Glossary of areas of mathematics

Short description: none

Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers.

This glossary is alphabetically sorted. This hides a large part of the relationships between areas. For the broadest areas of mathematics, see Mathematics § Areas of mathematics. The Mathematics Subject Classification is a hierarchical list of areas and subjects of study that has been elaborated by the community of mathematicians. It is used by most publishers for classifying mathematical articles and books.

A

Absolute differential calculus: An older name of Ricci calculus
Absolute geometry: Also called neutral geometry,^[1] a synthetic geometry similar to Euclidean geometry but without the parallel postulate.^[2]
Abstract algebra: The part of algebra devoted to the study of algebraic structures in themselves.^[3] Occasionally named modern algebra in course titles.
Abstract analytic number theory: The study of arithmetic semigroups as a means to extend notions from classical analytic number theory.^[4]
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).
Actuarial science: The discipline that applies mathematical and statistical methods to assess risk in insurance, finance and other industries and professions. More generally, actuaries apply rigorous mathematics to model matters of uncertainty.
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 deals with properties that are independent from distances and angles, such as alignment and parallelism.
Affine geometry of curves: The study of curve properties that are invariant under affine transformations.
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: One of the major areas of mathematics. Roughly speaking, it is the art of manipulating and computing with operations acting on symbols called variables that represent indeterminate numbers or other mathematical objects, such as vectors, matrices, or elements of algebraic structures.
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: An older name of computer algebra.
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: The part of number theory devoted to the use of algebraic methods, mainly those of commutative algebra, for the study of number fields and their rings of 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 wide area of mathematics centered on the study of continuous functions and including such topics as differentiation, integration, limits, and series.^[5]
Analytic combinatorics: part of enumerative combinatorics where methods of complex analysis are applied to generating functions.
Analytic geometry: 1. Also known as Cartesian geometry, the study of Euclidean geometry using Cartesian coordinates.; 2. Analogue to differential geometry, where differentiable functions are replaced with analytic functions. It is a subarea of both complex analysis and algebraic geometry.
Analytic number theory: An area of number theory that applies methods from mathematical analysis to solve problems about integers.^[6]
Analytic theory of L-functions
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: 1. Also known as elementary arithmetic, the methods and rules for computing with addition, subtraction, multiplication and division of numbers.; 2. Also known as higher arithmetic, another name for 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 use of algebraic geometry and more specially scheme theory for solving problems of number theory.
Arithmetic topology: a combination of algebraic number theory and topology studying analogies between prime ideals and knots
Arithmetical algebraic geometry: Another 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 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 set theory: the study of systems of axioms in a context relevant to set theory and mathematical logic.

B

Bifurcation theory: the study of changes in the qualitative or topological structure of a given family. It is a part of dynamical systems theory
Biostatistics: the development and application of statistical methods to a wide range of topics in biology.
Birational geometry: a part of algebraic geometry that deals with the geometry (of an algebraic variety) that is dependent only on its function field.
Bolyai–Lobachevskian geometry: see hyperbolic geometry

C

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: An area of mathematics connected by the fundamental theorem of calculus.^[7]
Calculus of infinitesimals: A foundation of calculus, first developed in the 17th century,^[8] that makes use of infinitesimal numbers.
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: see 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 algebra
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
Coding theory: the study of the properties of codes and their respective fitness for specific applications.
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 finite 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: an area primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.
Combinatorial number theory
Combinatorial optimization
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 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: a branch of computer science devoted to the study of algorithms which can be stated in terms of 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 statistics
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: an approach to mathematical constructivism following the program of axiomatic set theory, using the usual first-order language of classical 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.
Cryptography

