View source for Glossary of areas of mathematics

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{{List missing criteria|date=July 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|analytic number theory]] is a subarea of [[Number theory|number theory]] devoted to the use of methods of [[Mathematical analysis|analysis]] for the study of [[Natural number|natural number]]s.

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.

==A==

{{glossary}}
{{term|[[Ricci calculus|Absolute differential calculus]]}}
{{defn| An older name of {{gli|Ricci calculus}} }}
{{term|[[Absolute geometry]]}}
{{defn| Also called {{gli|neutral geometry}},<ref>{{Citation | 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|synthetic geometry]] similar to [[Euclidean geometry]] but without the [[Parallel postulate|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>}}
{{term|[[Abstract algebra]]}}
{{defn| The part of {{gli|algebra}} devoted to the study of [[Algebraic structure|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.}}
{{term|[[Abstract analytic number theory]]}}
{{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>}}
{{term|[[Abstract differential geometry]]}}
{{defn| A form of [[Differential geometry|differential geometry]] without the notion of [[Smoothness|smoothness]] from [[Calculus|calculus]]. Instead it is built using sheaf theory and [[Sheaf cohomology|sheaf cohomology]].}}
{{term|Abstract harmonic analysis}}
{{defn| A modern branch of [[Physics:Harmonic analysis|harmonic analysis]] that extends upon the generalized [[Fourier transform]]s that can be defined on [[Locally compact group|locally compact group]]s.}}
{{term|[[Homotopy theory|Abstract homotopy theory]]}}
{{defn| A part of [[Topology|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|[[Actuarial science]]}}
{{defn| The discipline that applies [[Mathematics|mathematical]] and [[Statistics|statistical]] methods to [[Risk assessment|assess risk]] in [[Finance:Insurance|insurance]], [[Finance:Finance|finance]] and other industries and professions. More generally, actuaries apply rigorous mathematics to model matters of uncertainty.}}
{{term|[[Additive combinatorics]]}}
{{defn| The part of [[Arithmetic combinatorics|arithmetic combinatorics]] devoted to the operations of [[Addition|addition]] and [[Subtraction|subtraction]].}}
{{term|[[Additive number theory]]}}
{{defn| A part of [[Number theory|number theory]] that studies subsets of [[Integer|integer]]s and their behaviour under addition.}}
{{term|[[Affine geometry]]}}
{{defn| A branch of [[Geometry|geometry]] that deals with properties that are independent from distances and angles, such as [[Collinearity|alignment]] and [[Parallel (geometry)|parallelism]].}}
{{term|[[Affine geometry of curves]]}}
{{defn| The study of [[Curve|curve]] properties that are invariant under [[Affine transformation|affine transformation]]s.}}
{{term|[[Affine differential geometry]]}}
{{defn| A type of [[Differential geometry|differential geometry]] dedicated to differential [[Invariant (mathematics)|invariant]]s under [[Physics:Volume|volume]]-preserving [[Affine transformation|affine transformation]]s.}}
{{term|[[Ahlfors theory]]}}
{{defn| A part of [[Complex analysis|complex analysis]] being the geometric counterpart of [[Nevanlinna theory]]. It was invented by [[Biography:Lars Ahlfors|Lars Ahlfors]].}}
{{term|[[Algebra]]}}
{{defn| One of the major [[Areas of mathematics|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|number]]s or other [[Philosophy:Mathematical object|mathematical object]]s, such as [[Vector (mathematics and physics)|vector]]s, [[Matrix (mathematics)|matrices]], or elements of [[Algebraic structure|algebraic structure]]s.}}
{{term|[[Algebraic analysis]]}}
{{defn| motivated by systems of [[Linear differential equation|linear]] [[Partial differential equation|partial differential equation]]s, it is a branch of [[Algebraic geometry|algebraic geometry]] and [[Algebraic topology|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.}}
{{term|[[Algebraic combinatorics]]}}
