytic number theory: a branch of mathematics that takes ideas from classical analytic number theory and applies them to various other areas of mathematics.
- Abstract differential geometry: a form of differential geometry without the notion of smoothness from calculus. Instead it is built using sheaf theory and sheaf cohomology.
- Abstract harmonic analysis: a modern branch of harmonic analysis that extends upon the generalized Fourier transforms that can be defined on locally compact groups.
- Abstract homotopy theory: a part of topology that deals with homotopic functions, i.e. functions from one topological space to another which are homotopic (the functions can be deformed into one another).
- Additive combinatorics: the part of arithmetic combinatorics devoted to the operations of addition and subtraction.
- Additive number theory: a part of number theory that studies subsets of integers and their behaviour under addition.
- Affine geometry: a branch of geometry that is centered on the study of geometric properties that remain unchanged by affine transformations. It can be described as a generalization of Euclidean geometry.
- Affine geometry of curves: the study of curves in affine space.
- Affine differential geometry: a type of differential geometry dedicated to differential invariants under volume-preserving affine transformations.
- Ahlfors theory: a part of complex analysis being the geometric counterpart of Nevanlinna theory. It was invented by Lars Ahlfors
- Algebra: a major part of pure mathematics centered on operations and relations. Beginning with elementary algebra, it introduces the concept of variables and how these can be manipulated towards problem solving; known as equation solving. Generalizations of operations and relations defined on sets have led to the idea of an algebraic structure which are studied in abstract algebra. Other branches of algebra include universal algebra, linear algebra and multilinear algebra.
- Algebraic analysis: motivated by systems of linear partial differential equations, it is a branch of algebraic geometry and algebraic topology that uses methods from sheaf theory and complex analysis, to study the properties and generalizations of functions. It was started by Mikio Sato.
- Algebraic combinatorics: an area that employs methods of abstract algebra to problems of combinatorics. It also refers to the application of methods from combinatorics to problems in abstract algebra.
- Algebraic computation: see symbolic computation.
- Algebraic geometry: a branch that combines techniques from abstract algebra with the language and problems of geometry. Fundamentally, it studies algebraic varieties.
- Algebraic graph theory: a branch of graph theory in which methods are taken from algebra and employed to problems about graphs. The methods are commonly taken from group theory and linear algebra.
- Algebraic K-theory: an important part of homological algebra concerned with defining and applying a certain sequence of functors from rings to abelian groups.
- Algebraic number theory: a part of algebraic geometry devoted to the study of the points of the algebraic varieties whose coordinates belong to an algebraic number field. It is a major branch of number theory and is also said to study algebraic structures related to algebraic integers.
- Algebraic statistics: the use of algebra to advance statistics, although the term is sometimes restricted to label the use of algebraic geometry and commutative algebra in statistics.
- Algebraic topology: a branch that uses tools from abstract algebra for topology to study topological spaces.
- Algorithmic number theory: also known as computational number theory, it is the study of algorithms for performing number theoretic computations.
- Anabelian geometry: an area of study based on the theory proposed by Alexander Grothendieck in the 1980s that describes the way a geometric object of an algebraic variety (such as an algebraic fundamental group) can be mapped into another object, without it being an abelian group.
- Analysis: a rigorous branch of pure mathematics that had its beginnings in the formulation of infinitesimal calculus. (Then it was known as infinitesimal analysis.) The classical forms of analysis are real analysis and its extension complex analysis, whilst more modern forms are those such as functional analysis.
- Analytic combinatorics: part of enumerative combinatorics where methods of complex analysis are applied to generating functions.
- Analytic geometry: usually this refer to the study of geometry using a coordinate system (also known as Cartesian geometry). Alternatively it can refer to the geometry of analytic varieties. In this respect it is essentially equivalent to real and complex algebraic geometry.
- Analytic number theory: part of number theory using methods of analysis (as opposed to algebraic number theory)
- Applied mathematics: a combination of various parts of mathematics that concern a variety of mathematical methods that can be applied to practical and theoretical problems. Typically the methods used are for science, engineering, finance, economics and logistics.
- Approximation theory: part of analysis that studies how well functions can be approximated by simpler ones (such as polynomials or trigonometric polynomials)
- Arakelov geometry: also known as Arakelov theory
- Arakelov theory: an approach to Diophantine geometry used to study Diophantine equations in higher dimensions (using techniques from algebraic geometry). It is named after Suren Arakelov.
- Arithmetic: to most people this refers to the branch known as elementary arithmetic dedicated to the usage of addition, subtraction, multiplication and division. However arithmetic also includes higher arithmetic referring to advanced results from number theory.
- Arithmetic algebraic geometry: see arithmetic geometry
- Arithmetic combinatorics: the study of the estimates from combinatorics that are associated with arithmetic operations such as addition, subtraction, multiplication and division.
- Arithmetic dynamics:Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, and/or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.