D

Decision analysis
Decision theory
Derived noncommutative algebraic geometry
Descriptive set theory: a part of mathematical logic, more specifically a part of set theory dedicated to the study of Polish spaces.
Differential algebraic geometry: the adaption of methods and concepts from algebraic geometry to systems of algebraic differential equations.
Differential calculus: A branch of calculus that's contrasted to integral calculus,^[9] and concerned with derivatives.^[10]
Differential Galois theory: the study of the Galois groups of differential fields.
Differential geometry: a form of geometry that uses techniques from integral and differential calculus as well as linear and multilinear algebra to study problems in geometry. Classically, these were problems of Euclidean geometry, although now it has been expanded. It is generally concerned with geometric structures on differentiable manifolds. It is closely related to differential topology.
Differential geometry of curves: the study of smooth curves in Euclidean space by using techniques from differential geometry
Differential geometry of surfaces: the study of smooth surfaces with various additional structures using the techniques of differential geometry.
Differential topology: a branch of topology that deals with differentiable functions on differentiable manifolds.
Diffiety theory
Diophantine geometry: in general the study of algebraic varieties over fields that are finitely generated over their prime fields.
Discrepancy theory
Discrete differential geometry
Discrete exterior calculus
Discrete geometry: a branch of geometry that studies combinatorial properties and constructive methods of discrete geometric objects.
Discrete mathematics: the study of mathematical structures that are fundamentally discrete rather than continuous.
Discrete Morse theory: a combinatorial adaption of Morse theory.
Distance geometry
Domain theory: a branch that studies special kinds of partially ordered sets (posets) commonly called domains.
Donaldson theory: the study of smooth 4-manifolds using gauge theory.
Dyadic algebra
Dynamical systems theory: an area used to describe the behavior of the complex dynamical systems, usually by employing differential equations or difference equations.

E

Econometrics: the application of mathematical and statistical methods to economic data.
Effective descriptive set theory: a branch of descriptive set theory dealing with set of real numbers that have lightface definitions. It uses aspects of computability theory.
Elementary algebra: a fundamental form of algebra extending on elementary arithmetic to include the concept of variables.
Elementary arithmetic: the simplified portion of arithmetic considered necessary for primary education. It includes the usage addition, subtraction, multiplication and division of the natural numbers. It also includes the concept of fractions and negative numbers.
Elementary mathematics: parts of mathematics frequently taught at the primary and secondary school levels. This includes elementary arithmetic, geometry, probability and statistics, elementary algebra and trigonometry. (calculus is not usually considered a part)
Elementary group theory: the study of the basics of group theory
Elimination theory: the classical name for algorithmic approaches to eliminating between polynomials of several variables. It is a part of commutative algebra and algebraic geometry.
Elliptic geometry: a type of non-Euclidean geometry (it violates Euclid's parallel postulate) and is based on spherical geometry. It is constructed in elliptic space.
Enumerative combinatorics: an area of combinatorics that deals with the number of ways that certain patterns can be formed.
Enumerative geometry: a branch of algebraic geometry concerned with counting the number of solutions to geometric questions. This is usually done by means of intersection theory.
Epidemiology
Equivariant noncommutative algebraic geometry
Ergodic Ramsey theory: a branch where problems are motivated by additive combinatorics and solved using ergodic theory.
Ergodic theory: the study of dynamical systems with an invariant measure, and related problems.
Euclidean geometry: An area of geometry based on the axiom system and synthetic methods of the ancient Greek mathematician Euclid.^[11]
Euclidean differential geometry: also known as classical differential geometry. See differential geometry.
Euler calculus: a methodology from applied algebraic topology and integral geometry that integrates constructible functions and more recently definable functions by integrating with respect to the Euler characteristic as a finitely-additive measure.
Experimental mathematics: an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns.
Exterior algebra
Exterior calculus
Extraordinary cohomology theory
Extremal combinatorics: a branch of combinatorics, it is the study of the possible sizes of a collection of finite objects given certain restrictions.
Extremal graph theory: a branch of mathematics that studies how global properties of a graph influence local substructure.

