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{{more footnotes|date=August 2024}}
{{Short description|Mathematics glossary}}
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{{OR1|reason=Most area descriptions are inventions of editors, contradicts the content of the linked article, or express POV on the subject of the area. Many are blatantly wrong.|date=May 2022}}
[[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 [[mathematical analysis|analysis]] for the study of [[natural number]]s.
This '''glossary of areas of mathematics''' is a list of sub-discliplines within [[Pure mathematics|pure]] and [[applied mathematics]]. Some entries are broad topics, like [[algebra]], while others are narrower in scope, like [[convex geometry]].{{Compact ToC|short1|sym=yes|seealso=yes|refs=no}}

This glossary is alphabetically sorted. This hides a large part of the relationships between areas. For the broadest areas of mathematics, see {{slink|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.
{{Compact ToC|side=yes|center=yes|top=yes|num=no|extlinks=no|seealso=yes|refs=yes|nobreak=yes|}}


==A==
==A==


{{glossary}}
* '''[[Tensor|Absolute differential calculus]]''': the original name for [[tensor calculus]] developed around 1890.
{{term|[[Ricci calculus|Absolute differential calculus]]}}
* '''[[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]].
{{defn| An older name of {{gli|Ricci calculus}} }}
* '''[[Abstract algebra]]''': the study of [[algebraic structures]] and their properties. Originally it was known as ''modern algebra''.
{{term|[[Absolute geometry]]}}
* '''[[Abstract analytic number theory]]''': a branch of mathematics that takes ideas from [[analytic number theory|classical analytic number theory]] and applies them to various other areas of mathematics.
{{defn| Also called {{gli|neutral geometry}},<ref>{{Citation | author-link=Marvin Greenberg|last=Greenberg|first=Marvin Jay| title=Euclidean and Non-Euclidean Geometries: Development and History|edition=4th|place=New York|publisher=W. H. Freeman|year=2007|isbn=978-0-7167-9948-1}}</ref> a [[synthetic geometry]] similar to [[Euclidean geometry]] but without the [[parallel postulate]].<ref>{{Citation |title=Foundations of Euclidean and Non-Euclidean Geometry |last=Faber |first=Richard L. |year=1983 |publisher=Marcel Dekker |location=New York|isbn=0-8247-1748-1 }}</ref>}}
* '''[[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]].
{{term|[[Abstract algebra]]}}
* '''[[Abstract harmonic analysis]]''': a modern branch of [[harmonic analysis]] that extends upon the generalized [[Fourier transforms]] that can be defined on [[locally compact group]]s.
{{defn| The part of {{gli|algebra}} devoted to the study of [[algebraic structure]]s in themselves.<ref>{{Citation | last1=Whitehead | first1=C. | title=Guide to Abstract Algebra | edition=2nd | isbn=978-0-333-79447-0 | year=2002 | publisher=Palgrave | location=Houndmills}}</ref> Occasionally named {{gli|modern algebra}} in course titles.}}
* '''[[homotopy theory|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).
{{term|[[Abstract analytic number theory]]}}
* '''[[Actuarial science]]''': the discipline that applies [[mathematics|mathematical]] and [[statistics|statistical]] methods to [[Risk assessment|assess risk]] in [[insurance]], [[finance]] and other industries and professions. More generally, actuaries apply rigorous mathematics to model matters of uncertainty.
{{defn| The study of [[Abstract analytic number theory|arithmetic semigroup]]s as a means to extend notions from [[analytic number theory|classical analytic number theory]].<ref>{{cite book | title=Abstract Analytic Number Theory | first=John | last=Knopfmacher | edition=2nd | publisher=Dover Publishing | location=New York, NY | year=1990 | orig-year=1975 | isbn=0-486-66344-2 | zbl=0743.11002 }}</ref>}}
* '''[[Additive combinatorics]]''': the part of [[arithmetic combinatorics]] devoted to the operations of [[addition]] and [[subtraction]].
{{term|Abstract differential geometry}}
* '''[[Additive number theory]]''': a part of [[number theory]] that studies subsets of [[integer]]s and their behaviour under addition.
{{defn| A form of [[differential geometry]] without the notion of [[smoothness]] from [[calculus]]. Instead it is built using [[sheaf theory]] and [[sheaf cohomology]].}}
* '''[[Affine geometry]]''': a branch of [[geometry]] that is centered on the study of geometric properties that remain unchanged by [[affine transformation]]s. It can be described as a generalization of Euclidean geometry.
{{term|[[Abstract harmonic analysis]]}}
* '''[[Affine geometry of curves]]''': the study of [[curve]]s in [[affine space]].
{{defn| A modern branch of [[harmonic analysis]] that extends upon the generalized [[Fourier transform]]s that can be defined on [[locally compact group]]s.}}
* '''[[Affine differential geometry]]''': a type of [[differential geometry]] dedicated to differential [[invariant (mathematics)|invariants]] under [[volume]]-preserving [[affine transformation]]s.
{{term|[[homotopy theory|Abstract homotopy theory]]}}
* '''[[Ahlfors theory]]''': a part of [[complex analysis]] being the geometric counterpart of [[Nevanlinna theory]]. It was invented by [[Lars Ahlfors]]
{{defn| 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).}}
* '''[[Algebra]]''': a major part of [[pure mathematics]] centered on [[operation (mathematics)|operations]] and [[relation (mathematics)|relations]]. Beginning with [[elementary algebra]], it introduces the concept of [[variable (mathematics)|variables]] and how these can be manipulated towards [[problem solving]]; known as [[equation solving]]. Generalizations of [[operation (mathematics)|operations]] and [[relation (mathematics)|relations]] defined on [[set (mathematics)|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]].
{{term|[[Actuarial science]]}}
* '''[[Algebraic analysis]]''': motivated by systems of [[Linear differential equation|linear]] [[partial differential equation]]s, 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 [[function (mathematics)|function]]s. It was started by [[Mikio Sato]].
{{defn| The discipline that applies [[mathematics|mathematical]] and [[statistics|statistical]] methods to [[Risk assessment|assess risk]] in [[insurance]], [[finance]] and other industries and professions. More generally, actuaries apply rigorous mathematics to model matters of uncertainty.}}
* '''[[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.
{{term|[[Additive combinatorics]]}}
* '''[[Algebraic computation]]''': see ''symbolic computation''.
{{defn| The part of [[arithmetic combinatorics]] devoted to the operations of [[addition]] and [[subtraction]].}}
* '''[[Algebraic geometry]]''': a branch that combines techniques from abstract algebra with the language and problems of geometry. Fundamentally, it studies [[algebraic varieties]].
{{term|[[Additive number theory]]}}
* '''[[Algebraic graph theory]]''': a branch of [[graph theory]] in which methods are taken from algebra and employed to problems about [[Graph (discrete mathematics)|graphs]]. The methods are commonly taken from [[group theory]] and linear algebra.
{{defn| A part of [[number theory]] that studies subsets of [[integer]]s and their behaviour under addition.}}
* '''[[Algebraic K-theory]]''': an important part of [[homological algebra]] concerned with defining and applying a certain sequence of [[functor]]s from [[ring (mathematics)|rings]] to [[abelian group]]s.
{{term|[[Affine geometry]]}}
* '''[[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 integer]]s.
{{defn| A branch of [[geometry]] that deals with properties that are independent from distances and angles, such as [[collinearity|alignment]] and [[parallel (geometry)|parallelism]].}}
* '''[[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]].
{{term|[[Affine geometry of curves]]}}
* '''[[Algebraic topology]]''': a branch that uses tools from [[abstract algebra]] for [[topology]] to study [[topological space]]s.
{{defn| The study of [[curve]] properties that are invariant under [[affine transformation]]s.}}
* '''[[Algorithmic number theory]]''': also known as ''computational number theory'', it is the study of [[algorithm]]s for performing [[number theory|number theoretic]] [[computations]].
{{term|[[Affine differential geometry]]}}
* '''[[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]].
{{defn| A type of [[differential geometry]] dedicated to differential [[invariant (mathematics)|invariant]]s under [[volume]]-preserving [[affine transformation]]s.}}
* '''[[Mathematical analysis|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]].
{{term|[[Ahlfors theory]]}}
* '''[[Analytic combinatorics]]''': part of [[enumerative combinatorics]] where methods of complex analysis are applied to [[generating function]]s.
{{defn| A part of [[complex analysis]] being the geometric counterpart of [[Nevanlinna theory]]. It was invented by [[Lars Ahlfors]].}}
* '''[[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 variety|analytic varieties]]. In this respect it is essentially equivalent to [[real algebraic geometry|real]] and [[complex algebraic geometry]].
{{term|[[Algebra]]}}
* '''[[Analytic number theory]]''': part of [[number theory]] using methods of analysis (as opposed to [[algebraic number theory]])
{{defn| One of the major [[areas of mathematics]]. Roughly speaking, it is the art of manipulating and computing with [[operation (mathematics)|operation]]s acting on symbols called [[variable (mathematics)|variable]]s that represent indeterminate [[number]]s or other [[mathematical object]]s, such as [[vector (mathematics and physics)|vector]]s, [[matrix (mathematics)|matrices]], or elements of [[algebraic structure]]s.}}
* '''[[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]].
{{term|[[Algebraic analysis]]}}
* '''[[Approximation theory]]''': part of [[analysis]] that studies how well functions can be approximated by simpler ones (such as [[polynomial]]s or [[trigonometric polynomial]]s)
{{defn| motivated by systems of [[Linear differential equation|linear]] [[partial differential equation]]s, 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 [[function (mathematics)|function]]s. It was started by [[Mikio Sato]].}}
* '''[[Arakelov geometry]]''': also known as ''Arakelov theory''
{{term|[[Algebraic combinatorics]]}}
* '''[[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]].
{{defn| 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.}}
* '''[[Arithmetic]]''': to most people this refers to the branch known as [[elementary arithmetic]] dedicated to the usage of [[addition]], [[subtraction]], [[multiplication]] and [[division (mathematics)|division]]. However arithmetic also includes [[higher arithmetic]] referring to advanced results from [[number theory]].
{{term|[[Algebraic computation]]}}
* '''[[Arithmetic algebraic geometry]]''': see ''arithmetic geometry''
{{defn| An older name of [[computer algebra]].}}
* '''[[Arithmetic combinatorics]]''': the study of the estimates from [[combinatorics]] that are associated with [[arithmetic operation]]s such as addition, [[subtraction]], [[multiplication]] and [[division (mathematics)|division]].
{{term|[[Algebraic geometry]]}}
* '''[[Arithmetic dynamics]]''':Arithmetic dynamics is the study of the number-theoretic properties of [[integer point|integer]], [[rational point|rational]], {{mvar|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.
{{defn| a branch that combines techniques from abstract algebra with the language and problems of geometry. Fundamentally, it studies [[algebraic varieties]].}}
* '''[[Arithmetic geometry]]''': the study of [[scheme (mathematics)|schemes]] of finite type over the [[spectrum of a ring|spectrum]] of the [[ring of integers]]
{{term|[[Algebraic graph theory]]}}
* '''[[Arithmetic topology]]''': a combination of [[algebraic number theory]] and [[topology]] studying analogies between [[prime ideals]] and [[knot (mathematics)|knots]]
{{defn| a branch of [[graph theory]] in which methods are taken from algebra and employed to problems about [[Graph (discrete mathematics)|graph]]s. The methods are commonly taken from [[group theory]] and linear algebra.}}
* '''[[Arithmetical algebraic geometry]]''': an alternative name for ''arithmetic algebraic geometry''
{{term| [[Algebraic K-theory]]}}
* '''[[Assignment problem]]'''
{{defn| an important part of [[homological algebra]] concerned with defining and applying a certain sequence of [[functor]]s from [[ring (mathematics)|ring]]s to [[abelian group]]s.}}
* '''[[Asymptotic combinatorics]]''':It uses the internal structure of the objects to derive formulas for their [[generating function]]s and then complex analysis techniques to get asymptotics.
{{term| [[Algebraic number theory]]}}
* '''[[Asymptotic theory]]''': the study of [[asymptotic expansions]]
{{defn| The part of {{gli|number theory}} devoted to the use of algebraic methods, mainly those of {{gli|commutative algebra}}, for the study of [[number field]]s and their [[rings of integers]].}}
* '''[[Auslander–Reiten theory]]''': the study of the [[representation theory]] of [[Artinian ring]]s
{{term|[[Algebraic statistics]]}}
* '''Axiomatic geometry''': also known as ''[[synthetic geometry]]'': it is a branch of geometry that uses [[axioms]] and [[logical argument]]s to draw conclusions as opposed to [[analytic geometry|analytic]] and algebraic methods.
{{defn| 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]].}}
* '''[[Axiomatic set theory]]''': the study of systems of [[axiom]]s in a context relevant to [[set theory]] and [[mathematical logic]].
{{term|[[Algebraic topology]]}}
{{defn |a branch that uses tools from [[abstract algebra]] for [[topology]] to study [[topological space]]s.}}
{{term|[[Algorithmic number theory]]}}
{{defn| also known as ''computational number theory'', it is the study of [[algorithm]]s for performing [[number theory|number theoretic]] [[computation]]s.}}
{{term|[[Anabelian geometry]]}}
{{defn| 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]].}}
{{term|[[Mathematical analysis|Analysis]]}}
{{defn | A wide area of mathematics centered on the study of [[continuous function]]s and including such topics as [[differentiation (mathematics)|differentiation]], [[integration (mathematics)|integration]], [[limit (mathematics)|limit]]s, and [[series (mathematics)|series]].<ref>{{cite book| first=Tom M. | last=Apostol | author-link= Tom M. Apostol | title=Mathematical Analysis: A Modern Approach to Advanced Calculus| edition= 2 |publisher=Addison-Wesley }}</ref>}}
{{term |[[Analytic combinatorics]]}}
{{defn| part of [[enumerative combinatorics]] where methods of complex analysis are applied to [[generating function]]s.}}
{{term |[[Analytic geometry]]}}
{{defn|no=1|Also known as {{gli|Cartesian geometry}}, the study of {{gli|Euclidean geometry}} using [[Cartesian coordinates]].}}
{{defn|no=2|Analogue to {{gli|differential geometry}}, where [[differentiable function]]s are replaced with [[analytic function]]s. It is a subarea of both {{gli|complex analysis}} and {{gli|algebraic geometry}}.}}
{{term| [[Analytic number theory]]}}
{{defn | An area of [[number theory]] that applies methods from [[mathematical analysis]] to solve problems about [[integer]]s.<ref>{{Apostol IANT}}</ref>}}
{{term| [[L-functions|Analytic theory of L-function]]s}}
{{term |[[Applied mathematics]]}}
{{defn | 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]].}}
{{term|[[Approximation theory]]}}
{{defn| part of [[analysis]] that studies how well functions can be approximated by simpler ones (such as [[polynomial]]s or [[trigonometric polynomial]]s)}}
{{term|[[Arakelov geometry]]}}
{{defn| also known as ''Arakelov theory''}}
{{term| [[Arakelov theory]]}}
{{defn| an approach to [[Diophantine geometry]] used to study [[Diophantine equation]]s in higher dimensions (using techniques from algebraic geometry). It is named after [[Suren Arakelov]].}}
{{term| [[Arithmetic]]}}
{{defn|no=1| Also known as {{gli|elementary arithmetic}}, the methods and rules for computing with [[addition]], [[subtraction]], [[multiplication]] and [[division (mathematics)|division]] of numbers.}}
{{defn|no=2| Also known as {{gli|higher arithmetic}}, another name for [[number theory]].}}
{{term| [[Arithmetic algebraic geometry]]}}
{{defn| See {{gli|arithmetic geometry}}.}}
{{term|[[Arithmetic combinatorics]]}}
{{defn| the study of the estimates from [[combinatorics]] that are associated with [[arithmetic operation]]s such as addition, [[subtraction]], [[multiplication]] and [[division (mathematics)|division]].}}
{{term |[[Arithmetic dynamics]]}}
{{defn| Arithmetic dynamics is the study of the number-theoretic properties of [[integer point|integer]], [[rational point|rational]], {{mvar|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.}}
{{term| [[Arithmetic geometry]]}}
{{defn| The use of [[algebraic geometry]] and more specially [[scheme theory]] for solving problems of number theory.}}
{{term| [[Arithmetic topology]]}}
{{defn | a combination of [[algebraic number theory]] and [[topology]] studying analogies between [[prime ideal]]s and [[knot (mathematics)|knot]]s}}
{{term| [[Arithmetical algebraic geometry]]}}
{{defn| Another name for {{gli|arithmetic algebraic geometry}}}}
{{term| [[Asymptotic combinatorics]]}}
{{defn| It uses the internal structure of the objects to derive formulas for their [[generating function]]s and then complex analysis techniques to get asymptotics.}}
{{term| [[Asymptotic theory]]}}
{{defn| the study of [[asymptotic expansion]]s}}
{{term| [[Auslander–Reiten theory]]}}
{{defn| the study of the [[representation theory]] of [[Artinian ring]]s}}
{{term| Axiomatic geometry}}
{{defn| also known as ''[[synthetic geometry]]'': it is a branch of geometry that uses [[axiom]]s and [[logical argument]]s to draw conclusions as opposed to [[analytic geometry|analytic]] and algebraic methods.}}
{{term| [[Axiomatic set theory]]}}
{{defn| the study of systems of [[axiom]]s in a context relevant to [[set theory]] and [[mathematical logic]].}}
{{glossary end}}