- Arithmetic geometry: the study of schemes of finite type over the spectrum of the ring of integers
- Arithmetic topology: a combination of algebraic number theory and topology studying analogies between prime ideals and knots
- Arithmetical algebraic geometry: an alternative name for arithmetic algebraic geometry
- Asymptotic combinatorics:It uses the internal structure of the objects to derive formulas for their generating functions and then complex analysis techniques to get asymptotics.
- Asymptotic geometric analysis
- Asymptotic theory: the study of asymptotic expansions
- Auslander–Reiten theory: the study of the representation theory of Artinian rings
- Axiomatic geometry: also known as synthetic geometry: it is a branch of geometry that uses axioms and logical arguments to draw conclusions as opposed to analytic and algebraic methods.
- Axiomatic homology theory
- Axiomatic set theory: the study of systems of axioms in a context relevant to set theory and mathematical logic.
- C*-algebra theory: a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties-(i) A is a topologically closed set in the norm topology of operators.(ii)A is closed under the operation of taking adjoints of operators.
- Cartesian geometry: see analytic geometry
- Calculus: a branch usually associated with limits, functions, derivatives, integrals and infinite series. It forms the basis of classical analysis, and historically was called the calculus of infinitesimals or infinitesimal calculus. Now it can refer to a system of calculation guided by symbolic manipulation.
- Calculus of infinitesimals: also known as infinitesimal calculus. It is a branch of calculus built upon the concepts of infinitesimals.
- Calculus of moving surfaces: an extension of the theory of tensor calculus to include deforming manifolds.
- Calculus of variations: the field dedicated to maximizing or minimizing functionals. It used to be called functional calculus.
- Catastrophe theory: a branch of bifurcation theory from dynamical systems theory, and also a special case of the more general singularity theory from geometry. It analyses the germs of the catastrophe geometries.
- Categorical logic: a branch of category theory adjacent to the mathematical logic. It is based on type theory for intuitionistic logics.
- Category theory: the study of the properties of particular mathematical concepts by formalising them as collections of objects and arrows.
- Chaos theory: the study of the behaviour of dynamical systems that are highly sensitive to their initial conditions.
- Character theory: a branch of group theory that studies the characters of group representations or modular representations.
- Class field theory: a branch of algebraic number theory that studies abelian extensions of number fields.
- Classical differential geometry: also known as Euclidean differential geometry. see Euclidean differential geometry.
- Classical algebraic topology
- Classical analysis: usually refers to the more traditional topics of analysis such as real analysis and complex analysis. It includes any work that does not use techniques from functional analysis and is sometimes called hard analysis. However it may also refer to mathematical analysis done according to the principles of classical mathematics.
- Classical analytic number theory
- Classical differential calculus
- Classical Diophantine geometry
- Classical Euclidean geometry: see Euclidean geometry
- Classical geometry: may refer to solid geometry or classical Euclidean geometry. See geometry
- Classical invariant theory: the form of invariant theory that deals with describing polynomial functions that are invariant under transformations from a given linear group.
- Classical mathematics: the standard approach to mathematics based on classical logic and ZFC set theory.
- Classical projective geometry
- Classical tensor calculus
- Clifford analysis: the study of Dirac operators and Dirac type operators from geometry and analysis using clifford algebras.
- Clifford theory is a branch of representation theory spawned from Cliffords theorem.
- Cobordism theory
- Cohomology theory
- Combinatorial analysis
- Combinatorial commutative algebra: a discipline viewed as the intersection between commutative algebra and combinatorics. It frequently employs methods from one to address problems arising in the other. Polyhedral geometry also plays a significant role.
- Combinatorial design theory: a part of combinatorial mathematics that deals with the existence and construction of systems of fintie sets whose intersections have certain properties.
- Combinatorial game theory
- Combinatorial geometry: see discrete geometry
- Combinatorial group theory: the theory of free groups and the presentation of a group. It is closely related to geometric group theory and is applied in geometric topology.
- Combinatorial mathematics
- Combinatorial number theory
- Combinatorial set theory: also known as Infinitary combinatorics. see infinitary combinatorics
- Combinatorial theory
- Combinatorial topology: an old name for algebraic topology, when topological invariants of spaces were regarded as derived from combinatorial decompositions.
- Combinatorics: a branch of discrete mathematics concerned with countable structures. Branches of it include enumerative combinatorics, combinatorial design theory, matroid theory, extremal combinatorics and algebraic combinatorics, as well as many more.
- Commutative algebra: a branch of abstract algebra studying commutative rings.
- Complex algebra
- Complex algebraic geometry: the mainstream of algebraic geometry devoted to the study of the complex points of algebraic varieties.
- Complex analysis: a part of analysis that deals with functions of a complex variable.
- Complex analytic dynamics: a subdivision of complex dynamics being the study of the dynamic systems defined by analytic functions.
- Complex analytic geometry: the application of complex numbers to plane geometry.