F

Field theory: The branch of algebra dedicated to fields, a type of algebraic structure.^[12]
Finite geometry
Finite model theory: a restriction of model theory to interpretations on finite structures, which have a finite universe.
Finsler geometry: a branch of differential geometry whose main object of study is Finsler manifolds, a generalisation of a Riemannian manifolds.
First order arithmetic
Fourier analysis: the study of the way general functions may be represented or approximated by sums of trigonometric functions.
Fractal geometry
Fractional calculus: a branch of analysis that studies the possibility of taking real or complex powers of the differentiation operator.
Fractional dynamics: investigates the behaviour of objects and systems that are described by differentiation and integration of fractional orders using methods of fractional calculus.
Fredholm theory: part of spectral theory studying integral equations.
Function theory: an ambiguous term that generally refers to mathematical analysis.
Functional analysis: a branch of mathematical analysis, the core of which is formed by the study of function spaces, which are some sort of topological vector spaces.
Functional calculus: historically the term was used synonymously with calculus of variations, but now refers to a branch of functional analysis connected with spectral theory
Fuzzy mathematics: a branch of mathematics based on fuzzy set theory and fuzzy logic.
Fuzzy measure theory
Fuzzy set theory: a form of set theory that studies fuzzy sets, that is sets that have degrees of membership.

G

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: the study of mathematical models of strategic interaction among rational decision-makers.
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, and universal hyperbolic 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: extends the geometric algebra to include differentiation and integration.
Geometric combinatorics: a branch of combinatorics. It includes a number of subareas such as polyhedral combinatorics (the study of faces of convex polyhedra), convex geometry (the study of convex sets, in particular combinatorics of their intersections), and discrete geometry, which in turn has many applications to computational geometry.
Geometric function theory: the study of geometric properties of analytic functions.
Geometric invariant theory: a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces.
Geometric graph theory: a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means.
Geometric group theory: the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).
Geometric measure theory: the study of geometric properties of sets (typically in Euclidean space) through measure theory.
Geometric number 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: the study of algebraic structures known as groups.
Gyrotrigonometry: a form of trigonometry used in gyrovector space for hyperbolic geometry. (An analogy of the vector space in Euclidean geometry.)

H

Hard analysis: see classical analysis; part of analysis concerned with the representations of functions in terms of waves. It generalizes the notions of Fourier series and Fourier transforms from the Fourier analysis.; the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities.; the study of categorified structures.; a method for studying the cohomology groups of a smooth manifold M using partial differential equations.; a branch of functional calculus starting with holomorphic functions.; the study of homology in general algebraic settings.; also known as Lobachevskian geometry or Bolyai-Lobachevskian geometry. It is a non-Euclidean geometry looking at hyperbolic space.; the study of hyperbolic triangles in hyperbolic geometry, or hyperbolic functions in Euclidean geometry. Other forms include gyrotrigonometry and universal hyperbolic trigonometry.; the extension of real analysis and complex analysis to the study of functions where the argument is a hypercomplex number.

I

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Ideal theory

once the precursor name for what is now known as commutative algebra; it is the theory of ideals in commutative rings.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Idempotent analysis

the study of idempotent semirings, such as the tropical semiring.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Incidence geometry

the study of relations of incidence between various geometric objects, like curves and lines.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Inconsistent mathematics

see paraconsistent mathematics.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Infinitary combinatorics

an expansion of ideas in combinatorics to account for infinite sets.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Infinitesimal analysis

once a synonym for infinitesimal calculus

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Infinitesimal calculus

See calculus of infinitesimals

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Information geometry

an interdisciplinary field that applies the techniques of differential geometry to study probability theory and statistics. It studies statistical manifolds, which are Riemannian manifolds whose points correspond to probability distributions.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Integral calculus <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Integral geometry

the theory of measures on a geometrical space invariant under the symmetry group of that space.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Intersection theory

a branch of algebraic geometry and algebraic topology

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Intuitionistic type theory

a type theory and an alternative foundation of mathematics.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Invariant theory

studies how group actions on algebraic varieties affect functions.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Inventory theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Inversive geometry

the study of invariants preserved by a type of transformation known as inversion

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Inversive plane geometry

inversive geometry that is limited to two dimensions

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Inversive ring geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Itô calculus

extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Iwasawa theory

the study of objects of arithmetic interest over infinite towers of number fields.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Iwasawa-Tate theory

J

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;"> Job shop scheduling

K

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">K-theory

originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry it is referred to as algebraic K-theory. In physics, K-theory has appeared in type II string theory. (In particular twisted K-theory.)