{{defn| an area that employs methods of abstract algebra to problems of [[Combinatorics|combinatorics]]. It also refers to the application of methods from combinatorics to problems in abstract algebra.}}
{{term|Algebraic computation}}
{{defn| An older name of [[Computer algebra|computer algebra]].}}
{{term|[[Algebraic geometry]]}}
{{defn| a branch that combines techniques from abstract algebra with the language and problems of geometry. Fundamentally, it studies algebraic varieties.}}
{{term|[[Algebraic graph theory]]}}
{{defn| a branch of [[Graph theory|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|group theory]] and linear algebra.}}
{{term| [[Algebraic K-theory]]}}
{{defn| an important part of [[Homological algebra|homological algebra]] concerned with defining and applying a certain sequence of [[Functor|functor]]s from [[Ring (mathematics)|ring]]s to [[Abelian group|abelian group]]s.}}
{{term| [[Algebraic number theory]]}}
{{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 fields and their rings of integers.}}
{{term|[[Algebraic statistics]]}}
{{defn| the use of algebra to advance [[Statistics|statistics]], although the term is sometimes restricted to label the use of algebraic geometry and [[Commutative algebra|commutative algebra]] in [[Statistics|statistics]].}}
{{term|[[Algebraic topology]]}}
{{defn |a branch that uses tools from [[Abstract algebra|abstract algebra]] for [[Topology|topology]] to study [[Topological space|topological space]]s.}}
{{term|Algorithmic number theory}}
{{defn| also known as ''computational number theory'', it is the study of [[Algorithm|algorithm]]s for performing [[Number theory|number theoretic]] [[Computation|computation]]s.}}
{{term|[[Anabelian geometry]]}}
{{defn| an area of study based on the theory proposed by [[Biography:Alexander Grothendieck|Alexander Grothendieck]] in the 1980s that describes the way a geometric object of an [[Algebraic variety|algebraic variety]] (such as an algebraic fundamental group) can be mapped into another object, without it being an [[Abelian group|abelian group]].}}
{{term|[[Mathematical analysis|Analysis]]}}
{{defn | A wide area of mathematics centered on the study of [[Continuous function|continuous function]]s and including such topics as differentiation, integration, [[Limit (mathematics)|limit]]s, and [[Series (mathematics)|series]].<ref>{{cite book| first=Tom M. | last=Apostol |  title=Mathematical Analysis: A Modern Approach to Advanced Calculus| edition= 2 |publisher=Addison-Wesley | id={{ASIN|0201002884|country=ca}} }}</ref>}}
{{term |[[Analytic combinatorics]]}}
{{defn| part of [[Enumerative combinatorics|enumerative combinatorics]] where methods of complex analysis are applied to [[Generating function|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|differentiable function]]s are replaced with [[Analytic function|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|number theory]] that applies methods from [[Mathematical analysis|mathematical analysis]] to solve problems about [[Integer|integer]]s.<ref>{{Apostol IANT}}</ref>}}
{{term| Analytic theory of L-functions}}
{{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|science]], [[Engineering:Engineering|engineering]], [[Finance:Finance|finance]], [[Finance:Economics|economics]] and [[Finance:Logistics|logistics]].}}
{{term|[[Approximation theory]]}}
{{defn| part of [[Philosophy:Analysis|analysis]] that studies how well functions can be approximated by simpler ones (such as [[Polynomial|polynomial]]s or [[Trigonometric polynomial|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 [[Biography:Suren Arakelov|Suren Arakelov]].}}
{{term| [[Arithmetic]]}}
{{defn|no=1| Also known as {{gli|elementary arithmetic}}, the methods and rules for computing with [[Addition|addition]], [[Subtraction|subtraction]], [[Multiplication|multiplication]] and [[Division (mathematics)|division]] of numbers.}}
{{defn|no=2| Also known as {{gli|higher arithmetic}}, another name for [[Number theory|number theory]].}}
{{term| Arithmetic algebraic geometry}}