{{Compact ToC|side=yes|center=yes|top=yes|num=yes|extlinks=yes|seealso=yes|refs=yes|nobreak=yes|}}
{{Compact ToC|side=yes|center=yes|top=yes|num=no|extlinks=no|seealso=yes|refs=yes|nobreak=yes|}}


==B==
==B==


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

{{Compact ToC|side=yes|center=yes|top=yes|num=no|extlinks=no|seealso=yes|refs=yes|nobreak=yes|}}


==C==
==C==


{{glossary}}
* '''[[C*-algebra|C*-algebra theory]]''': a [[complex number|complex]] [[algebra over a field|algebra]] ''A'' of [[continuous linear operator]]s on a [[complex number|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 [[adjoint of an operator|adjoint]]s of operators.
{{term| [[C*-algebra|C*-algebra theory]]}}
* '''[[Cartesian geometry]]''': see ''analytic geometry''
{{defn | a [[complex number|complex]] [[algebra over a field|algebra]] ''A'' of [[continuous linear operator]]s on a [[complex number|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 [[adjoint of an operator|adjoint]]s of operators.}}
* '''[[Calculus]]''': a branch usually associated with [[limit (mathematics)|limits]], [[function (mathematics)|functions]], [[derivative]]s, [[integrals]] and [[series (mathematics)|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.
{{term | [[Cartesian geometry]]}}
* '''[[Infinitesimal calculus|Calculus of infinitesimals]]''': also known as ''infinitesimal calculus''. It is a branch of calculus built upon the concepts of [[infinitesimal]]s.
{{defn | see ''analytic geometry''}}
* '''[[Calculus of moving surfaces]]''': an extension of the theory of [[tensor|tensor calculus]] to include deforming [[manifold]]s.
{{term | [[Calculus]]}}
* '''[[Calculus of variations]]''': the field dedicated to maximizing or minimizing [[functional (mathematics)|functionals]]. It used to be called ''functional calculus''.
{{defn | An area of mathematics connected by the [[fundamental theorem of calculus]].<ref>{{cite book |first1=Henry F. |last1=DeBaggis |first2=Kenneth S. |last2=Miller |title=Foundations of the Calculus |location=Philadelphia |publisher=Saunders |year=1966 |oclc=527896 }}</ref>}}
* '''[[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 [[germ (mathematics)|germs]] of the catastrophe geometries.
{{term | [[Infinitesimal calculus|Calculus of infinitesimal]]s}}
* '''[[Categorical logic]]''': a branch of [[category theory]] adjacent to the [[mathematical logic]]. It is based on [[type theory]] for [[intuitionistic logic]]s.
{{ghat|Also called '''infinitesimal calculus'''}}
* '''[[Category theory]]''': the study of the properties of particular mathematical concepts by formalising them as collections of objects and arrows.
{{defn | A foundation of [[calculus]], first developed in the 17th century,<ref>{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=The History of the Calculus and its Conceptual Development |url=https://archive.org/details/historyofcalculu0000boye |url-access=registration |location=New York |publisher=Dover |year=1959 |oclc=643872 }}</ref> that makes use of [[infinitesimal]] numbers.}}
* '''[[Chaos theory]]''': the study of the behaviour of [[dynamical systems]] that are highly sensitive to their initial conditions.
{{term | [[Calculus of moving surfaces]]}}
* '''[[Character theory]]''': a branch of [[group theory]] that studies the characters of [[group representation]]s or [[modular representation theory|modular representation]]s.
{{defn| an extension of the theory of [[tensor|tensor calculus]] to include deforming [[manifold]]s.}}
* '''[[Class field theory]]''': a branch of [[algebraic number theory]] that studies [[abelian extension]]s of [[number field]]s.
{{term | [[Calculus of variation]]s}}
* '''[[Differential geometry|Classical differential geometry]]''': also known as [[differential geometry|Euclidean differential geometry]]. see ''Euclidean differential geometry''.
{{defn | the field dedicated to maximizing or minimizing [[functional (mathematics)|functional]]s. It used to be called ''functional calculus''.}}
* '''[[Algebraic topology|Classical algebraic topology]]''' see ''algebraic topology''
{{term | [[Catastrophe theory]]}}
* '''[[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]].
{{defn | 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 [[germ (mathematics)|germ]]s of the catastrophe geometries.}}
* '''[[Analytic number theory|Classical analytic number theory]]'''
{{term| [[Categorical logic]]}}
* '''[[Differential calculus|Classical differential calculus]]'''
{{defn | a branch of [[category theory]] adjacent to the [[mathematical logic]]. It is based on [[type theory]] for [[intuitionistic logic]]s.}}
* '''[[Diophantine geometry|Classical Diophantine geometry]]'''
{{term| [[Category theory]]}}
* '''[[Euclidean geometry|Classical Euclidean geometry]]''': see ''Euclidean geometry''
{{defn | the study of the properties of particular mathematical concepts by formalising them as collections of objects and arrows.}}
* '''Classical geometry''': may refer to [[solid geometry]] or classical Euclidean geometry. See ''geometry''
{{term | [[Chaos theory]]}}
* '''[[Invariant theory|Classical invariant theory]]''': the form of [[invariant theory]] that deals with describing [[polynomial function]]s that are [[invariant (mathematics)|invariant]] under transformations from a given [[linear group]].
{{defn | the study of the behaviour of [[dynamical system]]s that are highly sensitive to their initial conditions.}}
* '''[[Classical mathematics]]''': the standard approach to mathematics based on [[classical logic]] and [[ZFC set theory]].
{{term | [[Character theory]]}}
* '''[[Projective geometry|Classical projective geometry]]'''
{{defn | a branch of [[group theory]] that studies the characters of [[group representation]]s or [[modular representation theory|modular representation]]s.}}
* '''[[tensor|Classical tensor calculus]]'''
{{term | [[Class field theory]]}}
* '''[[Clifford analysis]]''': the study of [[Dirac operator]]s and [[Dirac operator|Dirac type operators]] from geometry and analysis using [[clifford algebra]]s.
* '''[[Clifford theory]]''' is a branch of [[representation theory]] spawned from [[Clifford theory|Cliffords theorem]].
{{defn | a branch of [[algebraic number theory]] that studies [[abelian extension]]s of [[number field]]s.}}
{{term | [[Differential geometry|Classical differential geometry]]}}
* '''[[Cobordism theory]]'''
{{defn | also known as [[differential geometry|Euclidean differential geometry]]. see ''Euclidean differential geometry''.}}
* '''[[Coding theory]]''': the study of the properties of [[code]]s and their respective fitness for specific applications.
{{term | [[Algebraic topology|Classical algebraic topology]]}}
* '''[[Cohomology theory]]'''
{{defn | see ''algebraic topology''}}
* '''[[Combinatorial analysis]]'''
{{term | [[Classical 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.
{{defn | 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]].}}
* '''[[Combinatorial design theory]]''': a part of combinatorial mathematics that deals with the existence and construction of [[Set system|systems of finite sets]] whose intersections have certain properties.
{{term | [[Analytic number theory|Classical analytic number theory]]}}
* '''[[Combinatorial game theory]]'''
{{term | [[Differential calculus|Classical differential calculus]]}}
* '''[[Combinatorial geometry]]''': see ''discrete geometry''
{{term | [[Diophantine geometry|Classical Diophantine geometry]]}}
* '''[[Combinatorial group theory]]''': the theory of [[free group]]s and the [[presentation of a group]]. It is closely related to [[geometric group theory]] and is applied in [[geometric topology]].
{{term | [[Euclidean geometry|Classical Euclidean geometry]]}}
* '''[[Combinatorial mathematics]]''': an area primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of [[finite set|finite]] [[Mathematical structure|structures]].
{{ defn | see ''Euclidean geometry''}}
* '''[[Combinatorial number theory]]'''
{{ term | Classical geometry}}
* '''[[Combinatorial optimization]]'''
{{defn | may refer to [[solid geometry]] or classical Euclidean geometry. See ''geometry''}}
* '''[[Combinatorial set theory]]''': also known as [[Infinitary combinatorics]]. see ''infinitary combinatorics''
{{term | [[Invariant theory|Classical invariant theory]]}}
* '''[[Combinatorial theory]]'''
{{defn | the form of [[invariant theory]] that deals with describing [[polynomial function]]s that are [[invariant (mathematics)|invariant]] under transformations from a given [[linear group]].}}
* '''[[Combinatorial topology]]''': an old name for algebraic topology, when [[topological invariant]]s of spaces were regarded as derived from combinatorial decompositions.
{{term | [[Classical mathematics]]}}
* '''[[Combinatorics]]''': a branch of [[discrete mathematics]] concerned with [[countable set|countable]] [[mathematical structure|structures]]. Branches of it include [[enumerative combinatorics]], [[combinatorial design theory]], [[matroid theory]], [[extremal combinatorics]] and [[algebraic combinatorics]], as well as many more.
{{defn | the standard approach to mathematics based on [[classical logic]] and [[ZFC set theory]].}}
* '''[[Commutative algebra]]''': a branch of abstract algebra studying [[commutative ring]]s.
{{term | [[Projective geometry|Classical projective geometry]]}}
* '''[[Complex algebra]]'''
{{term | [[tensor|Classical tensor calculus]]}}
* '''[[Complex algebraic geometry]]''': the mainstream of algebraic geometry devoted to the study of the [[complex number|complex]] points of [[algebraic varieties]].
{{term | [[Clifford algebra]]}}
* {{anchor|complex-analysis}}'''[[Complex analysis]]''': a part of [[analysis]] that deals with functions of a [[complex numbers|complex]] variable.
{{term | [[Clifford analysis]]}}
* '''[[Complex analytic dynamics]]''': a subdivision of [[complex dynamics]] being the study of the [[dynamic system]]s defined by [[analytic function]]s.
{{defn|the study of [[Dirac operator]]s and [[Dirac operator|Dirac type operator]]s from geometry and analysis using [[clifford algebra]]s.}}
* '''[[Complex analytic geometry]]''': the application of complex numbers to [[plane geometry]].
{{term | [[Clifford theory]]}}
* '''[[Differential geometry|Complex differential geometry]]''': a branch of [[differential geometry]] that studies [[complex manifolds]].
{{defn | is a branch of [[representation theory]] spawned from [[Clifford theory|Cliffords theorem]].}}
* '''[[Complex dynamics]]''': the study of [[dynamical system]]s defined by [[iterated function]]s on complex [[number space]]s.
{{term | [[Cobordism theory]]}}
* '''[[Complex geometry]]''': the study of [[complex manifolds]] and functions of [[complex numbers|complex]] variables. It includes [[complex algebraic geometry]] and [[complex analytic geometry]].
{{term | [[Coding theory]]}}
* '''[[Complexity theory (disambiguation)|Complexity theory]]''': the study of [[complex system]]s with the inclusion of the theory of [[complex systems]].
{{defn | the study of the properties of [[code]]s and their respective fitness for specific applications.}}
* '''[[Computable analysis]]''': the study of which parts of [[real analysis]] and [[functional analysis]] can be carried out in a [[computability theory|computable]] manner. It is closely related to [[constructive analysis]].
{{term | [[Cohomology theory]]}}
* '''[[Computable model theory]]''': a branch of [[model theory]] dealing with the relevant questions [[computability]].
{{term| [[Combinatorial analysis]]}}
* '''[[Computability theory]]''': a branch of [[mathematical logic]] originating in the 1930s with the study of [[computable function]]s and [[Turing degree]]s, but now includes the study of generalized computability and definability. It overlaps with [[proof theory]] and [[effective descriptive set theory]].
{{term | [[Combinatorial commutative algebra]]}}
* '''[[Computational algebraic geometry]]'''
{{defn | 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.}}
* '''[[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 [[complexity class|classes]] to each other.
{{term |[[Combinatorial design theory]]}}
* '''[[Computational geometry]]''': a branch of [[computer science]] devoted to the study of algorithms which can be stated in terms of [[geometry]].
{{defn | a part of combinatorial mathematics that deals with the existence and construction of [[Set system|systems of finite set]]s whose intersections have certain properties.}}
* '''[[Computational group theory]]''': the study of [[group (mathematics)|groups]] by means of computers.
{{term | [[Combinatorial game theory]]}}
* '''[[Computational mathematics]]''': the mathematical research in areas of [[science]] where [[computation|computing]] plays an essential role.
{{term | [[Combinatorial geometry]]}}
* '''[[Computational number theory]]''': also known as ''algorithmic number theory'', it is the study of [[algorithm]]s for performing [[number theory|number theoretic]] [[computations]].
{{defn | see ''discrete geometry''}}
* '''[[Computational statistics]]'''
{{term |[[Combinatorial group theory]]}}
* '''[[Computational synthetic geometry]]'''
{{defn | the theory of [[free group]]s and the [[presentation of a group]]. It is closely related to [[geometric group theory]] and is applied in [[geometric topology]].}}
* '''[[Computational topology]]'''
{{term |[[Combinatorial mathematics]]}}
* '''[[Computer algebra]]''': see ''symbolic computation''
{{defn| an area primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of [[finite set|finite]] [[Mathematical structure|structure]]s.}}
* '''[[Conformal geometry]]''': the study of [[conformal map|conformal]] transformations on a space.
{{term |[[Combinatorial number theory]]}}
* '''[[Constructive analysis]]''': mathematical analysis done according to the principles of [[constructive mathematics]]. This differs from ''classical analysis''.
{{term |[[Combinatorial optimization]]}}
* '''[[Constructive function theory]]''': a branch of analysis that is closely related to [[approximation theory]], studying the connection between the [[smooth function|smoothness of a function]] and its [[approximation theory|degree of approximation]]
{{term |[[Combinatorial set theory]]}}
* '''[[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]].
{{defn | also known as [[Infinitary combinatorics]]. see ''infinitary combinatorics''}}
* '''[[Constructive quantum field theory]]''': a branch of [[mathematical physics]] that is devoted to showing that [[quantum mechanics|quantum theory]] is mathematically compatible with [[special relativity]].
{{term |[[Combinatorial theory]]}}
* '''[[Constructive set theory]]''': an approach to [[constructivism (mathematics)|mathematical constructivism]] following the program of [[axiomatic set theory]], using the usual [[first-order logic|first-order]] language of classical set theory.
{{term |[[Combinatorial topology]]}}
* '''[[Contact geometry]]''': a branch of [[differential geometry]] and [[differential topology|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]].
{{defn | an old name for algebraic topology, when [[topological invariant]]s of spaces were regarded as derived from combinatorial decompositions.}}
* '''[[Convex analysis]]''': the study of properties of [[convex function]]s and [[convex set]]s.
{{term |[[Combinatorics]]}}
* '''[[Convex geometry]]''': part of geometry devoted to the study of [[convex set]]s.
{{defn | a branch of [[discrete mathematics]] concerned with [[countable set|countable]] [[mathematical structure|structure]]s. Branches of it include [[enumerative combinatorics]], [[combinatorial design theory]], [[matroid theory]], [[extremal combinatorics]] and [[algebraic combinatorics]], as well as many more.}}
* '''[[Coordinate geometry]]''': see ''analytic geometry''
{{term |[[Commutative algebra]]}}
* '''[[CR geometry]]''': a branch of [[differential geometry]], being the study of [[CR manifold]]s.
{{defn | a branch of abstract algebra studying [[commutative ring]]s.}}
* '''[[Cryptography]]'''
{{term |[[Complex algebraic geometry]]}}
{{defn | the mainstream of algebraic geometry devoted to the study of the [[complex number|complex]] points of [[algebraic varieties]].}}
{{term |[[Complex analysis]]}}
{{defn | a part of [[analysis]] that deals with functions of a [[complex numbers|complex]] variable.}}
{{term |[[Complex analytic dynamics]]}}
{{defn | a subdivision of [[complex dynamics]] being the study of the [[dynamic system]]s defined by [[analytic function]]s.}}
{{term |[[Complex analytic geometry]]}}
{{defn | the application of complex numbers to [[plane geometry]].}}
{{term |[[Differential geometry|Complex differential geometry]]}}
{{defn | a branch of [[differential geometry]] that studies [[complex manifold]]s.}}
{{term |[[Complex dynamics]]}}
{{defn | the study of [[dynamical system]]s defined by [[iterated function]]s on complex [[number space]]s.}}
{{term |[[Complex geometry]]}}
{{defn | the study of [[complex manifold]]s and functions of [[complex numbers|complex]] variables. It includes [[complex algebraic geometry]] and [[complex analytic geometry]].}}
{{term |[[Complexity theory (disambiguation)|Complexity theory]]}}
{{defn | the study of [[complex system]]s with the inclusion of the theory of [[complex system]]s.}}
{{term |[[Computable analysis]]}}
{{defn | the study of which parts of [[real analysis]] and [[functional analysis]] can be carried out in a [[computability theory|computable]] manner. It is closely related to [[constructive analysis]].}}
{{term |[[Computable model theory]]}}
{{defn | a branch of [[model theory]] dealing with the relevant questions [[computability]].}}
{{term |[[Computability theory]]}}
{{defn | a branch of [[mathematical logic]] originating in the 1930s with the study of [[computable function]]s and [[Turing degree]]s, but now includes the study of generalized computability and definability. It overlaps with [[proof theory]] and [[effective descriptive set theory]].}}
{{term |[[Computational algebraic geometry]]}}
{{term |[[Computational complexity theory]]}}
{{defn | a branch of mathematics and [[theoretical computer science]] that focuses on classifying [[computational problem]]s according to their inherent difficulty, and relating those [[complexity class|classes]] to each other.}}
{{term |[[Computational geometry]]}}
{{defn | a branch of [[computer science]] devoted to the study of algorithms which can be stated in terms of [[geometry]].}}
{{term |[[Computational group theory]]}}
{{defn | the study of [[group (mathematics)|group]]s by means of computers.}}
{{term |[[Computational mathematics]]}}
{{defn | the mathematical research in areas of [[science]] where [[computation|computing]] plays an essential role.}}
{{term |[[Computational number theory]]}}
{{defn | also known as ''algorithmic number theory'', it is the study of [[algorithm]]s for performing [[number theory|number theoretic]] [[computation]]s.}}
{{term |[[Computational statistics]]}}
{{term |[[Computational synthetic geometry]]}}
{{term |[[Computational topology]]}}
{{term |[[Computer algebra]]}}
{{defn | see ''symbolic computation''}}
{{term |[[Conformal geometry]]}}
{{defn | the study of [[conformal map|conformal]] transformations on a space.}}
{{term |[[Constructive analysis]]}}
{{defn | mathematical analysis done according to the principles of [[constructive mathematics]]. This differs from ''classical analysis''.}}
{{term |[[Constructive function theory]]}}
{{defn | a branch of analysis that is closely related to [[approximation theory]], studying the connection between the [[smooth function|smoothness of a function]] and its [[approximation theory|degree of approximation]]}}
{{term |[[Constructive mathematics]]}}
{{defn | 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]].}}
{{term |[[Constructive quantum field theory]]}}
{{defn | a branch of [[mathematical physics]] that is devoted to showing that [[quantum mechanics|quantum theory]] is mathematically compatible with [[special relativity]].}}
{{term |[[Constructive set theory]]}}
{{defn | an approach to [[constructivism (mathematics)|mathematical constructivism]] following the program of [[axiomatic set theory]], using the usual [[first-order logic|first-order]] language of classical set theory.}}
{{term |[[Contact geometry]]}}
{{defn | a branch of [[differential geometry]] and [[differential topology|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]].}}
{{term |[[Convex analysis]]}}
{{defn | the study of properties of [[convex function]]s and [[convex set]]s.}}
{{term |[[Convex geometry]]}}
{{defn | part of geometry devoted to the study of [[convex set]]s.}}
{{term |[[Coordinate geometry]]}}
{{defn | see ''analytic geometry''}}
{{term |[[CR geometry]]}}
{{defn | a branch of [[differential geometry]], being the study of [[CR manifold]]s.}}
{{term |[[Cryptography]]}}
{{glossary end}}