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">K-homology

a homology theory on the category of locally compact Hausdorff spaces.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Kähler geometry

a branch of differential geometry, more specifically a union of Riemannian geometry, complex differential geometry and symplectic geometry. It is the study of Kähler manifolds. (named after Erich Kähler)

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">KK-theory

a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Klein geometry

More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G, which acts as the symmetry group of the geometry.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Knot theory

part of topology dealing with knots

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Kummer theory

provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field

L

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">L-theory

the K-theory of quadratic forms.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Large deviations theory

part of probability theory studying events of small probability (tail events).

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Large sample theory

also known as asymptotic theory

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Lattice theory

the study of lattices, being important in order theory and universal algebra

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Lie algebra theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Lie group theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Lie sphere geometry

geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Lie theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Line geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Linear algebra

a branch of algebra studying linear spaces and linear maps. It has applications in fields such as abstract algebra and functional analysis; it can be represented in analytic geometry and it is generalized in operator theory and in module theory. Sometimes matrix theory is considered a branch, although linear algebra is restricted to only finite dimensions. Extensions of the methods used belong to multilinear algebra.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Linear functional analysis <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Linear programming

a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">List of graphical methods

Included are diagram techniques, chart techniques, plot techniques, and other forms of visualization.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Local algebra

a term sometimes applied to the theory of local rings.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Local class field theory

the study of abelian extensions of local fields.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Low-dimensional topology

the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions.

M

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Malliavin calculus

a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Mathematical biology

the mathematical modeling of biological phenomena.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Mathematical chemistry

the mathematical modeling of chemical phenomena.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Mathematical economics

the application of mathematical methods to represent theories and analyze problems in economics.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Mathematical finance

a field of applied mathematics, concerned with mathematical modeling of financial markets.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Mathematical logic

a subfield of mathematics exploring the applications of formal logic to mathematics.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Mathematical optimization <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Mathematical physics

The development of mathematical methods suitable for application to problems in physics.^[13]

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Mathematical psychology

an approach to psychological research that is based on mathematical modeling of perceptual, thought, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characteristics with quantifiable behavior.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Mathematical sciences

refers to academic disciplines that are mathematical in nature, but are not considered proper subfields of mathematics. Examples include statistics, cryptography, game theory and actuarial science.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Mathematical sociology

the area of sociology that uses mathematics to construct social theories.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Mathematical statistics

the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Mathematical system theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Matrix algebra <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Matrix calculus <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Matrix theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Matroid theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Measure theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Metric geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Microlocal analysis <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Model theory

the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Modern algebra

Occasionally used for abstract algebra. The term was coined by van der Waerden as the title of his book Moderne Algebra, which was renamed Algebra in the latest editions.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Modern algebraic geometry

the form of algebraic geometry given by Alexander Grothendieck and Jean-Pierre Serre drawing on sheaf theory.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Modern invariant theory

the form of invariant theory that analyses the decomposition of representations into irreducibles.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Modular representation theory

a part of representation theory that studies linear representations of finite groups over a field K of positive characteristic p, necessarily a prime number.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Module theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Molecular geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Morse theory

a part of differential topology, it analyzes the topological space of a manifold by studying differentiable functions on that manifold.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Motivic cohomology <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Multilinear algebra

an extension of linear algebra building upon concepts of p-vectors and multivectors with Grassmann algebra.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Multiplicative number theory

a subfield of analytic number theory that deals with prime numbers, factorization and divisors.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Multivariable calculus

the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather than just one.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Multiple-scale analysis