- Complex differential geometry: a branch of differential geometry that studies complex manifolds.
- Complex dynamics: the study of dynamical systems defined by iterated functions on complex number spaces.
- Complex geometry: the study of complex manifolds and functions of complex variables. It includes complex algebraic geometry and complex analytic geometry.
- Complexity theory: the study of complex systems with the inclusion of the theory of complex systems.
- Computable analysis: the study of which parts of real analysis and functional analysis can be carried out in a computable manner. It is closely related to constructive analysis.
- Computable model theory: a branch of model theory dealing with the relevant questions computability.
- Computability theory: a branch of mathematical logic originating in the 1930s with the study of computable functions and Turing degrees, but now includes the study of generalized computability and definability. It overlaps with proof theory and effective descriptive set theory.
- Computational algebraic geometry
- Computational complexity theory: a branch of mathematics and theoretical computer science that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other.
- Computational geometry
- Computational group theory: the study of groups by means of computers.
- Computational mathematics: the mathematical research in areas of science where computing plays an essential role.
- Computational number theory: also known as algorithmic number theory, it is the study of algorithms for performing number theoretic computations.
- Computational real algebraic geometry
- Computational synthetic geometry
- Computational topology
- Computer algebra: see symbolic computation
- Conformal geometry: the study of conformal transformations on a space.
- Constructive analysis: mathematical analysis done according to the principles of constructive mathematics. This differs from classical analysis.
- Constructive function theory: a branch of analysis that is closely related to approximation theory, studying the connection between the smoothness of a function and its degree of approximation
- Constructive mathematics: mathematics which tends to use intuitionistic logic. Essentially that is classical logic but without the assumption that the law of the excluded middle is an axiom.
- Constructive quantum field theory: a branch of mathematical physics that is devoted to showing that quantum theory is mathematically compatible with special relativity.
- Constructive set theory
- Contact geometry: a branch of differential geometry and topology, closely related to and considered the odd-dimensional counterpart of symplectic geometry. It is the study of a geometric structure called a contact structure on a differentiable manifold.
- Convex analysis: the study of properties of convex functions and convex sets.
- Convex geometry: part of geometry devoted to the study of convex sets.
- Coordinate geometry: see analytic geometry
- CR geometry: a branch of differential geometry, being the study of CR manifolds.
- Galois cohomology: an application of homological algebra, it is the study of group cohomology of Galois modules.
- Galois theory: named after Évariste Galois, it is a branch of abstract algebra providing a connection between field theory and group theory.
- Galois geometry: a branch of finite geometry concerned with algebraic and analytic geometry over a Galois field.
- Game theory
- Gauge theory
- General topology: also known as point-set topology, it is a branch of topology studying the properties of topological spaces and structures defined on them. It differs from other branches of topology as the topological spaces do not have to be similar to manifolds.
- Generalized trigonometry: developments of trigonometric methods from the application to real numbers of Euclidean geometry to any geometry or space. This includes spherical trigonometry, hyperbolic trigonometry, gyrotrigonometry, rational trigonometry, universal hyperbolic trigonometry, fuzzy qualitative trigonometry, operator trigonometry and lattice trigonometry.
- Geometric algebra: an alternative approach to classical, computational and relativistic geometry. It shows a natural correspondence between geometric entities and elements of algebra.
- Geometric analysis: a discipline that uses methods from differential geometry to study partial differential equations as well as the applications to geometry.
- Geometric calculus
- Geometric combinatorics
- Geometric function theory: the study of geometric properties of analytic functions.
- Geometric homology theory
- Geometric invariant theory
- Geometric graph theory
- Geometric group theory
- Geometric measure theory
- Geometric topology: a branch of topology studying manifolds and mappings between them; in particular the embedding of one manifold into another.
- Geometry: a branch of mathematics concerned with shape and the properties of space. Classically it arose as what is now known as solid geometry; this was concerning practical knowledge of length, area and volume. It was then put into an axiomatic form by Euclid, giving rise to what is now known as classical Euclidean geometry. The use of coordinates by René Descartes gave rise to Cartesian geometry enabling a more analytical approach to geometric entities. Since then many other branches have appeared including projective geometry, differential geometry, non-Euclidean geometry, Fractal geometry and algebraic geometry. Geometry also gave rise to the modern discipline of topology.
- Geometry of numbers: initiated by Hermann Minkowski, it is a branch of number theory studying convex bodies and integer vectors.
- Global analysis: the study of differential equations on manifolds and the relationship between differential equations and topology.
- Global arithmetic dynamics
- Graph theory: a branch of discrete mathematics devoted to the study of graphs. It has many applications in physical, biological and social systems.
- Group-character theory: the part of character theory dedicated to the study of characters of group representations.
- Group representation theory
- Group theory
- Gyrotrigonometry: a form of trigonometry used in gyrovector space for hyperbolic geometry. (An analogy of the vector space in Euclidean geometry.)