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==D==
==D==
{{glossary}}
* '''[[Decision analysis]]'''
* '''[[Decision theory]]'''
{{term |[[Decision analysis]]}}
{{term |[[Decision theory]]}}
* '''[[Noncommutative algebraic geometry|Derived noncommutative algebraic geometry]]'''
{{term |[[Noncommutative algebraic geometry|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 space]]s.
{{term |[[Descriptive set theory]]}}
* '''[[Differential algebraic geometry]]''': the adaption of methods and concepts from algebraic geometry to systems of [[algebraic differential equation]]s.
{{defn |a part of [[mathematical logic]], more specifically a part of [[set theory]] dedicated to the study of [[Polish space]]s.}}
* '''[[Differential calculus]]''': a subfield of calculus concerned with [[derivative]]s or the rates that quantities change. It is one of two traditional divisions of calculus, the other being [[integral calculus]].
{{term |[[Differential algebraic geometry]]}}
* '''[[Differential Galois theory]]''': the study of the [[Galois group]]s of [[differential field]]s.
{{defn |the adaption of methods and concepts from algebraic geometry to systems of [[algebraic differential equation]]s.}}
* '''[[Differential geometry]]''': a form of geometry that uses techniques from [[integral calculus|integral]] and [[differential calculus]] as well as [[linear algebra|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.
{{term |[[Differential calculus]]}}
* '''[[Differential geometry of curves]]''': the study of [[curve|smooth curves]] in [[Euclidean space]] by using techniques from [[differential geometry]]
{{defn |A branch of [[calculus]] that's contrasted to [[integral calculus]],<ref>{{Citation| first = R.| last = Courant| translator-first = E. J.| translator-last = McShane| title = Differential and Integral Calculus| volume = I| year= 1937| edition = 2nd| publisher = Interscience| place = New York| isbn = 978-4-87187-838-8}}</ref> and concerned with [[derivative]]s.<ref>{{Cite web|title=Definition of DIFFERENTIAL CALCULUS|url=https://merriam-webster.com/dictionary/differential+calculus|website=www.merriam-webster.com|language=en|access-date=2020-05-09}}</ref>}}
* '''[[Differential geometry of surfaces]]''': the study of [[smooth manifold|smooth]] [[Surface (topology)|surface]]s with various additional structures using the techniques of [[differential geometry]].
{{term |[[Differential Galois theory]]}}
* '''[[Differential topology]]''': a branch of [[topology]] that deals with [[differentiable function]]s on [[differentiable manifold]]s.
{{defn |the study of the [[Galois group]]s of [[differential field]]s.}}
* '''[[Diffiety|Diffiety theory]]'''
{{term |[[Differential geometry]]}}
* '''[[Diophantine geometry]]''': in general the study of algebraic varieties over [[field (mathematics)|fields]] that are finitely generated over their [[prime field]]s.
{{defn |a form of geometry that uses techniques from [[integral calculus|integral]] and [[differential calculus]] as well as [[linear algebra|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 manifold]]s. It is closely related to differential topology.}}
* '''[[Discrepancy theory]]'''
* '''[[Discrete differential geometry]]'''
{{term |[[Differential geometry of curves]]}}
{{defn |the study of [[curve|smooth curve]]s in [[Euclidean space]] by using techniques from [[differential geometry]]}}
* '''[[Discrete exterior calculus]]'''
{{term |[[Differential geometry of surfaces]]}}
* '''[[Discrete geometry]]''': a branch of [[geometry]] that studies [[Combinatorics|combinatorial]] properties and constructive methods of [[discrete mathematics|discrete]] geometric objects.
{{defn |the study of [[smooth manifold|smooth]] [[Surface (topology)|surface]]s with various additional structures using the techniques of [[differential geometry]].}}
* '''[[Discrete mathematics]]''': the study of [[mathematical structures]] that are fundamentally [[discrete space|discrete]] rather than [[continuous function|continuous]].
{{term |[[Differential topology]]}}
* '''[[Discrete Morse theory]]''': a [[combinatorial]] adaption of [[Morse theory]].
{{defn |a branch of [[topology]] that deals with [[differentiable function]]s on [[differentiable manifold]]s.}}
* '''[[Distance geometry]]'''
{{term |[[Diffiety|Diffiety theory]]}}
* '''[[Domain theory]]''' a branch that studies special kinds of [[partially ordered set]]s (posets) commonly called domains.
{{term |[[Diophantine geometry]]}}
* '''[[Donaldson theory]]''': the study of smooth [[4-manifold]]s using [[gauge theory]].
{{defn |in general the study of algebraic varieties over [[field (mathematics)|field]]s that are finitely generated over their [[prime field]]s.}}
* '''[[Dynamical systems theory]]''': an area used to describe the behavior of the [[complex systems|complex]] [[dynamical system]]s, usually by employing [[differential equations]] or [[difference equations]].
{{term |[[Discrepancy theory]]}}
{{term |[[Discrete differential geometry]]}}
{{term |[[Discrete exterior calculus]]}}
{{term |[[Discrete geometry]]}}
{{defn |a branch of [[geometry]] that studies [[Combinatorics|combinatorial]] properties and constructive methods of [[discrete mathematics|discrete]] geometric objects.}}
{{term |[[Discrete mathematics]]}}
{{defn |the study of [[mathematical structure]]s that are fundamentally [[discrete space|discrete]] rather than [[continuous function|continuous]].}}
{{term |[[Discrete Morse theory]]}}
{{defn |a [[combinatorial]] adaption of [[Morse theory]].}}
{{term |[[Distance geometry]]}}
{{term |[[Domain theory]]}}
{{defn |a branch that studies special kinds of [[partially ordered set]]s (posets) commonly called domains.}}
{{term |[[Donaldson theory]]}}
{{defn |the study of smooth [[4-manifold]]s using [[gauge theory]].}}
{{term |[[Dyadics|Dyadic algebra]]}}
{{term |[[Dynamical systems theory]]}}
{{defn |an area used to describe the behavior of the [[complex systems|complex]] [[dynamical system]]s, usually by employing [[differential equation]]s or [[difference equation]]s.}}
{{glossary end}}

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==E==
==E==


{{glossary}}
* '''[[Econometrics]]''': the application of mathematical and [[statistics|statistical]] methods to [[Economics|economic]] [[data]].
{{term |[[Econometrics]]}}
* '''[[Effective descriptive set theory]]''': a branch of [[descriptive set theory]] dealing with [[set (mathematics)|set]] of [[real number]]s that have [[lightface]] definitions. It uses aspects of [[computability theory]].
{{defn |the application of mathematical and [[statistics|statistical]] methods to [[Economics|economic]] [[data]].}}
* '''[[Elementary algebra]]''': a fundamental form of [[algebra]] extending on [[elementary arithmetic]] to include the concept of [[variable (mathematics)|variable]]s.
{{term |[[Effective descriptive set theory]]}}
* '''[[Elementary arithmetic]]''': the simplified portion of arithmetic considered necessary for [[primary education]]. It includes the usage addition, [[subtraction]], [[multiplication]] and [[division (mathematics)|division]] of the [[natural number]]s. It also includes the concept of [[fraction (mathematics)|fractions]] and [[negative number]]s.
{{defn |a branch of [[descriptive set theory]] dealing with [[set (mathematics)|set]] of [[real number]]s that have [[lightface]] definitions. It uses aspects of [[computability theory]].}}
* '''[[Elementary mathematics]]''': parts of mathematics frequently taught at the [[primary school|primary]] and [[secondary school]] levels. This includes [[elementary arithmetic]], geometry, [[probability]] and [[statistics]], [[elementary algebra]] and [[trigonometry]]. (calculus is not usually considered a part)
{{term |[[Elementary algebra]]}}
* '''[[Elementary group theory]]''': the study of the basics of [[group theory]]
{{defn |a fundamental form of [[algebra]] extending on [[elementary arithmetic]] to include the concept of [[variable (mathematics)|variable]]s.}}
* '''[[Elimination theory]]''': the classical name for algorithmic approaches to eliminating between [[polynomial]]s of several variables. It is a part of [[commutative algebra]] and algebraic geometry.
{{term |[[Elementary arithmetic]]}}
* '''[[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]].
{{defn |the simplified portion of arithmetic considered necessary for [[primary education]]. It includes the usage addition, [[subtraction]], [[multiplication]] and [[division (mathematics)|division]] of the [[natural number]]s. It also includes the concept of [[fraction (mathematics)|fraction]]s and [[negative number]]s.}}
* '''[[Enumerative combinatorics]]''': an area of combinatorics that deals with the number of ways that certain patterns can be formed.
{{term |[[Elementary mathematics]]}}
* '''[[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]].
{{defn |parts of mathematics frequently taught at the [[primary school|primary]] and [[secondary school]] levels. This includes [[elementary arithmetic]], geometry, [[probability]] and [[statistics]], [[elementary algebra]] and [[trigonometry]]. (calculus is not usually considered a part)}}
* '''[[Epidemiology]]'''
{{term |[[Elementary group theory]]}}
* '''[[Noncommutative algebraic geometry|Equivariant noncommutative algebraic geometry]]'''
{{defn |the study of the basics of [[group theory]]}}
* '''[[Ergodic Ramsey theory]]''': a branch where problems are motivated by [[additive combinatorics]] and solved using [[ergodic theory]].
{{term |[[Elimination theory]]}}
* '''[[Ergodic theory]]''': the study of [[dynamical systems]] with an [[invariant measure]], and related problems.
{{defn |the classical name for algorithmic approaches to eliminating between [[polynomial]]s of several variables. It is a part of [[commutative algebra]] and algebraic geometry.}}
* '''[[Euclidean geometry]]'''
{{term |[[Elliptic geometry]]}}
* '''[[Differential geometry|Euclidean differential geometry]]''': also known as ''classical differential geometry''. See ''differential geometry''.
{{defn |a type of [[non-Euclidean geometry]] (it violates [[Euclid]]'s [[parallel postulate]]) and is based on [[spherical geometry]]. It is constructed in [[elliptic space]].}}
* '''[[Euler calculus]]''': a methodology from applied [[algebraic topology]] and [[integral geometry]] that integrates [[constructible function]]s and more recently [[definable function]]s by integrating with respect to the [[Euler characteristic]] as a finitely-additive [[measure theory|measure]].
{{term |[[Enumerative combinatorics]]}}
* '''[[Experimental mathematics]]''': an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns.
{{defn |an area of combinatorics that deals with the number of ways that certain patterns can be formed.}}
* '''[[Extraordinary cohomology theory]]'''
{{term |[[Enumerative geometry]]}}
* '''[[Extremal combinatorics]]''': a branch of combinatorics, it is the study of the possible sizes of a collection of finite objects given certain restrictions.
{{defn |a branch of algebraic geometry concerned with counting the number of solutions to geometric questions. This is usually done by means of [[intersection theory]].}}
* '''[[Extremal graph theory]]''': a branch of mathematics that studies how global properties of a graph influence local substructure.
{{term |[[Epidemiology]]}}
{{term |[[Noncommutative algebraic geometry|Equivariant noncommutative algebraic geometry]]}}
{{term |[[Ergodic Ramsey theory]]}}
{{defn |a branch where problems are motivated by [[additive combinatorics]] and solved using [[ergodic theory]].}}
{{term |[[Ergodic theory]]}}
{{defn |the study of [[dynamical system]]s with an [[invariant measure]], and related problems.}}
{{term |[[Euclidean geometry]]}}
{{defn | An area of [[geometry]] based on the [[axiom system]] and [[synthetic geometry|synthetic method]]s of the ancient Greek mathematician [[Euclid]].<ref>{{cite book|first=Howard|last= Eves|title=A Survey of Geometry (Volume One)|publisher=Allyn and Bacon|year=1963}}</ref>}}
{{term |[[Differential geometry|Euclidean differential geometry]]}}
{{defn |also known as ''classical differential geometry''. See ''differential geometry''.}}
{{term |[[Euler calculus]]}}
{{defn |a methodology from applied [[algebraic topology]] and [[integral geometry]] that integrates [[constructible function]]s and more recently [[definable function]]s by integrating with respect to the [[Euler characteristic]] as a finitely-additive [[measure theory|measure]].}}
{{term |[[Experimental mathematics]]}}
{{defn |an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns.}}
{{term | Exterior algebra}}
{{term | Exterior calculus}}
{{term |[[Extraordinary cohomology theory]]}}
{{term |[[Extremal combinatorics]]}}
{{defn |a branch of combinatorics, it is the study of the possible sizes of a collection of finite objects given certain restrictions.}}
{{term |[[Extremal graph theory]]}}
{{defn |a branch of mathematics that studies how global properties of a graph influence local substructure.}}
{{glossary end}}