N

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Neutral geometry

See absolute geometry.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Nevanlinna theory

part of complex analysis studying the value distribution of meromorphic functions. It is named after Rolf Nevanlinna

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Nielsen theory

an area of mathematical research with its origins in fixed point topology, developed by Jakob Nielsen

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Non-abelian class field theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Non-classical analysis <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Non-Euclidean geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Non-standard analysis <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Non-standard calculus <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Nonarchimedean dynamics

also known as p-adic analysis or local arithmetic dynamics

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Noncommutative algebra <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Noncommutative algebraic geometry

a direction in noncommutative geometry studying the geometric properties of formal duals of non-commutative algebraic objects.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Noncommutative geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Noncommutative harmonic analysis

see representation theory

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Noncommutative topology <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Nonlinear analysis <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Nonlinear functional analysis <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Number theory

a branch of pure mathematics primarily devoted to the study of the integers. Originally it was known as arithmetic or higher arithmetic.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Numerical analysis <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Numerical linear algebra

O

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;"> Operad theory

a type of abstract algebra concerned with prototypical algebras.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Operation research <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Operator K-theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Operator theory

part of functional analysis studying operators.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Optimal control theory

a generalization of the calculus of variations.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Optimal maintenance <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Orbifold theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Order theory

a branch that investigates the intuitive notion of order using binary relations.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Ordered geometry

a form of geometry omitting the notion of measurement but featuring the concept of intermediacy. It is a fundamental geometry forming a common framework for affine geometry, Euclidean geometry, absolute geometry and hyperbolic geometry.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Oscillation theory

P

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">p-adic analysis

a branch of number theory that deals with the analysis of functions of p-adic numbers.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">p-adic dynamics

an application of p-adic analysis looking at p-adic differential equations.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">p-adic Hodge theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Parabolic geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Paraconsistent mathematics

sometimes called inconsistent mathematics, it is an attempt to develop the classical infrastructure of mathematics based on a foundation of paraconsistent logic instead of classical logic.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Partition theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Perturbation theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Picard–Vessiot theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Plane geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Point-set topology

see general topology

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Pointless topology <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Poisson geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Polyhedral combinatorics

a branch within combinatorics and discrete geometry that studies the problems of describing convex polytopes.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Possibility theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Potential theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Precalculus <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Predicative mathematics <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Probability theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Probabilistic combinatorics <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Probabilistic graph theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Probabilistic number theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Projective geometry

a form of geometry that studies geometric properties that are invariant under a projective transformation.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Projective differential geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Proof theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Pseudo-Riemannian geometry

generalizes Riemannian geometry to the study of pseudo-Riemannian manifolds.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Pure mathematics

the part of mathematics that studies entirely abstract concepts.

Q

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Quantum calculus

a form of calculus without the notion of limits.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Quantum geometry

the generalization of concepts of geometry used to describe the physical phenomena of quantum physics

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Quaternionic analysis

R

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Ramsey theory

the study of the conditions in which order must appear. It is named after Frank P. Ramsey.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Rational geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Real algebra

the study of the part of algebra relevant to real algebraic geometry.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Real algebraic geometry

the part of algebraic geometry that studies real points of the algebraic varieties.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Real analysis

a branch of mathematical analysis; in particular hard analysis, that is the study of real numbers and functions of Real values. It provides a rigorous formulation of the calculus of real numbers in terms of continuity and smoothness, whilst the theory is extended to the complex numbers in complex analysis.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;"> Real Clifford algebra <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Real K-theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Recreational mathematics

the area dedicated to mathematical puzzles and mathematical games.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Recursion theory

see computability theory

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Representation theory

a subfield of abstract algebra; it studies algebraic structures by representing their elements as linear transformations of vector spaces. It also studies modules over these algebraic structures, providing a way of reducing problems in abstract algebra to problems in linear algebra.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Representation theory of groups <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Representation theory of the Galilean group <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Representation theory of the Lorentz group <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Representation theory of the Poincaré group <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Representation theory of the symmetric group <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Ribbon theory

a branch of topology studying ribbons.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Ricci calculus

Lua error: Internal error: The interpreter has terminated with signal "24".