{{defn| See {{gli|arithmetic geometry}}.}}
{{term|[[Arithmetic combinatorics]]}}
{{defn| the study of the estimates from [[Combinatorics|combinatorics]] that are associated with arithmetic operations such as addition, [[Subtraction|subtraction]], [[Multiplication|multiplication]] and [[Division (mathematics)|division]].}}
{{term |[[Arithmetic dynamics]]}}
{{defn| Arithmetic dynamics is the study of the number-theoretic properties of integer, [[Rational point|rational]], {{mvar|p}}-adic, and/or algebraic points under repeated application of a [[Polynomial|polynomial]] or [[Rational function|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|algebraic geometry]] and more specially scheme theory for solving problems of number theory.}}
{{term| [[Arithmetic topology]]}}
{{defn | a combination of [[Algebraic number theory|algebraic number theory]] and [[Topology|topology]] studying analogies between prime ideals and knots}}
{{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|generating function]]s and then complex analysis techniques to get asymptotics.}}
{{term| Asymptotic theory}} 
{{defn| the study of [[Asymptotic expansion|asymptotic expansion]]s}}
{{term| [[Auslander–Reiten theory]]}}
{{defn| the study of the [[Representation theory|representation theory]] of [[Artinian ring]]s}}
{{term| Axiomatic geometry}}
{{defn| also known as ''[[Synthetic geometry|synthetic geometry]]'': it is a branch of geometry that uses [[Axiom|axiom]]s and logical arguments to draw conclusions as opposed to [[Analytic geometry|analytic]] and algebraic methods.}}
{{term| Axiomatic set theory}}
{{defn| the study of systems of [[Axiom|axiom]]s in a context relevant to [[Set theory|set theory]] and [[Mathematical logic|mathematical logic]].}}
{{glossary end}}


==B==

{{glossary}}
{{term| [[Bifurcation theory]]}}
{{defn| the study of changes in the qualitative or topological structure of a given family. It is a part of [[Dynamical systems theory|dynamical systems theory]]}}
{{term| [[Biology:Biostatistics|Biostatistics]]}}
{{defn | the development and application of statistical methods to a wide range of topics in [[Biology:Biology|biology]].}}
{{term| [[Birational geometry]]}}
{{defn |a part of [[Algebraic geometry|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''}}
{{glossary end}}


==C==

{{glossary}}
{{term| [[C*-algebra|C*-algebra theory]]}}
{{defn | a [[Complex number|complex]] [[Algebra over a field|algebra]] ''A'' of [[Continuous linear operator|continuous linear operator]]s on a [[Complex number|complex]] [[Hilbert space]] with two additional properties-(i) ''A'' is a topologically [[Closed set|closed set]] in the norm topology of operators.(ii)''A'' is closed under the operation of taking adjoints of operators.}}
{{term | Cartesian geometry}}
{{defn | see ''analytic geometry''}}
{{term | [[Calculus]]}}
{{defn | An area of mathematics connected by the [[Fundamental theorem of calculus|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>}}
{{term | Calculus of infinitesimals}}
{{ghat|Also called '''infinitesimal calculus'''}}
{{defn | A foundation of [[Calculus|calculus]], first developed in the 17th century,<ref>{{cite book |first=Carl B. |last=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|infinitesimal]] numbers.}}
{{term | [[Calculus of moving surfaces]]}}
{{defn| an extension of the theory of [[Tensor|tensor calculus]] to include deforming [[Manifold|manifold]]s.}}
{{term | Calculus of variations}}
{{defn | the field dedicated to maximizing or minimizing [[Functional (mathematics)|functional]]s. It used to be called ''functional calculus''.}}
{{term | [[Catastrophe theory]]}}
{{defn | a branch of [[Bifurcation theory|bifurcation theory]] from [[Dynamical systems theory|dynamical systems theory]], and also a special case of the more general [[Singularity theory|singularity theory]] from geometry. It analyses the [[Germ (mathematics)|germ]]s of the catastrophe geometries.}}
{{term| [[Categorical logic]]}}