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==F==
==F==


{{glossary}}
* '''[[Field theory (mathematics)|Field theory]]''': branch of abstract algebra studying [[field (mathematics)|fields]].
{{term |[[Field theory (mathematics)|Field theory]]}}
* '''[[Finite geometry]]'''
{{defn | The branch of [[algebra]] dedicated to [[field (mathematics)|field]]s, a type of [[algebraic structure]].<ref>{{Citation|title=Introduction to Field Theory| last=Adamson|first=I. T.|isbn=978-0-486-46266-0|year=2007|publisher=Dover Publications}}</ref>}}
* '''[[Finite model theory]]''': a restriction of [[model theory]] to [[interpretation (logic)|interpretations]] on finite [[Structure (mathematical logic)|structures]], which have a finite universe.
{{term |[[Finite geometry]]}}
* '''[[Finsler geometry]]''': a branch of [[differential geometry]] whose main object of study is the [[Finsler manifold]] (a generalisation of a [[Riemannian manifold]]).
{{term |[[Finite model theory]]}}
* '''[[Arithmetic|First order arithmetic]]'''
{{defn |a restriction of [[model theory]] to [[interpretation (logic)|interpretation]]s on finite [[Structure (mathematical logic)|structure]]s, which have a finite universe.}}
* '''[[Fourier analysis]]''': the study of the way general [[function (mathematics)|functions]] may be represented or approximated by sums of simpler [[trigonometric functions]].
* '''[[Fractal geometry]]''':
{{term |[[Finsler geometry]]}}
* '''[[Fractional calculus]]''': a branch of analysis that studies the possibility of taking [[real number|real]] or complex powers of the [[differentiation operator]].
{{defn |a branch of [[differential geometry]] whose main object of study is [[Finsler manifold]]s, a generalisation of a [[Riemannian manifold]]s.}}
{{term |[[Arithmetic|First order arithmetic]]}}
* '''[[Fractional dynamics]]''': investigates the behaviour of objects and systems that are described by [[Derivative|differentiation]] and [[integral|integration]] of [[fraction (mathematics)|fractional]] orders using methods of [[fractional calculus]].
{{term |[[Fourier analysis]]}}
* '''[[Fredholm theory]]''': part of [[spectral theory]] studying [[integral equation]]s.
* '''Function theory''': part of analysis devoted to properties of [[function (mathematics)|functions]], especially functions of a complex variable (see ''[[#complex-analysis|complex analysis]]'').
{{defn |the study of the way general [[function (mathematics)|function]]s may be represented or approximated by sums of [[trigonometric function]]s.}}
{{term |[[Fractal geometry]]}}
* '''[[Functional analysis]]''': a branch of [[mathematical analysis]], the core of which is formed by the study of [[vector space]]s endowed with some kind of limit-related structure and the [[linear transformation|linear function]]s defined on these spaces and respecting these structures in a suitable sense.
{{term |[[Fractional calculus]]}}
* '''[[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]]
{{defn |a branch of analysis that studies the possibility of taking [[real number|real]] or complex powers of the [[differentiation operator]].}}
* '''[[Fuzzy mathematics]]''': a branch of mathematics based on [[fuzzy set theory]] and [[fuzzy logic]].
{{term |[[Fractional dynamics]]}}
* '''[[Fuzzy measure theory]]'''
{{defn |investigates the behaviour of objects and systems that are described by [[Derivative|differentiation]] and [[integral|integration]] of [[fraction (mathematics)|fractional]] orders using methods of [[fractional calculus]].}}
* '''[[Fuzzy set theory]]''': a form of [[set theory]] that studies [[fuzzy set]]s, that is [[set (mathematics)|sets]] that have degrees of membership.
{{term |[[Fredholm theory]]}}

{{defn |part of [[spectral theory]] studying [[integral equation]]s.}}
{{term |[[Function theory (disambiguation)|Function theory]]}}
{{defn |an ambiguous term that generally refers to [[mathematical analysis]].}}
{{term |[[Functional analysis]]}}
{{defn |a branch of [[mathematical analysis]], the core of which is formed by the study of [[function space]]s, which are some sort of [[topological vector space]]s.}}
{{term |[[Functional calculus]]}}
{{defn |historically the term was used synonymously with [[calculus of variations]], but now refers to a branch of [[functional analysis]] connected with [[spectral theory]]}}
{{term |[[Fuzzy mathematics]]}}
{{defn |a branch of mathematics based on [[fuzzy set theory]] and [[fuzzy logic]].}}
{{term |[[Fuzzy measure theory]]}}
{{term |[[Fuzzy set theory]]}}
{{defn |a form of [[set theory]] that studies [[fuzzy set]]s, that is [[set (mathematics)|set]]s that have degrees of membership.}}
{{glossary end}}


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==G==
==G==


{{glossary}}
* '''[[Galois cohomology]]''': an application of [[homological algebra]], it is the study of [[group cohomology]] of [[Galois module]]s.
{{term |[[Galois cohomology]]}}
* '''[[Galois theory]]''': named after [[Évariste Galois]], it is a branch of abstract algebra providing a connection between [[field theory (mathematics)|field theory]] and [[group theory]].
* '''[[Galois geometry]]''': a branch of [[finite geometry]] concerned with algebraic and [[analytic geometry|analytic]] geometry over a [[Galois field]].
{{defn |an application of [[homological algebra]], it is the study of [[group cohomology]] of [[Galois module]]s.}}
{{term |[[Galois theory]]}}
* '''[[Game theory]]''': the study of [[mathematical model]]s of strategic interaction among rational decision-makers.
{{defn |named after [[Évariste Galois]], it is a branch of abstract algebra providing a connection between [[field theory (mathematics)|field theory]] and [[group theory]].}}
* '''[[Gauge theory]]'''
{{term |[[Galois geometry]]}}
* '''[[General topology]]''': also known as ''point-set topology'', it is a branch of [[topology]] studying the properties of [[topological space]]s and structures defined on them. It differs from other branches of [[topology]] as the [[topological space]]s do not have to be similar to manifolds.
{{defn |a branch of [[finite geometry]] concerned with algebraic and [[analytic geometry|analytic]] geometry over a [[Galois field]].}}
* '''[[Generalized trigonometry]]''': developments of [[trigonometry|trigonometric]] methods from the application to [[real number]]s of Euclidean geometry to any geometry or [[space]]. This includes [[spherical trigonometry]], [[hyperbolic geometry|hyperbolic trigonometry]], [[gyrotrigonometry]], and [[rational trigonometry|universal hyperbolic trigonometry]].
{{term |[[Game theory]]}}
* '''[[Geometric algebra]]''': an alternative approach to classical, [[computational geometry|computational]] and [[Special relativity|relativistic geometry]]. It shows a natural correspondence between geometric entities and elements of algebra.
{{defn |the study of [[mathematical model]]s of strategic interaction among rational decision-makers.}}
* '''[[Geometric analysis]]''': a discipline that uses methods from [[differential geometry]] to study [[partial differential equations]] as well as the applications to geometry.
{{term |[[Gauge theory]]}}
* '''[[Geometric calculus]]''': extends the [[geometric algebra]] to include [[Derivative|differentiation]] and [[Integral|integration]].
{{term |[[General topology]]}}
* '''[[Geometric combinatorics]]''': a branch of [[combinatorics]]. It includes a number of subareas such as [[polyhedral combinatorics]] (the study of [[Face (geometry)|faces]] of [[convex polyhedron|convex polyhedra]]), [[convex geometry]] (the study of [[convex set]]s, in particular combinatorics of their intersections), and [[discrete geometry]], which in turn has many applications to [[computational geometry]].
{{defn |also known as ''point-set topology'', it is a branch of [[topology]] studying the properties of [[topological space]]s and structures defined on them. It differs from other branches of [[topology]] as the [[topological space]]s do not have to be similar to manifolds.}}
* '''[[Geometric function theory]]''': the study of geometric properties of [[analytic function]]s.
{{term |[[Generalized trigonometry]]}}
*
{{defn |developments of [[trigonometry|trigonometric]] methods from the application to [[real number]]s of Euclidean geometry to any geometry or [[space]]. This includes [[spherical trigonometry]], [[hyperbolic geometry|hyperbolic trigonometry]], [[gyrotrigonometry]], and [[rational trigonometry|universal hyperbolic trigonometry]].}}
* '''[[Geometric invariant theory]]''': a method for constructing quotients by [[Group action (mathematics)|group action]]s in [[algebraic geometry]], used to construct [[moduli space]]s.
{{term |[[Geometric algebra]]}}
* '''[[Geometric graph theory]]''': a large and amorphous subfield of [[graph theory]], concerned with graphs defined by geometric means.
{{defn |an alternative approach to classical, [[computational geometry|computational]] and [[Special relativity|relativistic geometry]]. It shows a natural correspondence between geometric entities and elements of algebra.}}
* '''[[Geometric group theory]]''': the study of [[finitely generated group]]s via exploring the connections between algebraic properties of such groups and [[topology|topological]] and [[geometry|geometric]] properties of spaces on which these groups [[Group action (mathematics)|act]] (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).
{{term |[[Geometric analysis]]}}
* '''[[Geometric measure theory]]''': the study of [[geometry|geometric]] properties of [[Set (mathematics)|sets]] (typically in [[Euclidean space]]) through [[measure (mathematics)|measure theory]].
{{defn |a discipline that uses methods from [[differential geometry]] to study [[partial differential equation]]s as well as the applications to geometry.}}
* '''[[Geometric topology]]''': a branch of [[topology]] studying manifolds and mappings between them; in particular the [[embedding]] of one manifold into another.
{{term |[[Geometric calculus]]}}
* '''[[Geometry]]''': a branch of mathematics concerned with [[shape]] and the properties of [[space (mathematics)|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 system|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]].
{{defn |extends the [[geometric algebra]] to include [[Derivative|differentiation]] and [[Integral|integration]].}}
* '''[[Geometry of numbers]]''': initiated by [[Hermann Minkowski]], it is a branch of [[number theory]] studying [[convex bodies]] and [[integer]] [[Euclidean vector|vector]]s.
{{term |[[Geometric combinatorics]]}}
* '''[[Global analysis]]''': the study of [[differential equation]]s on manifolds and the relationship between [[differential equation]]s and [[topology]].
{{defn |a branch of [[combinatorics]]. It includes a number of subareas such as [[polyhedral combinatorics]] (the study of [[Face (geometry)|face]]s of [[convex polyhedron|convex polyhedra]]), [[convex geometry]] (the study of [[convex set]]s, in particular combinatorics of their intersections), and [[discrete geometry]], which in turn has many applications to [[computational geometry]].}}
* '''[[Arithmetic dynamics|Global arithmetic dynamics]]'''
{{term |[[Geometric function theory]]}}
* '''[[Graph theory]]''': a branch of [[discrete mathematics]] devoted to the study of [[Graph (discrete mathematics)|graphs]]. It has many applications in [[physical science|physical]], [[biological science|biological]] and [[social science|social]] systems.
{{defn |the study of geometric properties of [[analytic function]]s.}}
* '''[[Character theory|Group-character theory]]''': the part of character theory dedicated to the study of characters of [[group representation]]s.
* '''[[Group representation theory]]'''
{{term |[[Geometric invariant theory]]}}
{{defn |a method for constructing quotients by [[Group action (mathematics)|group action]]s in [[algebraic geometry]], used to construct [[moduli space]]s.}}
* '''[[Group theory]]''': the study of [[algebraic structure]]s known as [[group (mathematics)|groups]].
{{term |[[Geometric graph theory]]}}
* '''[[Gyrotrigonometry]]''': a form of [[trigonometry]] used in [[gyrovector space]] for [[hyperbolic geometry]]. (An analogy of the [[vector space]] in Euclidean geometry.)
{{defn |a large and amorphous subfield of [[graph theory]], concerned with graphs defined by geometric means.}}
{{term |[[Geometric group theory]]}}
{{defn |the study of [[finitely generated group]]s via exploring the connections between algebraic properties of such groups and [[topology|topological]] and [[geometry|geometric]] properties of spaces on which these groups [[Group action (mathematics)|act]] (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).}}
{{term |[[Geometric measure theory]]}}
{{defn |the study of [[geometry|geometric]] properties of [[Set (mathematics)|set]]s (typically in [[Euclidean space]]) through [[measure (mathematics)|measure theory]].}}
{{term |[[Geometric number theory]]}}
{{term |[[Geometric topology]]}}
{{defn |a branch of [[topology]] studying manifolds and mappings between them; in particular the [[embedding]] of one manifold into another.}}
{{term |[[Geometry]]}}
{{defn |a branch of mathematics concerned with [[shape]] and the properties of [[space (mathematics)|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 system|axiomatic form]] by [[Euclid]], giving rise to what is now known as classical Euclidean geometry. The use of [[coordinate]]s 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]].}}
{{term |[[Geometry of number]]s}}
{{defn |initiated by [[Hermann Minkowski]], it is a branch of [[number theory]] studying [[convex bodies]] and [[integer]] [[Euclidean vector|vector]]s.}}
{{term |[[Global analysis]]}}
{{defn |the study of [[differential equation]]s on manifolds and the relationship between [[differential equation]]s and [[topology]].}}
{{term |[[Arithmetic dynamics|Global arithmetic dynamics]]}}
{{term |[[Graph theory]]}}
{{defn |a branch of [[discrete mathematics]] devoted to the study of [[Graph (discrete mathematics)|graph]]s. It has many applications in [[physical science|physical]], [[biological science|biological]] and [[social science|social]] systems.}}
{{term |[[Character theory|Group-character theory]]}}
{{defn |the part of character theory dedicated to the study of characters of [[group representation]]s.}}
{{term |[[Group representation theory]]}}
{{term |[[Group theory]]}}
{{defn |the study of [[algebraic structure]]s known as [[group (mathematics)|group]]s.}}
{{term |[[Gyrotrigonometry]]}}
{{defn |a form of [[trigonometry]] used in [[gyrovector space]] for [[hyperbolic geometry]]. (An analogy of the [[vector space]] in Euclidean geometry.)}}
{{glossary end}}


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==H==


{{glossary}}
{{Compact ToC|side=yes|center=yes|top=yes|num=yes|extlinks=yes|seealso=yes|refs=yes|nobreak=yes|}}
{{term |[[Hard analysis]]}}

{{defn |see ''classical analysis''}}
==H==
{{term |[[Harmonic analysis]]}}
{{defn |part of analysis concerned with the representations of [[function (mathematics)|function]]s in terms of [[wave]]s. It generalizes the notions of [[Fourier series]] and [[Fourier transform]]s from the [[Fourier analysis]].}}
{{term|[[Arithmetic|Higher arithmetic]]}}
{{term|[[Higher category theory]]}}
{{defn |the part of [[category theory]] at a ''higher order'', which means that some equalities are replaced by explicit [[morphism|arrow]]s in order to be able to explicitly study the structure behind those equalities.}}
{{term|[[Higher-dimensional algebra]]}}
{{defn |the study of [[Categorification|categorified]] structures.}}
{{term|[[Hodge theory]]}}
{{defn |a method for studying the [[cohomology group]]s of a [[smooth manifold]] ''M'' using [[partial differential equation]]s.}}
{{term |[[Hodge-Arakelov theory]]}}
{{term|[[Holomorphic functional calculus]]}}
{{defn |a branch of [[functional calculus]] starting with [[holomorphic function]]s.}}
{{term|[[Homological algebra]]}}
{{defn |the study of [[homology (mathematics)|homology]] in general algebraic settings.}}
{{term|[[Homology theory]]}}
{{term|[[Homotopy theory]]}}
{{term|[[Hyperbolic geometry]]}}
{{defn |also known as ''Lobachevskian geometry'' or ''Bolyai-Lobachevskian geometry''. It is a [[non-Euclidean geometry]] looking at [[hyperbolic space]].}}
{{term|[[Hyperbolic geometry|hyperbolic trigonometry]]}}
{{defn |the study of [[hyperbolic triangle]]s in [[hyperbolic geometry]], or [[hyperbolic function]]s in Euclidean geometry. Other forms include [[gyrotrigonometry]] and [[rational trigonometry|universal hyperbolic trigonometry]].}}
{{term|[[Hypercomplex analysis]]}}
{{defn |the extension of [[real analysis]] and [[complex analysis]] to the study of functions where the [[argument of a function|argument]] is a [[hypercomplex number]].}}
{{term|[[Hyperfunction|Hyperfunction theory]]}}
{{glossary end}}


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* '''[[Hard analysis]]''': see ''classical analysis''
* '''[[Harmonic analysis]]''': part of analysis concerned with the representations of [[function (mathematics)|function]]s in terms of [[wave|waves]]. It generalizes the notions of [[Fourier series]] and [[Fourier transforms]] from the [[Fourier analysis]].
* '''[[Arithmetic|Higher arithmetic]]'''
* '''[[Higher category theory]]''': the part of [[category theory]] at a ''higher order'', which means that some equalities are replaced by explicit [[morphism|arrows]] in order to be able to explicitly study the structure behind those equalities.
* '''[[Higher-dimensional algebra]]''': the study of [[Categorification|categorified]] structures.
* '''[[Hodge theory]]''': a method for studying the [[cohomology group]]s of a [[smooth manifold]] ''M'' using [[partial differential equation]]s.
* '''[[Holomorphic functional calculus]]''': a branch of [[functional calculus]] starting with [[holomorphic function]]s.
* '''[[Homological algebra]]''': the study of [[homology (mathematics)|homology]] in general algebraic settings.
* '''[[Homology theory]]'''
* '''[[Homotopy theory]]'''
* {{anchor|hyperbolic-geometry}}'''[[Hyperbolic geometry]]''': also known as ''Lobachevskian geometry'' or ''Bolyai-Lobachevskian geometry''. It is a [[non-Euclidean geometry]] looking at [[hyperbolic space]].
* '''[[Hyperbolic geometry|hyperbolic trigonometry]]''': the study of [[hyperbolic triangle]]s in [[hyperbolic geometry]], or [[hyperbolic function]]s in Euclidean geometry. Other forms include [[gyrotrigonometry]] and [[rational trigonometry|universal hyperbolic trigonometry]].
* '''[[Hypercomplex analysis]]''': the extension of [[real analysis]] and [[complex analysis]] to the study of functions where the [[argument of a function|argument]] is a [[hypercomplex number]].
* '''[[Hyperfunction|Hyperfunction theory]]'''