A foundation of tensor calculus, developed by Gregorio Ricci-Curbastro in 1887–1896,^[14] and later developed for its applications to general relativity and differential geometry.^[15]

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Ring theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Riemannian geometry

a branch of differential geometry that is more specifically, the study of Riemannian manifolds. It is named after Bernhard Riemann and it features many generalizations of concepts from Euclidean geometry, analysis and calculus.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Rough set theory

the a form of set theory based on rough sets.

S

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Sampling theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Scheme theory

the study of schemes introduced by Alexander Grothendieck. It allows the use of sheaf theory to study algebraic varieties and is considered the central part of modern algebraic geometry.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Secondary calculus <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Semialgebraic geometry

a part of algebraic geometry; more specifically a branch of real algebraic geometry that studies semialgebraic sets.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Set-theoretic topology <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Set theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Sheaf theory

The study of sheaves, which connect local and global properties of geometric objects.^[16]

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Sheaf cohomology <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Sieve theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Single operator theory

deals with the properties and classifications of single operators.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Singularity theory

a branch, notably of geometry; that studies the failure of manifold structure.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Smooth infinitesimal analysis

a rigorous reformation of infinitesimal calculus employing methods of category theory. As a theory, it is a subset of synthetic differential geometry.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Solid geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Spatial geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Spectral geometry

a field that concerns the relationships between geometric structures of manifolds and spectra of canonically defined differential operators.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Spectral graph theory

the study of properties of a graph using methods from matrix theory.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Spectral theory

part of operator theory extending the concepts of eigenvalues and eigenvectors from linear algebra and matrix theory.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Spectral theory of ordinary differential equations

part of spectral theory concerned with the spectrum and eigenfunction expansion associated with linear ordinary differential equations.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Spectrum continuation analysis

generalizes the concept of a Fourier series to non-periodic functions.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Spherical geometry

a branch of non-Euclidean geometry, studying the 2-dimensional surface of a sphere.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Spherical trigonometry

a branch of spherical geometry that studies polygons on the surface of a sphere. Usually the polygons are triangles.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Statistical mechanics <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Statistical modelling <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Statistical theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Statistics

although the term may refer to the more general study of statistics, the term is used in mathematics to refer to the mathematical study of statistics and related fields. This includes probability theory.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Steganography <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Stochastic calculus <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Stochastic calculus of variations <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Stochastic geometry

the study of random patterns of points

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Stochastic process <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Stratified Morse theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Super linear algebra <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Surgery theory

a part of geometric topology referring to methods used to produce one manifold from another (in a controlled way.)

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Survey sampling <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Survey methodology <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Symbolic computation

also known as algebraic computation and computer algebra. It refers to the techniques used to manipulate mathematical expressions and equations in symbolic form as opposed to manipulating them by the numerical quantities represented by them.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Symbolic dynamics <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Symplectic geometry

a branch of differential geometry and topology whose main object of study is the symplectic manifold.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Symplectic topology <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Synthetic differential geometry

a reformulation of differential geometry in the language of topos theory and in the context of an intuitionistic logic.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Synthetic geometry

also known as axiomatic geometry, it is a branch of geometry that uses axioms and logical arguments to draw conclusions as opposed to analytic and algebraic methods.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Systolic geometry

a branch of differential geometry studying systolic invariants of manifolds and polyhedra.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Systolic hyperbolic geometry

the study of systoles in hyperbolic geometry.