{{defn | a branch of [[Category theory|category theory]] adjacent to the [[Mathematical logic|mathematical logic]]. It is based on [[Type theory|type theory]] for [[Intuitionistic logic|intuitionistic logic]]s.}}
{{term| [[Category theory]]}}
{{defn | the study of the properties of particular mathematical concepts by formalising them as collections of objects and arrows.}}
{{term | [[Chaos theory]]}}
{{defn | the study of the behaviour of [[Dynamical system|dynamical system]]s that are highly sensitive to their initial conditions.}}
{{term | [[Character theory]]}}
{{defn | a branch of [[Group theory|group theory]] that studies the characters of [[Group representation|group representation]]s or [[Modular representation theory|modular representation]]s.}}
{{term | [[Class field theory]]}}
{{defn | a branch of [[Algebraic number theory|algebraic number theory]] that studies [[Abelian extension|abelian extension]]s of number fields.}}
{{term | [[Differential geometry|Classical differential geometry]]}}
{{defn | also known as [[Differential geometry|Euclidean differential geometry]]. see ''Euclidean differential geometry''.}}
{{term | [[Algebraic topology|Classical algebraic topology]]}}
{{defn | see ''algebraic topology''}}
{{term | Classical analysis}}
{{defn | usually refers to the more traditional topics of analysis such as [[Real analysis|real analysis]] and complex analysis. It includes any work that does not use techniques from [[Functional analysis|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 mathematics]].}}
{{term | [[Analytic number theory|Classical analytic number theory]]}}
{{term  | [[Differential calculus|Classical differential calculus]]}}
{{term | [[Diophantine geometry|Classical Diophantine geometry]]}}
{{term | [[Euclidean geometry|Classical Euclidean geometry]]}} 
{{ defn | see ''Euclidean geometry''}}
{{ term | Classical geometry}}
{{defn | may refer to [[Solid geometry|solid geometry]] or classical Euclidean geometry. See ''geometry''}}
{{term | [[Invariant theory|Classical invariant theory]]}}
{{defn | the form of [[Invariant theory|invariant theory]] that deals with describing [[Polynomial function|polynomial function]]s that are [[Invariant (mathematics)|invariant]] under transformations from a given [[Linear group|linear group]].}}
{{term | [[Classical mathematics]]}}
{{defn | the standard approach to mathematics based on [[Philosophy:Classical logic|classical logic]] and ZFC set theory.}}
{{term | [[Projective geometry|Classical projective geometry]]}}
{{term | [[Tensor|Classical tensor calculus]]}}
{{term | [[Clifford algebra]]}}
{{term | [[Clifford analysis]]}}
{{defn|the study of [[Physics:Dirac operator|Dirac operator]]s and [[Physics:Dirac operator|Dirac type operator]]s from geometry and analysis using [[Clifford algebra|clifford algebra]]s.}}
{{term | [[Clifford theory]]}}
{{defn | is a branch of [[Representation theory|representation theory]] spawned from [[Clifford theory|Cliffords theorem]].}}
{{term | Cobordism theory}}
{{term | [[Coding theory]]}}
{{defn | the study of the properties of [[Code|code]]s and their respective fitness for specific applications.}}
{{term | Cohomology theory}}
{{term| Combinatorial analysis}}
{{term | [[Combinatorial commutative algebra]]}}
{{defn | a discipline viewed as the intersection between [[Commutative algebra|commutative algebra]] and combinatorics. It frequently employs methods from one to address problems arising in the other. Polyhedral geometry also plays a significant role.}}
{{term |Combinatorial design theory}}
{{defn | a part of combinatorial mathematics that deals with the existence and construction of systems of finite sets whose intersections have certain properties.}}
{{term | [[Combinatorial game theory]]}}
{{term | Combinatorial geometry}}
{{defn | see ''discrete geometry''}}
{{term |Combinatorial group theory}}
{{defn | the theory of [[Free group|free group]]s and the presentation of a group. It is closely related to [[Geometric group theory|geometric group theory]] and is applied in [[Geometric topology|geometric topology]].}}
{{term |Combinatorial mathematics}}