==I==
==I==


{{glossary}}
* '''[[Ideal theory]]''': once the precursor name for what is now known as [[commutative algebra]]; it is the theory of [[ideal (ring theory)|ideals]] in [[commutative ring]]s.
{{term |[[Ideal theory]]}}
* '''[[Idempotent analysis]]''': the study of [[idempotent semiring]]s, such as the [[tropical semiring]].
{{defn |once the precursor name for what is now known as [[commutative algebra]]; it is the theory of [[ideal (ring theory)|ideal]]s in [[commutative ring]]s.}}
* '''[[Incidence geometry]]''': the study of relations of [[incidence (geometry)|incidence]] between various geometric objects, like [[curve (mathematics)|curves]] and [[line (mathematics)|lines]].
{{term |[[Idempotent analysis]]}}
* '''[[Inconsistent mathematics]]''': see ''paraconsistent mathematics''.
{{defn |the study of [[idempotent semiring]]s, such as the [[tropical semiring]].}}
* '''[[Infinitary combinatorics]]''': an expansion of ideas in combinatorics to account for [[infinite set]]s.
{{term |[[Incidence geometry]]}}
* '''[[Infinitesimal calculus|Infinitesimal analysis]]''': once a synonym for ''infinitesimal calculus''
{{defn |the study of relations of [[incidence (geometry)|incidence]] between various geometric objects, like [[curve (mathematics)|curve]]s and [[line (mathematics)|line]]s.}}
* '''[[Infinitesimal calculus]]''': see ''calculus of infinitesimals''
{{term |[[Inconsistent mathematics]]}}
* '''[[Information geometry]]''': an interdisciplinary field that applies the techniques of [[differential geometry]] to study [[probability theory]] and [[statistics]]. It studies [[statistical manifold]]s, which are [[Riemannian manifold]]s whose points correspond to [[probability distributions]].
{{defn |see ''paraconsistent mathematics''.}}
* '''[[Integral calculus]]'''
{{term |[[Infinitary combinatorics]]}}
* '''[[Integral geometry]]''': the theory of [[measure (mathematics)|measures]] on a geometrical space invariant under the [[symmetry group]] of that space.
{{defn |an expansion of ideas in combinatorics to account for [[infinite set]]s.}}
* '''[[Intersection theory]]''': a branch of algebraic geometry and algebraic topology
{{term |[[Infinitesimal calculus|Infinitesimal analysis]]}}
* '''[[Intuitionistic type theory]]''': a [[type theory]] and an alternative [[Foundations of mathematics|foundation of mathematics]].
{{defn |once a synonym for ''infinitesimal calculus''}}
* '''[[Invariant theory]]''': studies how [[Group action (mathematics)|group action]]s on algebraic varieties affect functions.
{{term |[[Infinitesimal calculus]]}}
* '''[[Inventory theory]]''':
{{defn| See ''{{gli|calculus of infinitesimals}}'' }}
* '''[[Inversive geometry]]''': the study of invariants preserved by a type of transformation known as inversion
{{term |[[Information geometry]]}}
* '''[[Inversive geometry|Inversive plane geometry]]''': inversive geometry that is limited to two dimensions
{{defn |an interdisciplinary field that applies the techniques of [[differential geometry]] to study [[probability theory]] and [[statistics]]. It studies [[statistical manifold]]s, which are [[Riemannian manifold]]s whose points correspond to [[probability distribution]]s.}}
* '''[[Inversive ring geometry]]'''
{{term |[[Integral calculus]]}}
* '''[[Itô calculus]]''': extends the methods of calculus to [[stochastic process]]es such as [[Brownian motion]] (see [[Wiener process]]). It has important applications in [[mathematical finance]] and [[stochastic differential equation]]s.
{{term |[[Integral geometry]]}}
* '''[[Iwasawa theory]]''': the study of objects of arithmetic interest over infinite [[Tower of fields|towers]] of [[number field]]s.
{{defn |the theory of [[measure (mathematics)|measure]]s on a geometrical space invariant under the [[symmetry group]] of that space.}}
{{term |[[Intersection theory]]}}
{{defn |a branch of algebraic geometry and algebraic topology}}
{{term |[[Intuitionistic type theory]]}}
{{defn |a [[type theory]] and an alternative [[Foundations of mathematics|foundation of mathematics]].}}
{{term |[[Invariant theory]]}}
{{defn |studies how [[Group action (mathematics)|group action]]s on algebraic varieties affect functions.}}
{{term |[[Inventory theory]]}}
{{term |[[Inversive geometry]]}}
{{defn |the study of invariants preserved by a type of transformation known as inversion}}
{{term |[[Inversive geometry|Inversive plane geometry]]}}
{{defn |inversive geometry that is limited to two dimensions}}
{{term |[[Inversive ring geometry]]}}
{{term |[[Itô calculus]]}}
{{defn |extends the methods of calculus to [[stochastic process]]es such as [[Brownian motion]] (see [[Wiener process]]). It has important applications in [[mathematical finance]] and [[stochastic differential equation]]s.}}
{{term |[[Iwasawa theory]]}}
{{defn |the study of objects of arithmetic interest over infinite [[Tower of fields|tower]]s of [[number field]]s.}}
{{term |[[Iwasawa-Tate theory]]}}
{{glossary end}}

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==J==
==J==


{{glossary}}
* '''[[Job shop scheduling]]'''
{{term | [[Job shop scheduling]]}}
{{glossary end}}


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==K==
==K==
{{glossary}}

{{term |[[K-theory]]}}
* '''[[K-theory]]''': originated as the study of a [[ring (mathematics)|ring]] generated by [[vector bundle]]s over a [[topological space]] or [[scheme (mathematics)|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 [[mathematical physics|physics]], [[K-theory (physics)|K-theory]] has appeared in [[type II string theory]]. (In particular [[twisted K-theory]].)
{{defn |originated as the study of a [[ring (mathematics)|ring]] generated by [[vector bundle]]s over a [[topological space]] or [[scheme (mathematics)|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 [[mathematical physics|physics]], [[K-theory (physics)|K-theory]] has appeared in [[type II string theory]]. (In particular [[twisted K-theory]].)}}
* '''[[K-homology]]''': a [[homology (mathematics)|homology]] theory on the [[Category (mathematics)|category]] of locally [[compact space|compact]] [[Hausdorff space]]s.
{{term |[[K-homology]]}}
* '''[[Kähler manifold|Kähler geometry]]''': a branch of [[differential geometry]], more specifically a union of [[Riemannian geometry]], [[differential geometry|complex differential geometry]] and [[symplectic geometry]]. It is the study of [[Kähler manifold]]s. (named after [[Erich Kähler]])
{{defn |a [[homology (mathematics)|homology]] theory on the [[Category (mathematics)|category]] of locally [[compact space|compact]] [[Hausdorff space]]s. }}
* '''[[KK-theory]]''': a common generalization both of [[K-homology]] and [[operator K-theory|K-theory]] as an additive [[functor#Bifunctors|bivariant functor]] on [[separable space|separable]] [[C*-algebras]].
{{term |[[Kähler manifold|Kähler geometry]]}}
* '''[[Klein geometry]]''': More specifically, it is a [[homogeneous space]] ''X'' together with a [[Group action (mathematics)|transitive action]] on ''X'' by a [[Lie group]] ''G'', which acts as the [[symmetry group]] of the geometry.
{{defn |a branch of [[differential geometry]], more specifically a union of [[Riemannian geometry]], [[differential geometry|complex differential geometry]] and [[symplectic geometry]]. It is the study of [[Kähler manifold]]s. (named after [[Erich Kähler]])}}
* '''[[Knot theory]]''': part of [[topology]] dealing with [[knot (mathematics)|knots]]
{{term |[[KK-theory]]}}
* '''[[Kummer theory]]''': provides a description of certain types of [[field extension]]s involving the [[adjunction (field theory)|adjunction]] of ''n''th roots of elements of the base [[field (mathematics)|field]]
{{defn |a common generalization both of [[K-homology]] and [[operator K-theory|K-theory]] as an additive [[functor#Bifunctors|bivariant functor]] on [[separable space|separable]] [[C*-algebra]]s.}}
{{term |[[Klein geometry]]}}
{{defn |More specifically, it is a [[homogeneous space]] ''X'' together with a [[Group action (mathematics)|transitive action]] on ''X'' by a [[Lie group]] ''G'', which acts as the [[symmetry group]] of the geometry.}}
{{term |[[Knot theory]]}}
{{defn |part of [[topology]] dealing with [[knot (mathematics)|knot]]s}}
{{term |[[Kummer theory]]}}
{{defn |provides a description of certain types of [[field extension]]s involving the [[adjunction (field theory)|adjunction]] of ''n''th roots of elements of the base [[field (mathematics)|field]]}}
{{glossary end}}


==L==
==L==


{{glossary}}
* '''[[L-theory]]''': the [[K-theory]] of [[quadratic form]]s.
{{term |[[L-theory]]}}
* '''[[Large deviations theory]]''': part of [[probability theory]] studying [[event (probability theory)|events]] of small probability ([[event (probability theory)|tail events]]).
{{defn | the [[K-theory]] of [[quadratic form]]s.}}
* '''[[Large sample theory]]''': also known as ''asymptotic theory''
{{term |[[Large deviations theory]]}}
* '''[[Lattice theory]]''': the study of [[lattice (order)|lattices]], being important in [[order theory]] and [[universal algebra]]
{{defn | part of [[probability theory]] studying [[event (probability theory)|event]]s of small probability ([[event (probability theory)|tail event]]s).}}
* '''[[Lie algebra|Lie algebra theory]]'''
* '''[[Lie group theory]]'''
{{term |[[Large sample theory]]}}
{{defn | also known as ''asymptotic theory''}}
* '''[[Lie sphere geometry]]''': a [[geometry|geometrical]] theory of [[plane geometry|planar]] or [[spatial geometry]] in which the fundamental concept is the [[circle]] or [[sphere]].
* '''[[Lie theory]]'''
{{term |[[Lattice theory]]}}
{{defn | the study of [[lattice (order)|lattice]]s, being important in [[order theory]] and [[universal algebra]]}}
* '''[[Line geometry]]'''
{{term |[[Lie algebra|Lie algebra theory]]}}
* '''[[Linear algebra]]''' &ndash; a branch of algebra studying [[linear space]]s and [[linear map]]s. 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 (mathematics)|matrix theory]] is considered a branch, although linear algebra is restricted to only finite dimensions. Extensions of the methods used belong to [[multilinear algebra]].
{{term |[[Lie group theory]]}}
* '''[[Functional analysis|Linear functional analysis]]'''
{{term |[[Lie sphere geometry]]}}
* '''[[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 function#As a polynomial function|linear relationships]].
{{defn |[[geometry|geometrical]] theory of [[plane geometry|planar]] or [[spatial geometry]] in which the fundamental concept is the [[circle]] or [[sphere]].}}
* '''[[List of graphical methods]]''' Included are diagram techniques, chart techniques, plot techniques, and other forms of visualization.
{{term |[[Lie theory]]}}
* '''[[Local algebra]]''': a term sometimes applied to the theory of [[local ring]]s.
{{term |[[Line geometry]]}}
* '''[[Local class field theory]]''': the study of [[abelian extension]]s of [[local field]]s.
{{term |[[Linear algebra]]}}
* '''[[Low-dimensional topology]]''': the branch of [[topology]] that studies [[manifold]]s, or more generally topological spaces, of four or fewer [[dimension]]s.
{{defn |a branch of algebra studying [[linear space]]s and [[linear map]]s. 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 (mathematics)|matrix theory]] is considered a branch, although linear algebra is restricted to only finite dimensions. Extensions of the methods used belong to [[multilinear algebra]].}}
{{term |[[Functional analysis|Linear functional analysis]]}}
{{term |[[Linear programming]]}}
{{defn |a method to achieve the best outcome (such as maximum profit or lowest cost) in a [[mathematical model]] whose requirements are represented by [[linear function#As a polynomial function|linear relationship]]s.}}
{{term |[[List of graphical method]]s}}
{{defn |Included are diagram techniques, chart techniques, plot techniques, and other forms of visualization.}}
{{term |[[Local algebra]]}}
{{defn |a term sometimes applied to the theory of [[local ring]]s.}}
{{term |[[Local class field theory]]}}
{{defn |the study of [[abelian extension]]s of [[local field]]s.}}
{{term |[[Low-dimensional topology]]}}
{{defn |the branch of [[topology]] that studies [[manifold]]s, or more generally topological spaces, of four or fewer [[dimension]]s.}}
{{glossary end}}


{{Compact ToC|side=yes|center=yes|top=yes|num=no|extlinks=no|seealso=yes|refs=yes|nobreak=yes|}}

{{Compact ToC|side=yes|center=yes|top=yes|num=yes|extlinks=yes|seealso=yes|refs=yes|nobreak=yes|}}


==M==
==M==
{{glossary}}
{{term |[[Malliavin calculus]]}}
{{defn |a set of mathematical techniques and ideas that extend the mathematical field of [[calculus of variation]]s from deterministic functions to [[stochastic processes]].}}
{{term |[[Mathematical biology]]}}
{{defn |the [[mathematical modeling]] of biological phenomena.}}
{{term |[[Mathematical chemistry]]}}
{{defn |the [[mathematical modeling]] of chemical phenomena.}}
{{term |[[Mathematical economics]]}}
{{defn |the application of mathematical methods to represent theories and analyze problems in [[economics]].}}
{{term |[[Mathematical finance]]}}
{{defn |a field of [[applied mathematics]], concerned with mathematical modeling of [[financial market]]s. }}
{{term |[[Mathematical logic]]}}
{{defn |a subfield of [[mathematics]] exploring the applications of formal [[logic]] to mathematics.}}
{{term |[[Mathematical optimization]]}}
{{term |[[Mathematical physics]]}}
{{defn | The development of mathematical methods suitable for application to problems in [[physics]].<ref>Definition from the ''Journal of Mathematical Physics''. {{cite web |url=http://jmp.aip.org/jmp/staff.jsp |title=Archived copy |access-date=2006-10-03 |url-status=dead |archive-url=https://web.archive.org/web/20061003233339/http://jmp.aip.org/jmp/staff.jsp |archive-date=2006-10-03 }}</ref>}}
{{term |[[Mathematical psychology]]}}
{{defn |an approach to [[psychology|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.}}
{{term |[[Mathematical science]]s}}
{{defn |refers to [[academic discipline]]s that are mathematical in nature, but are not considered proper subfields of mathematics. Examples include [[statistics]], [[cryptography]], [[game theory]] and [[actuarial science]].}}
{{term |[[Mathematical sociology]]}}
{{defn |the area of sociology that uses mathematics to construct social theories.}}
{{term |[[Mathematical statistics]]}}
{{defn |the application of [[probability theory]], a branch of [[mathematics]], to [[statistics]], as opposed to techniques for collecting statistical data.}}
{{term |[[Mathematical system theory]]}}
{{term |[[Matrix ring|Matrix algebra]]}}
{{term |[[Matrix calculus]]}}
{{term |[[Matrix (mathematics)|Matrix theory]]}}
{{term |[[Matroid theory]]}}
{{term |[[Measure theory]]}}
{{term |[[Metric geometry]]}}
{{term |[[Microlocal analysis]]}}
{{term |[[Model theory]]}}
{{defn |the study of classes of mathematical [[Structure (mathematical logic)|structure]]s (e.g. [[Group (mathematics)|group]]s, [[Field (mathematics)|field]]s, [[Graph (discrete mathematics)|graph]]s, universes of [[set theory]]) from the perspective of [[mathematical logic]].}}
{{term |[[Abstract algebra|Modern algebra]]}}
{{defn| Occasionally used for ''{{gli|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.}}
{{term |[[Scheme (mathematics)|Modern algebraic geometry]]}}
{{defn |the form of algebraic geometry given by [[Alexander Grothendieck]] and [[Jean-Pierre Serre]] drawing on [[sheaf theory]].}}
{{term |[[Invariant theory|Modern invariant theory]]}}
{{defn |the form of [[invariant theory]] that analyses the decomposition of [[representation (mathematics)|representation]]s into irreducibles.}}
{{term |[[Modular representation theory]]}}
{{defn |a part of [[representation theory]] that studies [[linear representation]]s of [[finite group]]s over a [[field (mathematics)|field]] ''K'' of positive [[characteristic (algebra)|characteristic]] ''p'', necessarily a prime number.}}
{{term |[[Module theory]]}}
{{term |[[Molecular geometry]]}}
{{term |[[Morse theory]]}}
{{defn |a part of differential topology, it analyzes the [[topological space]] of a manifold by studying [[differentiable function]]s on that manifold.}}
{{term |[[Motivic cohomology]]}}
{{term |[[Multilinear algebra]]}}
{{defn |an extension of linear algebra building upon concepts of [[p-vector]]s and [[multivector]]s with [[Grassmann algebra]].}}
{{term |[[Multiplicative number theory]]}}
{{defn |a subfield of analytic number theory that deals with [[prime number]]s, [[factorization]] and [[divisor]]s.}}
{{term |[[Multivariable calculus]]}}
{{defn |the extension of [[calculus]] in one [[Variable (mathematics)|variable]] to calculus with [[function of several variables|functions of several variable]]s: the [[Differential calculus|differentiation]] and [[integral|integration]] of functions involving several variables, rather than just one.}}
{{term |[[Multiple-scale analysis]]}}
{{glossary end}}