T

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Tensor algebra, Tensor analysis, Tensor calculus, Tensor theory

the study and use of tensors, which are generalizations of vectors. A tensor algebra is also an algebraic structure that is used in the formal definition of tensors.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Tessellation

when periodic tiling has a repeating pattern.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Theoretical physics

a branch primarily of the science physics that uses mathematical models and abstraction of physics to rationalize and predict phenomena.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Theory of computation <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Time-scale calculus <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Topology <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Topological combinatorics

the application of methods from algebraic topology to solve problems in combinatorics.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Topological degree theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Topological graph theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Topological K-theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Topos theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Toric geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Transcendental number theory

a branch of number theory that revolves around the transcendental numbers.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Transformation geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Trigonometry

the study of triangles and the relationships between the length of their sides, and the angles between them. It is essential to many parts of applied mathematics.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Tropical analysis

see idempotent analysis

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Tropical geometry <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Twisted K-theory

a variation on K-theory, spanning abstract algebra, algebraic topology and operator theory.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Type theory

U

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Umbral calculus

the study of Sheffer sequences

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Uncertainty theory

a new branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Universal algebra

a field studying the formalization of algebraic structures itself.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Universal hyperbolic trigonometry

an approach to hyperbolic trigonometry based on rational geometry.

V

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Valuation theory <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Variational analysis <dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Vector algebra

a part of linear algebra concerned with the operations of vector addition and scalar multiplication, although it may also refer to vector operations of vector calculus, including the dot and cross product. In this case it can be contrasted with geometric algebra which generalizes into higher dimensions.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Vector analysis

also known as vector calculus, see vector calculus.

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Vector calculus

a branch of multivariable calculus concerned with differentiation and integration of vector fields. Primarily it is concerned with 3-dimensional Euclidean space.

W

<dt class="glossary " id="Lua error: Internal error: The interpreter has terminated with signal "24"." style="margin-top: 0.4em;">Wavelets

References

↑ Greenberg, Marvin Jay (2007), Euclidean and Non-Euclidean Geometries: Development and History (4th ed.), New York: W. H. Freeman, ISBN 978-0-7167-9948-1
↑ Faber, Richard L. (1983), Foundations of Euclidean and Non-Euclidean Geometry, New York: Marcel Dekker, ISBN 0-8247-1748-1
↑ Whitehead, C. (2002), Guide to Abstract Algebra (2nd ed.), Houndmills: Palgrave, ISBN 978-0-333-79447-0
↑ Knopfmacher, John (1990). Abstract Analytic Number Theory (2nd ed.). New York, NY: Dover Publishing. ISBN 0-486-66344-2.
↑ Apostol, Tom M.. Mathematical Analysis: A Modern Approach to Advanced Calculus (2 ed.). Addison-Wesley. ASIN 0201002884.
↑ Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3
↑ DeBaggis, Henry F.; Miller, Kenneth S. (1966). Foundations of the Calculus. Philadelphia: Saunders. OCLC 527896.
↑ Boyer, Carl B. (1959). The History of the Calculus and its Conceptual Development. New York: Dover. OCLC 643872. https://archive.org/details/historyofcalculu0000boye.
↑ Courant, R. (1937), Differential and Integral Calculus, I (2nd ed.), New York: Interscience, ISBN 978-4-87187-838-8
↑ "Definition of DIFFERENTIAL CALCULUS" (in en). https://merriam-webster.com/dictionary/differential+calculus.
↑ Eves, Howard (1963). A Survey of Geometry (Volume One). Allyn and Bacon.
↑ Adamson, I. T. (2007), Introduction to Field Theory, Dover Publications, ISBN 978-0-486-46266-0
↑ Definition from the Journal of Mathematical Physics. "Archived copy". http://jmp.aip.org/jmp/staff.jsp.
↑ Ricci, Gregorio; Levi-Civita, Tullio (March 1900), "Méthodes de calcul différentiel absolu et leurs applications" (in fr), Mathematische Annalen (Springer) 54 (1–2): 125–201, doi:10.1007/BF01454201, https://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002258102, retrieved 19 October 2019
↑ Schouten, Jan A. (1924). R. Courant. ed (in de). Der Ricci-Kalkül – Eine Einführung in die neueren Methoden und Probleme der mehrdimensionalen Differentialgeometrie (Ricci Calculus – An introduction in the latest methods and problems in multi-dimensional differential geometry). Grundlehren der mathematischen Wissenschaften. 10. Berlin: Springer Verlag. https://resolver.sub.uni-goettingen.de/purl?PPN373339186.
↑ Tennison, Barry R. (1975), Sheaf theory, London Mathematical Society Lecture Note Series, 20, Cambridge University Press, ISBN 978-0-521-20784-3, https://books.google.com/books?id=JOglpKIw6kIC