{{Compact ToC|side=yes|center=yes|top=yes|num=no|extlinks=no|seealso=yes|refs=yes|nobreak=yes|}}
* '''[[Malliavin calculus]]''': a set of mathematical techniques and ideas that extend the mathematical field of [[calculus of variations]] from deterministic functions to [[stochastic processes]].
* '''[[Mathematical biology]]''': the [[mathematical modeling]] of biological phenomena.
* '''[[Mathematical chemistry]]''': the [[mathematical modeling]] of chemical phenomena.
* '''[[Mathematical economics]]''': the application of mathematical methods to represent theories and analyze problems in [[economics]].
* '''[[Mathematical finance]]''': a field of [[applied mathematics]], concerned with mathematical modeling of [[financial market]]s.
* '''[[Mathematical logic]]''': a subfield of [[mathematics]] exploring the applications of formal [[logic]] to mathematics.
* '''[[Mathematical optimization]]'''
* '''[[Mathematical physics]]''': a part of mathematics that develops mathematical methods motivated by problems in [[physics]].
* '''[[Mathematical psychology]]''': an approach to [[psychology|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.
* '''[[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]].
* '''[[Mathematical sociology]]''': the area of sociology that uses mathematics to construct social theories.
* '''[[Mathematical statistics]]''': the application of [[probability theory]], a branch of [[mathematics]], to [[statistics]], as opposed to techniques for collecting statistical data.
* '''[[Mathematical system theory]]'''
* '''[[Matrix ring|Matrix algebra]]'''
* '''[[Matrix calculus]]'''
* '''[[Matrix (mathematics)|Matrix theory]]'''
* '''[[Matroid theory]]'''
* '''[[Measure theory]]'''
* '''[[Metric geometry]]'''
* '''[[Microlocal analysis]]'''
* '''[[Model theory]]''': the study of classes of mathematical [[Structure (mathematical logic)|structures]] (e.g. [[Group (mathematics)|groups]], [[Field (mathematics)|fields]], [[Graph (discrete mathematics)|graphs]], universes of [[set theory]]) from the perspective of [[mathematical logic]].
* '''[[Abstract algebra|Modern algebra]]''': see ''abstract algebra''
* '''[[Scheme (mathematics)|Modern algebraic geometry]]''': the form of algebraic geometry given by [[Alexander Grothendieck]] and [[Jean-Pierre Serre]] drawing on [[sheaf theory]].
* '''[[Invariant theory|Modern invariant theory]]''': the form of [[invariant theory]] that analyses the decomposition of [[representation (mathematics)|representations]] into irreducibles.
* '''[[Modular representation theory]]''': a part of [[representation theory]] that studies [[linear representation]]s of [[finite group]]s over a [[field (mathematics)|field]] ''K'' of positive [[characteristic (algebra)|characteristic]] ''p'', necessarily a prime number.
* '''[[Module theory]]'''
* '''[[Molecular geometry]]'''
* '''[[Morse theory]]''': a part of differential topology, it analyzes the [[topological space]] of a manifold by studying [[differentiable function]]s on that manifold.
* '''[[Motivic cohomology]]'''
* '''[[Multilinear algebra]]''': an extension of linear algebra building upon concepts of [[p-vector]]s and [[multivector]]s with [[Grassmann algebra]].
* '''[[Multiplicative number theory]]''': a subfield of analytic number theory that deals with [[prime number]]s, [[factorization]] and [[divisor]]s.
* '''[[Multivariable calculus]]''': the extension of [[calculus]] in one [[Variable (mathematics)|variable]] to calculus with [[function of several variables|functions of several variables]]: the [[Differential calculus|differentiation]] and [[integral|integration]] of functions involving several variables, rather than just one.
* '''[[Multiple-scale analysis]]'''


==N==
==N==
{{glossary}}
{{term |[[Neutral geometry]]}}
{{defn |See ''{{gli|absolute geometry}}''. }}
{{term |[[Nevanlinna theory]]}}
{{defn |part of complex analysis studying the value distribution of [[meromorphic function]]s. It is named after [[Rolf Nevanlinna]]}}
{{term |[[Nielsen theory]]}}
{{defn |an area of mathematical research with its origins in [[fixed point topology]], developed by [[Jakob Nielsen (mathematician)|Jakob Nielsen]]}}
{{term |[[Non-abelian class field theory]]}}
{{term |[[Non-classical analysis]]}}
{{term |[[Non-Euclidean geometry]]}}
{{term |[[Non-standard analysis]]}}
{{term |[[Non-standard calculus]]}}
{{term |[[Arithmetic dynamics|Nonarchimedean dynamics]]}}
{{defn |also known as ''p-adic analysis'' or ''local arithmetic dynamics''}}
{{term |[[Noncommutative algebra]]}}
{{term |[[Noncommutative algebraic geometry]]}}
{{defn |a direction in [[noncommutative geometry]] studying the geometric properties of formal duals of non-commutative algebraic objects.}}
{{term |[[Noncommutative geometry]]}}
{{term |[[Noncommutative harmonic analysis]]}}
{{defn |see ''representation theory''}}
{{term |[[Noncommutative topology]]}}
{{term |[[Nonlinear analysis]]}}
{{term |[[Nonlinear functional analysis]]}}
{{term |[[Number theory]]}}
{{defn |a branch of [[pure mathematics]] primarily devoted to the study of the [[integer]]s. Originally it was known as ''arithmetic'' or ''higher arithmetic''.}}
{{term |[[Numerical analysis]]}}
{{term |[[Numerical linear algebra]]}}
{{glossary end}}


{{Compact ToC|side=yes|center=yes|top=yes|num=no|extlinks=no|seealso=yes|refs=yes|nobreak=yes|}}
* '''[[Neutral geometry]]''': see ''absolute geometry''
* '''[[Nevanlinna theory]]''': part of complex analysis studying the value distribution of [[meromorphic function]]s. It is named after [[Rolf Nevanlinna]]
* '''[[Nielsen theory]]''': an area of mathematical research with its origins in [[fixed point topology]], developed by [[Jakob Nielsen (mathematician)|Jakob Nielsen]]
* '''[[Non-abelian class field theory]]'''
* '''[[Non-classical analysis]]'''
* '''[[Non-Euclidean geometry]]'''
* '''[[Non-standard analysis]]'''
* '''[[Non-standard calculus]]'''
* '''[[Arithmetic dynamics|Nonarchimedean dynamics]]''': also known as ''p-adic analysis'' or ''local arithmetic dynamics''
* '''[[Noncommutative algebraic geometry]]''': a direction in [[noncommutative geometry]] studying the geometric properties of formal duals of non-commutative algebraic objects.
* '''[[Noncommutative geometry]]'''
* '''[[Noncommutative harmonic analysis]]''': see ''representation theory''
* '''[[Noncommutative topology]]'''
* '''[[Nonlinear analysis]]'''
* '''[[Nonlinear functional analysis]]'''
* '''[[Number theory]]''': a branch of [[pure mathematics]] primarily devoted to the study of the [[integer]]s. Originally it was known as ''arithmetic'' or ''higher arithmetic''.
* '''[[Numerical analysis]]'''
* '''[[Numerical linear algebra]]'''


{{Compact ToC|side=yes|center=yes|top=yes|num=yes|extlinks=yes|seealso=yes|refs=yes|nobreak=yes|}}


==O==
==O==
{{glossary}}

* '''[[Operad theory]]''': a type of abstract algebra concerned with prototypical [[algebra over a field|algebras]].
{{term| [[Operad theory]]}}
{{defn| a type of abstract algebra concerned with prototypical [[algebra over a field|algebra]]s.}}
* '''[[Operation research]]'''
{{term|[[Operation research]]}}
* '''[[Operator K-theory]]'''
{{term|[[Operator K-theory]]}}
* '''[[Operator theory]]''': part of [[functional analysis]] studying [[operator (mathematics)|operators]].
{{term|[[Operator theory]]}}
{{defn|part of [[functional analysis]] studying [[operator (mathematics)|operator]]s.}}
* '''[[Optimal control theory]]''': a generalization of the [[calculus of variations]].
{{term|[[Optimal control theory]]}}
{{defn|a generalization of the [[calculus of variation]]s.}}
* '''[[Optimal maintenance]]'''
{{term|[[Optimal maintenance]]}}
* '''[[Orbifold|Orbifold theory]]'''
{{term|[[Orbifold|Orbifold theory]]}}
* '''[[Order theory]]''': a branch that investigates the intuitive notion of [[wikt:order|order]] using [[binary relations]].
{{term|[[Order theory]]}}
{{defn|a branch that investigates the intuitive notion of [[wikt:order|order]] using [[binary relation]]s.}}
{{term|[[Ordered geometry]]}}
* '''[[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]].
{{defn|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]].}}
* '''[[Oscillation theory]]'''
{{term|[[Oscillation theory]]}}
{{glossary end}}


==P==
==P==
{{glossary}}
{{term|[[p-adic analysis]]}}
{{defn|a branch of [[number theory]] that deals with the analysis of functions of [[p-adic number]]s.}}
{{term|[[p-adic analysis|p-adic dynamics]]}}
{{defn|an application of [[p-adic analysis]] looking at [[P-adic number|p-adic]] [[differential equation]]s.}}
{{term|[[p-adic Hodge theory]]}}
{{term|[[Parabolic geometry (disambiguation)|Parabolic geometry]]}}
{{term|[[Paraconsistent mathematics]]}}
{{defn|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]].}}
{{term|[[Partition theory]]}}
{{term|[[Perturbation theory]]}}
{{term|[[Picard&ndash;Vessiot theory]]}}
{{term|[[Plane geometry]]}}
{{term|[[Point-set topology]]}}
{{defn|see ''general topology''}}
{{term|[[Pointless topology]]}}
{{term|[[Poisson geometry]]}}
{{term|[[Polyhedral combinatorics]]}}
{{defn|a branch within combinatorics and [[discrete geometry]] that studies the problems of describing [[convex polytope]]s.}}
{{term|[[Possibility theory]]}}
{{term|[[Potential theory]]}}
{{term|[[Precalculus]]}}
{{term|[[Predicative mathematics]]}}
{{term|[[Probability theory]]}}
{{term|[[Probabilistic combinatorics]]}}
{{term|[[Probabilistic graph theory]]}}
{{term|[[Probabilistic number theory]]}}
{{term|[[Projective geometry]]}}
{{defn|a form of geometry that studies geometric properties that are [[invariant (mathematics)|invariant]] under a [[projective transformation]].}}
{{term|[[Projective differential geometry]]}}
{{term|[[Proof theory]]}}
{{term|[[Pseudo-Riemannian geometry]]}}
{{defn|generalizes [[Riemannian geometry]] to the study of [[pseudo-Riemannian manifold]]s.}}
{{term|[[Pure mathematics]]}}
{{defn|the part of mathematics that studies entirely abstract concepts.}}
{{glossary end}}


{{Compact ToC|side=yes|center=yes|top=yes|num=no|extlinks=no|seealso=yes|refs=yes|nobreak=yes|}}
* '''[[p-adic analysis]]''': a branch of [[number theory]] that deals with the analysis of functions of [[p-adic number]]s.
* '''[[p-adic analysis|p-adic dynamics]]''': an application of [[p-adic analysis]] looking at [[P-adic number|p-adic]] [[differential equation]]s.
* '''[[p-adic Hodge theory]]'''
* '''[[Parabolic geometry (disambiguation)|Parabolic geometry]]'''
* '''[[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]].
* '''[[Partition theory]]'''
* '''[[Perturbation theory]]'''
* '''[[Picard&ndash;Vessiot theory]]'''
* '''[[Plane geometry]]'''
* '''[[Point-set topology]]''': see ''general topology''
* '''[[Pointless topology]]'''
* '''[[Poisson geometry]]'''
* '''[[Polyhedral combinatorics]]''': a branch within combinatorics and [[discrete geometry]] that studies the problems of describing [[convex polytope]]s.
* '''[[Possibility theory]]'''
* '''[[Potential theory]]'''
* '''[[Precalculus]]'''
* '''[[Predicative mathematics]]'''
* '''[[Probability theory]]'''
* '''[[Probabilistic combinatorics]]'''
* '''[[Probabilistic graph theory]]'''
* '''[[Probabilistic number theory]]'''
* '''[[Projective geometry]]''': a form of geometry that studies geometric properties that are [[invariant (mathematics)|invariant]] under a [[projective transformation]].
* '''[[Projective differential geometry]]'''
* '''[[Proof theory]]'''
* '''[[Pseudo-Riemannian geometry]]''': generalizes [[Riemannian geometry]] to the study of [[pseudo-Riemannian manifold]]s.
* '''[[Pure mathematics]]''': the part of mathematics that studies entirely abstract concepts.



{{Compact ToC|side=yes|center=yes|top=yes|num=yes|extlinks=yes|seealso=yes|refs=yes|nobreak=yes|}}


==Q==
==Q==
{{glossary}}

* '''[[Quantum calculus]]''': a form of calculus without the notion of [[limit (mathematics)|limits]].
{{term|[[Quantum calculus]]}}
{{defn|a form of calculus without the notion of [[limit (mathematics)|limit]]s.}}
* '''[[Quantum geometry]]''': the generalization of concepts of geometry used to describe the [[physics|physical]] phenomena of [[quantum physics]]
{{term|[[Quantum geometry]]}}
{{defn|the generalization of concepts of geometry used to describe the [[physics|physical]] phenomena of [[quantum physics]]}}
* '''[[Quaternionic analysis]]'''
{{term|[[Quaternionic analysis]]}}
{{glossary end}}

==R==
==R==
{{glossary}}
{{term|[[Ramsey theory]]}}
{{defn|the study of the conditions in which order must appear. It is named after [[Frank P. Ramsey]].}}
{{term|[[Rational geometry]]}}
{{term|[[Real algebraic geometry|Real algebra]]}}
{{defn|the study of the part of algebra relevant to [[real algebraic geometry]].}}
{{term|[[Real algebraic geometry]]}}
{{defn|the part of algebraic geometry that studies [[real number|real]] points of the algebraic varieties.}}
{{term|[[Real analysis]]}}
{{defn|a branch of mathematical analysis; in particular ''hard analysis'', that is the study of [[real number]]s and [[function (mathematics)|function]]s of [[real number|Real]] values. It provides a rigorous formulation of the calculus of [[real number]]s in terms of [[continuous function|continuity]] and [[smooth function|smoothness]], whilst the theory is extended to the [[complex number]]s in [[complex analysis]].}}
{{term | Real Clifford algebra}}
{{term|[[Real K-theory]]}}
{{term|[[Recreational mathematics]]}}
{{defn|the area dedicated to [[mathematical puzzle]]s and [[mathematical game]]s.}}
{{term|[[Recursion theory]]}}
{{defn|see ''computability theory''}}
{{term|[[Representation theory]]}}
{{defn|a subfield of abstract algebra; it studies [[algebraic structure]]s by representing their elements as [[linear transformation]]s of [[vector space]]s. It also studies [[module (mathematics)|module]]s over these algebraic structures, providing a way of reducing problems in abstract algebra to problems in linear algebra.}}
{{term|[[Group representation|Representation theory of group]]s}}
{{term|[[Representation theory of the Galilean group]]}}
{{term|[[Representation theory of the Lorentz group]]}}
{{term|[[Representation theory of the Poincaré group]]}}
{{term|[[Representation theory of the symmetric group]]}}
{{term|[[Ribbon theory]]}}
{{defn|a branch of [[topology]] studying [[ribbon (mathematics)|ribbon]]s.}}
{{term|[[Ricci calculus]]}}
{{defn| {{ghat|Also called '''absolute differential calculus'''.}}
A foundation of [[tensor calculus]], developed by [[Gregorio Ricci-Curbastro]] in 1887–1896,<ref>{{Citation |last1=Ricci |first1=Gregorio |author-link1=Gregorio Ricci-Curbastro |last2=Levi-Civita |first2=Tullio |author-link2=Tullio Levi-Civita |title=Méthodes de calcul différentiel absolu et leurs applications |trans-title=Methods of the absolute differential calculus and their applications |journal=[[Mathematische Annalen]] |date=March 1900 |access-date=19 October 2019 |volume=54 |issue=1–2 |pages=125–201 |doi=10.1007/BF01454201 |url=https://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002258102 |publisher=Springer |s2cid=120009332 |language=fr}}</ref> and later developed for its applications to [[general relativity]] and [[differential geometry]].<ref>{{cite book|last=Schouten|first=Jan A.|title=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)|language=de|year=1924|series=Grundlehren der mathematischen Wissenschaften|volume=10|editor= R. Courant|publisher=Springer Verlag|location=Berlin|url=https://resolver.sub.uni-goettingen.de/purl?PPN373339186}}</ref>}}
{{term|[[Ring theory]]}}
{{term|[[Riemannian geometry]]}}
{{defn|a branch of [[differential geometry]] that is more specifically, the study of [[Riemannian manifold]]s. It is named after [[Bernhard Riemann]] and it features many generalizations of concepts from Euclidean geometry, analysis and calculus.}}
{{term|[[Rough set|Rough set theory]]}}
{{defn|the a form of [[set theory]] based on [[rough set]]s.}}
{{glossary end}}