Lua error: Internal error: The interpreter has terminated with signal "24".

[1] Greenberg, Marvin Jay (2007), Euclidean and Non-Euclidean Geometries: Development and History (4th ed.), New York: W. H. Freeman, ISBN 978-0-7167-9948-1

[2] Faber, Richard L. (1983), Foundations of Euclidean and Non-Euclidean Geometry, New York: Marcel Dekker, ISBN 0-8247-1748-1

[3] Whitehead, C. (2002), Guide to Abstract Algebra (2nd ed.), Houndmills: Palgrave, ISBN 978-0-333-79447-0

[4] Knopfmacher, John (1990). Abstract Analytic Number Theory (2nd ed.). New York, NY: Dover Publishing. ISBN 0-486-66344-2.

[5] Apostol, Tom M.. Mathematical Analysis: A Modern Approach to Advanced Calculus (2 ed.). Addison-Wesley. ASIN 0201002884.

[6] Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3

[7] DeBaggis, Henry F.; Miller, Kenneth S. (1966). Foundations of the Calculus. Philadelphia: Saunders. OCLC 527896.

[8] Boyer, Carl B. (1959). The History of the Calculus and its Conceptual Development. New York: Dover. OCLC 643872. https://archive.org/details/historyofcalculu0000boye.

[9] Courant, R. (1937), Differential and Integral Calculus, I (2nd ed.), New York: Interscience, ISBN 978-4-87187-838-8

[10] "Definition of DIFFERENTIAL CALCULUS" (in en). https://merriam-webster.com/dictionary/differential+calculus.

[11] Eves, Howard (1963). A Survey of Geometry (Volume One). Allyn and Bacon.

[12] Adamson, I. T. (2007), Introduction to Field Theory, Dover Publications, ISBN 978-0-486-46266-0

[13] Definition from the Journal of Mathematical Physics. "Archived copy". http://jmp.aip.org/jmp/staff.jsp.

[14] Ricci, Gregorio; Levi-Civita, Tullio (March 1900), "Méthodes de calcul différentiel absolu et leurs applications" (in fr), Mathematische Annalen (Springer) 54 (1–2): 125–201, doi:10.1007/BF01454201, https://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002258102, retrieved 19 October 2019

[15] Schouten, Jan A. (1924). R. Courant. ed (in de). Der Ricci-Kalkül – Eine Einführung in die neueren Methoden und Probleme der mehrdimensionalen Differentialgeometrie (Ricci Calculus – An introduction in the latest methods and problems in multi-dimensional differential geometry). Grundlehren der mathematischen Wissenschaften. 10. Berlin: Springer Verlag. https://resolver.sub.uni-goettingen.de/purl?PPN373339186.

[16] Tennison, Barry R. (1975), Sheaf theory, London Mathematical Society Lecture Note Series, 20, Cambridge University Press, ISBN 978-0-521-20784-3, https://books.google.com/books?id=JOglpKIw6kIC

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

Anonymous

Search

Glossary of areas of mathematics

Namespaces

More

Page actions

Contents

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

See also

References

Navigation

Navigation

Help

googletranslator

Navigation

Wiki tools

Wiki tools

Anonymous

Search

Glossary of areas of mathematics

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

See also

References

Navigation

Wiki tools

Page tools

Other projects

Categories