{{Compact ToC|side=yes|center=yes|top=yes|num=no|extlinks=no|seealso=yes|refs=yes|nobreak=yes|}}
* '''[[Ramsey theory]]''': the study of the conditions in which order must appear. It is named after [[Frank P. Ramsey]].
* '''[[Rational geometry]]'''
* '''[[Real algebraic geometry|Real algebra]]''': the study of the part of algebra relevant to [[real algebraic geometry]].
* '''[[Real algebraic geometry]]''': the part of algebraic geometry that studies [[real number|real]] points of the algebraic varieties.
* '''[[Real analysis]]''': a branch of mathematical analysis; in particular ''hard analysis'', that is the study of [[real number]]s and [[function (mathematics)|functions]] of [[real number|Real]] values. It provides a rigorous formulation of the calculus of [[real number]]s in terms of [[continuous function|continuity]] and [[smooth function|smoothness]], whilst the theory is extended to the [[complex number]]s in [[complex analysis]].
* '''[[Real K-theory]]'''
* '''[[Recreational mathematics]]''': the area dedicated to [[mathematical puzzle]]s and [[mathematical game]]s.
* '''[[Recursion theory]]''': see ''computability theory''
* '''[[Representation theory]]''': a subfield of abstract algebra; it studies [[algebraic structure]]s by representing their elements as [[linear transformation]]s of [[vector space]]s. It also studies [[module (mathematics)|modules]] over these algebraic structures, providing a way of reducing problems in abstract algebra to problems in linear algebra.
* '''[[Representation theory of algebras]]'''
* '''[[Representation theory of diffeomorphism groups]]'''
* '''[[Representation theory of finite groups]]'''
* '''[[Group representation|Representation theory of groups]]'''
* '''[[Representation theory of Hopf algebras]]'''
* '''[[Representation theory of Lie algebras]]'''
* '''[[Representation theory of Lie groups]]'''
* '''[[Representation theory of the Galilean group]]'''
* '''[[Representation theory of the Lorentz group]]'''
* '''[[Representation theory of the Poincaré group]]'''
* '''[[Representation theory of the symmetric group]]'''
* '''[[Ribbon theory]]''': a branch of [[topology]] studying [[ribbon (mathematics)|ribbons]].
* '''[[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.
* '''[[Rough set|Rough set theory]]''': the a form of [[set theory]] based on [[rough set]]s.


{{Compact ToC|side=yes|center=yes|top=yes|num=yes|extlinks=yes|seealso=yes|refs=yes|nobreak=yes|}}


==S==
==S==
{{glossary}}
{{term |[[Sampling (statistics)|Sampling theory]]}}
{{term |[[Scheme (mathematics)|Scheme theory]]}}
{{defn |the study of [[scheme (mathematics)|scheme]]s 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''.}}
{{term |[[Secondary calculus]]}}
{{term |[[Real algebraic geometry|Semialgebraic geometry]]}}
{{defn |a part of algebraic geometry; more specifically a branch of [[real algebraic geometry]] that studies [[semialgebraic set]]s.}}
{{term |[[Set-theoretic topology]]}}
{{term |[[Set theory]]}}
{{term |[[Sheaf theory]]}}
{{defn| The study of [[Sheaf (mathematics)|sheave]]s, which connect local and global properties of [[topological space|geometric object]]s.<ref>{{Citation | last1=Tennison | first1=Barry R. | title=Sheaf theory | publisher=[[Cambridge University Press]] | mr=0404390 | year=1975 |url=https://books.google.com/books?id=JOglpKIw6kIC |isbn=978-0-521-20784-3 |volume=20 |series=London Mathematical Society Lecture Note Series }}</ref>}}
{{term |[[Sheaf cohomology]]}}
{{term |[[Sieve theory]]}}
{{term |[[Operator theory|Single operator theory]]}}
{{defn |deals with the properties and classifications of single [[operator (mathematics)|operator]]s.}}
{{term |[[Singularity theory]]}}
{{defn |a branch, notably of geometry; that studies the failure of manifold structure.}}
{{term |[[Smooth infinitesimal analysis]]}}
{{defn |a rigorous reformation of [[infinitesimal calculus]] employing methods of [[category theory]]. As a theory, it is a subset of [[synthetic differential geometry]].}}
{{term |[[Solid geometry]]}}
{{term |[[Spatial geometry]]}}
{{term |[[Spectral geometry]]}}
{{defn |a field that concerns the relationships between geometric structures of manifolds and [[spectrum of an operator|spectra]] of canonically defined [[differential operator]]s.}}
{{term |[[Spectral graph theory]]}}
{{defn |the study of properties of a [[Graph (discrete mathematics)|graph]] using methods from [[Matrix (mathematics)|matrix theory]].}}
{{term |[[Spectral theory]]}}
{{defn |part of ''operator theory'' extending the concepts of [[eigenvalue]]s and [[eigenvector]]s from linear algebra and [[Matrix (mathematics)|matrix theory]].}}
{{term |[[Spectral theory of ordinary differential equation]]s}}
{{defn |part of [[spectral theory]] concerned with the [[spectrum of an operator|spectrum]] and [[eigenfunction]] expansion associated with [[linear differential equation|linear]] [[ordinary differential equation]]s.}}
{{term |[[Spectrum continuation analysis]]}}
{{defn |generalizes the concept of a [[Fourier series]] to non-periodic [[function (mathematics)|function]]s.}}
{{term |[[Spherical geometry]]}}
{{defn |a branch of [[non-Euclidean geometry]], studying the 2-dimensional surface of a [[sphere]].}}
{{term |[[Spherical trigonometry]]}}
{{defn |a branch of [[spherical geometry]] that studies [[polygon]]s on the surface of a [[sphere]]. Usually the [[polygon]]s are [[triangle]]s.}}
{{term |[[Statistical mechanics]]}}
{{term |[[Statistical modelling]]}}
{{term |[[Statistical theory]]}}
{{term |[[Mathematical statistics|Statistics]]}}
{{defn |although the term may refer to the more general study of [[statistics]], the term is used in mathematics to refer to the [[mathematical statistics|mathematical study of statistics and related field]]s. This includes [[probability theory]].}}
{{term |[[Steganography]]}}
{{term |[[Stochastic calculus]]}}
{{term |[[Malliavin calculus|Stochastic calculus of variation]]s}}
{{term |[[Stochastic geometry]]}}
{{defn |the study of random patterns of points}}
{{term |[[Stochastic process]]}}
{{term |[[Stratified Morse theory]]}}
{{term |[[Super linear algebra]]}}
{{term |[[Surgery theory]]}}
{{defn |a part of [[geometric topology]] referring to methods used to produce one manifold from another (in a controlled way.)}}
{{term |[[Survey sampling]]}}
{{term |[[Survey methodology]]}}
{{term |[[Symbolic computation]]}}
{{defn |also known as ''algebraic computation'' and ''computer algebra''. It refers to the techniques used to manipulate [[expression (mathematics)|mathematical expression]]s and [[equation]]s in [[symbol|symbolic form]] as opposed to manipulating them by the numerical quantities represented by them.}}
{{term |[[Symbolic dynamics]]}}
{{term |[[Symplectic geometry]]}}
{{defn |a branch of [[differential geometry]] and topology whose main object of study is the [[symplectic manifold]].}}
{{term |[[Symplectic topology]]}}
{{term |[[Synthetic differential geometry]]}}
{{defn |a reformulation of [[differential geometry]] in the language of [[topos theory]] and in the context of an [[intuitionistic logic]].}}
{{term |[[Synthetic geometry]]}}
{{defn |also known as '''axiomatic geometry''', it is a branch of geometry that uses [[axiom]]s and [[logical argument]]s to draw conclusions as opposed to [[analytic geometry|analytic]] and algebraic methods.}}
{{term |[[Systolic geometry]]}}
{{defn |a branch of [[differential geometry]] studying systolic [[invariant (mathematics)|invariant]]s of [[manifold]]s and [[polyhedra]].}}
{{term |[[Systolic geometry|Systolic hyperbolic geometry]]}}
{{defn |the study of [[systolic geometry|systole]]s in [[hyperbolic geometry]].}}
{{glossary end}}


{{Compact ToC|side=yes|center=yes|top=yes|num=no|extlinks=no|seealso=yes|refs=yes|nobreak=yes|}}
* '''[[Sampling (statistics)|Sampling theory]]'''
* '''[[Scheme (mathematics)|Scheme theory]]''': the study of [[scheme (mathematics)|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''.
* '''[[Secondary calculus]]'''
* '''[[Self-similarity]]''' an object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts).
* '''[[Real algebraic geometry|Semialgebraic geometry]]''': a part of algebraic geometry; more specifically a branch of [[real algebraic geometry]] that studies [[semialgebraic set]]s.
* '''[[Set-theoretic topology]]'''
* '''[[Set theory]]'''
* '''[[Sheaf theory]]'''
* '''[[Sheaf cohomology]]'''
* '''[[Sieve theory]]'''
* '''[[Operator theory|Single operator theory]]''': deals with the properties and classifications of single [[operator (mathematics)|operators]].
* '''[[Singularity theory]]''': a branch, notably of geometry; that studies the failure of manifold structure.
* '''[[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]].
* '''[[Solid geometry]]'''
* '''[[Spatial geometry]]'''
* '''[[Spectral geometry]]''': a field that concerns the relationships between geometric structures of manifolds and [[spectrum of an operator|spectra]] of canonically defined [[differential operator]]s.
* '''[[Spectral graph theory]]''': the study of properties of a [[Graph (discrete mathematics)|graph]] using methods from [[Matrix (mathematics)|matrix theory]].
* '''[[Spectral theory]]''': part of ''operator theory'' extending the concepts of [[eigenvalue]]s and [[eigenvector]]s from linear algebra and [[Matrix (mathematics)|matrix theory]].
* '''[[Spectral theory of ordinary differential equations]]''': part of [[spectral theory]] concerned with the [[spectrum of an operator|spectrum]] and [[eigenfunction]] expansion associated with [[linear differential equation|linear]] [[ordinary differential equation]]s.
* '''[[Spectrum continuation analysis]]''': generalizes the concept of a [[Fourier series]] to non-periodic [[function (mathematics)|functions]].
* '''[[Spherical geometry]]''': a branch of [[non-Euclidean geometry]], studying the 2-dimensional surface of a [[sphere]].
* '''[[Spherical trigonometry]]''': a branch of [[spherical geometry]] that studies [[polygon]]s on the surface of a [[sphere]]. Usually the [[polygon]]s are [[triangle]]s.
* '''[[Statistical mechanics]]'''
* '''[[Statistical modelling]]'''
* '''[[Statistical theory]]'''
* '''[[Mathematical statistics|Statistics]]''': although the term may refer to the more general study of [[statistics]], the term is used in mathematics to refer to the [[mathematical statistics|mathematical study of statistics and related fields]]. This includes [[probability theory]].
* '''[[Steganography]]'''
* '''[[Stochastic calculus]]'''
* '''[[Malliavin calculus|Stochastic calculus of variations]]'''
* '''[[Stochastic geometry]]''': the study of random patterns of points
* '''[[Stochastic process]]'''
* '''[[Stratified Morse theory]]'''
* '''[[Super linear algebra]]'''
* '''[[Surgery theory]]''': a part of [[geometric topology]] referring to methods used to produce one manifold from another (in a controlled way.)
* '''[[Survey sampling]]'''
* '''[[Survey methodology]]'''
* '''[[Symbolic computation]]''': also known as ''algebraic computation'' and ''computer algebra''. It refers to the techniques used to manipulate [[expression (mathematics)|mathematical expressions]] and [[equation]]s in [[symbol|symbolic form]] as opposed to manipulating them by the numerical quantities represented by them.
* '''[[Symbolic dynamics]]'''
* '''[[Symplectic geometry]]''': a branch of [[differential geometry]] and topology whose main object of study is the [[symplectic manifold]].
* '''[[Symplectic topology]]'''
* '''[[Synthetic differential geometry]]''': a reformulation of [[differential geometry]] in the language of [[topos theory]] and in the context of an [[intuitionistic logic]].
* {{anchor|synthetic-geometry}}'''[[Synthetic geometry]]''': also known as '''axiomatic geometry''', it is a branch of geometry that uses [[axioms]] and [[logical argument]]s to draw conclusions as opposed to [[analytic geometry|analytic]] and algebraic methods.
* '''[[Systolic geometry]]''': a branch of [[differential geometry]] studying systolic [[invariant (mathematics)|invariant]]s of [[manifold]]s and [[polyhedra]].
* '''[[Systolic geometry|Systolic hyperbolic geometry]]''': the study of [[systolic geometry|systoles]] in [[hyperbolic geometry]].


{{Compact ToC|side=yes|center=yes|top=yes|num=yes|extlinks=yes|seealso=yes|refs=yes|nobreak=yes|}}


==T==
==T==
{{glossary}}
{{term |tensor |[[tensor|Tensor algebra, Tensor analysis, Tensor calculus, Tensor theory]]}}
{{defn |the study and use of [[tensor]]s, which are generalizations of [[vector (mathematics and physics)|vectors]]. A [[tensor algebra]] is also an [[algebraic structure]] that is used in the [[mathematical jargon#formal|formal]] definition of tensors.}}
{{term |[[Tessellation]]}}
{{defn |when periodic tiling has a repeating pattern.}}
{{term |[[Theoretical physics]]}}
{{defn |a branch primarily of the [[science]] [[physics]] that uses [[mathematical model]]s and [[abstraction]] of [[mathematical physics|physics]] to rationalize and predict [[phenomena]].}}
{{term |[[Theory of computation]]}}
{{term |[[Time-scale calculus]]}}
{{term |[[Topology]]}}
{{term |[[Topological combinatorics]]}}
{{defn |the application of methods from algebraic topology to solve problems in combinatorics.}}
{{term |[[Topological degree theory]]}}
{{term |[[Topological graph theory]]}}
{{term |[[Topological K-theory]]}}
{{term |[[Topos theory]]}}
{{term |[[Toric geometry]]}}
{{term |[[Transcendental number theory]]}}
{{defn |a branch of [[number theory]] that revolves around the [[transcendental number]]s.}}
{{term |[[Transformation geometry]]}}
{{term |[[Trigonometry]]}}
{{defn |the study of [[triangle]]s and the relationships between the [[length]] of their sides, and the [[angle]]s between them. It is essential to many parts of [[applied mathematics]].}}
{{term |[[Tropical analysis]]}}
{{defn |see ''idempotent analysis''}}
{{term |[[Tropical geometry]]}}
{{term |[[Twisted K-theory]]}}
{{defn |a variation on [[K-theory]], spanning abstract algebra, algebraic topology and [[operator theory]].}}
{{term |[[Type theory]]}}
{{glossary end}}


{{Compact ToC|side=yes|center=yes|top=yes|num=no|extlinks=no|seealso=yes|refs=yes|nobreak=yes|}}
* '''[[Tensor analysis]]''': the study of [[tensor]]s, which play a role in subjects such as [[differential geometry]], [[mathematical physics]], algebraic topology, [[multilinear algebra]], [[homological algebra]] and [[representation theory]].
* '''[[Tensor calculus]]''': an older term for ''tensor analysis''.
* '''[[Tensor|Tensor theory]]''': an alternative name for ''tensor analysis''.
* '''[[Tessellation]]''': when periodic tiling has a repeating pattern.
* '''[[Theoretical physics]]''': a branch primarily of the [[science]] [[physics]] that uses [[mathematical model]]s and [[abstraction]] of [[mathematical physics|physics]] to rationalize and predict [[phenomena]].
* '''[[Theory of computation]]'''
* '''[[Time-scale calculus]]'''
* '''[[Topology]]'''
* '''[[Topological combinatorics]]''': the application of methods from algebraic topology to solve problems in combinatorics.
* '''[[Topological degree theory]]'''
* '''[[Topological graph theory]]'''
* '''[[Topological K-theory]]'''
* '''[[Topos theory]]'''
* '''[[Toric geometry]]'''
* '''[[Transcendental number theory]]''': a branch of [[number theory]] that revolves around the [[transcendental number]]s.
* '''[[Transformation geometry]]'''
* '''[[Trigonometry]]''': the study of [[triangle]]s and the relationships between the [[length]] of their sides, and the [[angle]]s between them. It is essential to many parts of [[applied mathematics]].
* '''[[Tropical analysis]]''': see ''idempotent analysis''
* '''[[Tropical geometry]]'''
* '''[[Twisted K-theory]]''': a variation on [[K-theory]], spanning abstract algebra, algebraic topology and [[operator theory]].
* '''[[Type theory]]'''


{{Compact ToC|side=yes|center=yes|top=yes|num=yes|extlinks=yes|seealso=yes|refs=yes|nobreak=yes|}}


==U==
==U==
{{glossary}}

* '''[[Umbral calculus]]''': the study of [[Sheffer sequences]]
{{term |[[Umbral calculus]]}}
{{defn |the study of [[Sheffer sequence]]s}}
* '''[[Uncertainty theory]]''': a new branch of [[mathematics]] based on normality, monotonicity, self-duality, countable subadditivity, and product measure [[axiom]]s.
{{term |[[Uncertainty theory]]}}
* '''[[Universal algebra]]''': a field studying the formalization of algebraic structures itself.
{{defn |a new branch of [[mathematics]] based on normality, monotonicity, self-duality, countable subadditivity, and product measure [[axiom]]s.}}
* '''[[Rational trigonometry|Universal hyperbolic trigonometry]]''': an approach to [[hyperbolic geometry|hyperbolic trigonometry]] based on [[rational geometry]].
{{term |[[Universal algebra]]}}
{{defn |a field studying the formalization of algebraic structures itself.}}
{{term |[[Rational trigonometry|Universal hyperbolic trigonometry]]}}
{{defn |an approach to [[hyperbolic geometry|hyperbolic trigonometry]] based on [[rational geometry]].}}
{{glossary end}}


==V==
==V==
{{glossary}}

* '''[[Valuation theory]]'''
{{term |[[Valuation theory]]}}
* '''[[Variational analysis]]'''
{{term |[[Variational analysis]]}}
{{term |Vector algebra}}
* '''Vector algebra''': a part of linear algebra concerned with the [[operation (mathematics)|operations]] of [[vector (mathematics)|vector]] addition and [[number (mathematics)|scalar]] [[multiplication]], although it may also refer to [[vector (mathematics)|vector]] [[operation (mathematics)|operation]]s of [[vector calculus]], including the [[dot product|dot]] and [[cross product]]. In this case it can be contrasted with [[geometric algebra]] which generalizes into higher dimensions.
{{defn |a part of linear algebra concerned with the [[operation (mathematics)|operation]]s of [[vector (mathematics)|vector]] addition and [[number (mathematics)|scalar]] [[multiplication]], although it may also refer to [[vector (mathematics)|vector]] [[operation (mathematics)|operation]]s of [[vector calculus]], including the [[dot product|dot]] and [[cross product]]. In this case it can be contrasted with [[geometric algebra]] which generalizes into higher dimensions.}}
* '''[[Vector analysis]]''': also known as [[vector calculus]], see ''vector calculus''.
{{term |[[Vector analysis]]}}
* '''[[Vector calculus]]''': a branch of [[multivariable calculus]] concerned with [[Derivative|differentiation]] and [[integral|integration]] of [[vector field]]s. Primarily it is concerned with 3-dimensional [[Euclidean space]].
{{defn |also known as [[vector calculus]], see ''vector calculus''.}}
{{term |[[Vector calculus]]}}
{{defn |a branch of [[multivariable calculus]] concerned with [[Derivative|differentiation]] and [[integral|integration]] of [[vector field]]s. Primarily it is concerned with 3-dimensional [[Euclidean space]].}}
{{glossary end}}


==W==
==W==
{{glossary}}
* '''[[Wavelet]]s'''
{{term |[[Wavelet]]s}}
* '''[[Windowed Fourier transform]]'''
{{glossary end}}
* '''[[Window function]]s'''


==See also==
==See also==
* [[Areas of mathematics]]
* [[Lists of mathematics topics]]
* [[Lists of mathematics topics]]
* [[Outline of mathematics]]
* [[Outline of mathematics]]
* [[:Category:Glossaries of mathematics]]


==References==
===Other glossaries===
{{reflist}}
* [[Glossary of astronomy]]
* [[Glossary of biology]]
* [[Glossary of calculus]]
* [[Glossary of chemistry terms|Glossary of chemistry]]
* [[Glossary of engineering]]
* [[Glossary of physics]]
* [[Glossary of probability and statistics]]


{{Glossaries of science and engineering}}
{{Glossaries of science and engineering}}


[[Category:Fields of mathematics|* ]]
[[Category:Fields of mathematics|*]]
[[Category:Glossaries of science|Areas of mathematics]]
[[Category:Glossaries of science|Areas of mathematics]]
[[Category:Glossaries of mathematics|Areas of mathematics]]
[[Category:Glossaries of mathematics|Areas of mathematics]]
[[Category:Wikipedia glossaries using description lists]]

Latest revision as of 12:05, 29 August 2024

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

[edit]
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

[edit]
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

[edit]
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

Also called infinitesimal calculus

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

[edit]
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

[edit]
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

[edit]
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

[edit]
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

[edit]
Hard analysis
see classical analysis
Harmonic 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.
Higher arithmetic
Higher category theory
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.
Higher-dimensional algebra
the study of categorified structures.
Hodge theory
a method for studying the cohomology groups of a smooth manifold M using partial differential equations.
Hodge-Arakelov theory
Holomorphic functional calculus
a branch of functional calculus starting with holomorphic functions.
Homological algebra
the study of homology in general algebraic settings.
Homology theory
Homotopy theory
Hyperbolic geometry
also known as Lobachevskian geometry or Bolyai-Lobachevskian geometry. It is a non-Euclidean geometry looking at hyperbolic space.
hyperbolic trigonometry
the study of hyperbolic triangles in hyperbolic geometry, or hyperbolic functions in Euclidean geometry. Other forms include gyrotrigonometry and universal hyperbolic trigonometry.
Hypercomplex analysis
the extension of real analysis and complex analysis to the study of functions where the argument is a hypercomplex number.
Hyperfunction theory

I

[edit]
Ideal theory
once the precursor name for what is now known as commutative algebra; it is the theory of ideals in commutative rings.
Idempotent analysis
the study of idempotent semirings, such as the tropical semiring.
Incidence geometry
the study of relations of incidence between various geometric objects, like curves and lines.
Inconsistent mathematics
see paraconsistent mathematics.
Infinitary combinatorics
an expansion of ideas in combinatorics to account for infinite sets.
Infinitesimal analysis
once a synonym for infinitesimal calculus
Infinitesimal calculus
See calculus of infinitesimals
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.
Integral calculus
Integral geometry
the theory of measures on a geometrical space invariant under the symmetry group of that space.
Intersection theory
a branch of algebraic geometry and algebraic topology
Intuitionistic type theory
a type theory and an alternative foundation of mathematics.
Invariant theory
studies how group actions on algebraic varieties affect functions.
Inventory theory
Inversive geometry
the study of invariants preserved by a type of transformation known as inversion
Inversive plane geometry
inversive geometry that is limited to two dimensions
Inversive ring geometry
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.
Iwasawa theory
the study of objects of arithmetic interest over infinite towers of number fields.
Iwasawa-Tate theory

J

[edit]
Job shop scheduling

K

[edit]
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.)
K-homology
a homology theory on the category of locally compact Hausdorff spaces.
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)
KK-theory
a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras.
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.
Knot theory
part of topology dealing with knots
Kummer theory
provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field

L

[edit]
L-theory
the K-theory of quadratic forms.
Large deviations theory
part of probability theory studying events of small probability (tail events).
Large sample theory
also known as asymptotic theory
Lattice theory
the study of lattices, being important in order theory and universal algebra
Lie algebra theory
Lie group theory
Lie sphere geometry
geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere.
Lie theory
Line geometry
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.
Linear functional analysis
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.
List of graphical methods
Included are diagram techniques, chart techniques, plot techniques, and other forms of visualization.
Local algebra
a term sometimes applied to the theory of local rings.
Local class field theory
the study of abelian extensions of local fields.
Low-dimensional topology
the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions.

M

[edit]
Malliavin calculus
a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes.
Mathematical biology
the mathematical modeling of biological phenomena.
Mathematical chemistry
the mathematical modeling of chemical phenomena.
Mathematical economics
the application of mathematical methods to represent theories and analyze problems in economics.
Mathematical finance
a field of applied mathematics, concerned with mathematical modeling of financial markets.
Mathematical logic
a subfield of mathematics exploring the applications of formal logic to mathematics.
Mathematical optimization
Mathematical physics
The development of mathematical methods suitable for application to problems in physics.[13]
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.
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.
Mathematical sociology
the area of sociology that uses mathematics to construct social theories.
Mathematical statistics
the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data.
Mathematical system theory
Matrix algebra
Matrix calculus
Matrix theory
Matroid theory
Measure theory
Metric geometry
Microlocal analysis
Model theory
the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic.
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.
Modern algebraic geometry
the form of algebraic geometry given by Alexander Grothendieck and Jean-Pierre Serre drawing on sheaf theory.
Modern invariant theory
the form of invariant theory that analyses the decomposition of representations into irreducibles.
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.
Module theory
Molecular geometry
Morse theory
a part of differential topology, it analyzes the topological space of a manifold by studying differentiable functions on that manifold.
Motivic cohomology
Multilinear algebra
an extension of linear algebra building upon concepts of p-vectors and multivectors with Grassmann algebra.
Multiplicative number theory
a subfield of analytic number theory that deals with prime numbers, factorization and divisors.
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.
Multiple-scale analysis

N

[edit]
Neutral geometry
See absolute geometry.
Nevanlinna theory
part of complex analysis studying the value distribution of meromorphic functions. It is named after Rolf Nevanlinna
Nielsen theory
an area of mathematical research with its origins in fixed point topology, developed by Jakob Nielsen
Non-abelian class field theory
Non-classical analysis
Non-Euclidean geometry
Non-standard analysis
Non-standard calculus
Nonarchimedean dynamics
also known as p-adic analysis or local arithmetic dynamics
Noncommutative algebra
Noncommutative algebraic geometry
a direction in noncommutative geometry studying the geometric properties of formal duals of non-commutative algebraic objects.
Noncommutative geometry
Noncommutative harmonic analysis
see representation theory
Noncommutative topology
Nonlinear analysis
Nonlinear functional analysis
Number theory
a branch of pure mathematics primarily devoted to the study of the integers. Originally it was known as arithmetic or higher arithmetic.
Numerical analysis
Numerical linear algebra

O

[edit]
Operad theory
a type of abstract algebra concerned with prototypical algebras.
Operation research
Operator K-theory
Operator theory
part of functional analysis studying operators.
Optimal control theory
a generalization of the calculus of variations.
Optimal maintenance
Orbifold theory
Order theory
a branch that investigates the intuitive notion of order using binary relations.
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.
Oscillation theory

P

[edit]
p-adic analysis
a branch of number theory that deals with the analysis of functions of p-adic numbers.
p-adic dynamics
an application of p-adic analysis looking at p-adic differential equations.
p-adic Hodge theory
Parabolic geometry
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.
Partition theory
Perturbation theory
Picard–Vessiot theory
Plane geometry
Point-set topology
see general topology
Pointless topology
Poisson geometry
Polyhedral combinatorics
a branch within combinatorics and discrete geometry that studies the problems of describing convex polytopes.
Possibility theory
Potential theory
Precalculus
Predicative mathematics
Probability theory
Probabilistic combinatorics
Probabilistic graph theory
Probabilistic number theory
Projective geometry
a form of geometry that studies geometric properties that are invariant under a projective transformation.
Projective differential geometry
Proof theory
Pseudo-Riemannian geometry
generalizes Riemannian geometry to the study of pseudo-Riemannian manifolds.
Pure mathematics
the part of mathematics that studies entirely abstract concepts.

Q

[edit]
Quantum calculus
a form of calculus without the notion of limits.
Quantum geometry
the generalization of concepts of geometry used to describe the physical phenomena of quantum physics
Quaternionic analysis

R

[edit]
Ramsey theory
the study of the conditions in which order must appear. It is named after Frank P. Ramsey.
Rational geometry
Real algebra
the study of the part of algebra relevant to real algebraic geometry.
Real algebraic geometry
the part of algebraic geometry that studies real points of the algebraic varieties.
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.
Real Clifford algebra
Real K-theory
Recreational mathematics
the area dedicated to mathematical puzzles and mathematical games.
Recursion theory
see computability theory
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.
Representation theory of groups
Representation theory of the Galilean group
Representation theory of the Lorentz group
Representation theory of the Poincaré group
Representation theory of the symmetric group
Ribbon theory
a branch of topology studying ribbons.
Ricci calculus

Also called absolute differential calculus.

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]
Ring theory
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.
Rough set theory
the a form of set theory based on rough sets.

S

[edit]
Sampling theory
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.
Secondary calculus
Semialgebraic geometry
a part of algebraic geometry; more specifically a branch of real algebraic geometry that studies semialgebraic sets.
Set-theoretic topology
Set theory
Sheaf theory
The study of sheaves, which connect local and global properties of geometric objects.[16]
Sheaf cohomology
Sieve theory
Single operator theory
deals with the properties and classifications of single operators.
Singularity theory
a branch, notably of geometry; that studies the failure of manifold structure.
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.
Solid geometry
Spatial geometry
Spectral geometry
a field that concerns the relationships between geometric structures of manifolds and spectra of canonically defined differential operators.
Spectral graph theory
the study of properties of a graph using methods from matrix theory.
Spectral theory
part of operator theory extending the concepts of eigenvalues and eigenvectors from linear algebra and matrix theory.
Spectral theory of ordinary differential equations
part of spectral theory concerned with the spectrum and eigenfunction expansion associated with linear ordinary differential equations.
Spectrum continuation analysis
generalizes the concept of a Fourier series to non-periodic functions.
Spherical geometry
a branch of non-Euclidean geometry, studying the 2-dimensional surface of a sphere.
Spherical trigonometry
a branch of spherical geometry that studies polygons on the surface of a sphere. Usually the polygons are triangles.
Statistical mechanics
Statistical modelling
Statistical theory
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.
Steganography
Stochastic calculus
Stochastic calculus of variations
Stochastic geometry
the study of random patterns of points
Stochastic process
Stratified Morse theory
Super linear algebra
Surgery theory
a part of geometric topology referring to methods used to produce one manifold from another (in a controlled way.)
Survey sampling
Survey methodology
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.
Symbolic dynamics
Symplectic geometry
a branch of differential geometry and topology whose main object of study is the symplectic manifold.
Symplectic topology
Synthetic differential geometry
a reformulation of differential geometry in the language of topos theory and in the context of an intuitionistic logic.
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.
Systolic geometry
a branch of differential geometry studying systolic invariants of manifolds and polyhedra.
Systolic hyperbolic geometry
the study of systoles in hyperbolic geometry.

T

[edit]
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.
Tessellation
when periodic tiling has a repeating pattern.
Theoretical physics
a branch primarily of the science physics that uses mathematical models and abstraction of physics to rationalize and predict phenomena.
Theory of computation
Time-scale calculus
Topology
Topological combinatorics
the application of methods from algebraic topology to solve problems in combinatorics.
Topological degree theory
Topological graph theory
Topological K-theory
Topos theory
Toric geometry
Transcendental number theory
a branch of number theory that revolves around the transcendental numbers.
Transformation geometry
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.
Tropical analysis
see idempotent analysis
Tropical geometry
Twisted K-theory
a variation on K-theory, spanning abstract algebra, algebraic topology and operator theory.
Type theory

U

[edit]
Umbral calculus
the study of Sheffer sequences
Uncertainty theory
a new branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms.
Universal algebra
a field studying the formalization of algebraic structures itself.
Universal hyperbolic trigonometry
an approach to hyperbolic trigonometry based on rational geometry.

V

[edit]
Valuation theory
Variational analysis
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.
Vector analysis
also known as vector calculus, see vector calculus.
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

[edit]
Wavelets

See also

[edit]

References

[edit]
  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) [1975]. Abstract Analytic Number Theory (2nd ed.). New York, NY: Dover Publishing. ISBN 0-486-66344-2. Zbl 0743.11002.
  5. ^ Apostol, Tom M. Mathematical Analysis: A Modern Approach to Advanced Calculus (2 ed.). Addison-Wesley.
  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, MR 0434929, Zbl 0335.10001
  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.
  9. ^ Courant, R. (1937), Differential and Integral Calculus, vol. I, translated by McShane, E. J. (2nd ed.), New York: Interscience, ISBN 978-4-87187-838-8
  10. ^ "Definition of DIFFERENTIAL CALCULUS". www.merriam-webster.com. Retrieved 2020-05-09.
  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". Archived from the original on 2006-10-03. Retrieved 2006-10-03.{{cite web}}: CS1 maint: archived copy as title (link)
  14. ^ Ricci, Gregorio; Levi-Civita, Tullio (March 1900), "Méthodes de calcul différentiel absolu et leurs applications" [Methods of the absolute differential calculus and their applications], Mathematische Annalen (in French), 54 (1–2), Springer: 125–201, doi:10.1007/BF01454201, S2CID 120009332, retrieved 19 October 2019
  15. ^ Schouten, Jan A. (1924). R. Courant (ed.). 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 (in German). Vol. 10. Berlin: Springer Verlag.
  16. ^ Tennison, Barry R. (1975), Sheaf theory, London Mathematical Society Lecture Note Series, vol. 20, Cambridge University Press, ISBN 978-0-521-20784-3, MR 0404390
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