{{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.}}
{{term |Combinatorial number theory}}
{{term |[[Combinatorial optimization]]}}
{{term |Combinatorial set theory}}
{{defn | also known as [[Infinitary combinatorics]]. see ''infinitary combinatorics''}}
{{term |Combinatorial theory}}
{{term |[[Combinatorial topology]]}}
{{defn | an old name for algebraic topology, when topological invariants of spaces were regarded as derived from combinatorial decompositions.}}
{{term |[[Combinatorics]]}}
{{defn | a branch of [[Discrete mathematics|discrete mathematics]] concerned with [[Countable set|countable]] [[Mathematical structure|structure]]s. Branches of it include [[Enumerative combinatorics|enumerative combinatorics]], combinatorial design theory, matroid theory, [[Extremal combinatorics|extremal combinatorics]] and [[Algebraic combinatorics|algebraic combinatorics]], as well as many more.}}
{{term |[[Commutative algebra]]}}
{{defn | a branch of abstract algebra studying [[Commutative ring|commutative ring]]s.}}
{{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 [[Philosophy:Analysis|analysis]] that deals with functions of a complex variable.}}
{{term |Complex analytic dynamics}}
{{defn | a subdivision of [[Complex dynamics|complex dynamics]] being the study of the dynamic systems defined by [[Analytic function|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|differential geometry]] that studies [[Complex manifold|complex manifold]]s.}}
{{term |[[Complex dynamics]]}}
{{defn | the study of [[Dynamical system|dynamical system]]s defined by [[Iterated function|iterated function]]s on complex number spaces.}}
{{term |[[Complex geometry]]}} 
{{defn | the study of [[Complex manifold|complex manifold]]s and functions of complex variables. It includes complex algebraic geometry and complex analytic geometry.}}
{{term |Complexity theory}}
{{defn | the study of [[Complex system|complex system]]s with the inclusion of the theory of [[Complex system|complex system]]s.}}
{{term |[[Computable analysis]]}}
{{defn | the study of which parts of [[Real analysis|real analysis]] and [[Functional analysis|functional analysis]] can be carried out in a [[Computability theory|computable]] manner. It is closely related to [[Constructive analysis|constructive analysis]].}}
{{term |[[Computable model theory]]}}
{{defn | a branch of [[Model theory|model theory]] dealing with the relevant questions [[Computability|computability]].}}
{{term |[[Computability theory]]}}
{{defn | a branch of [[Mathematical logic|mathematical logic]] originating in the 1930s with the study of [[Computable function|computable function]]s and [[Turing degree]]s, but now includes the study of generalized computability and definability. It overlaps with [[Proof theory|proof theory]] and [[Effective descriptive set theory|effective descriptive set theory]].}}
{{term |Computational algebraic geometry}}
{{term |[[Computational complexity theory]]}}
{{defn | a branch of mathematics and [[Theoretical computer science|theoretical computer science]] that focuses on classifying [[Computational problem|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|computer science]] devoted to the study of algorithms which can be stated in terms of [[Geometry|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|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|algorithm]]s for performing [[Number theory|number theoretic]] [[Computation|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|approximation theory]], studying the connection between the smoothness of a function and its [[Approximation theory|degree of approximation]]}}
{{term |Constructive mathematics}}
{{defn | mathematics which tends to use [[Intuitionistic logic|intuitionistic logic]]. Essentially that is classical logic but without the assumption that the law of the excluded middle is an [[Axiom|axiom]].}}
{{term |[[Constructive quantum field theory]]}}
{{defn | a branch of [[Physics:Mathematical physics|mathematical physics]] that is devoted to showing that [[Physics:Quantum mechanics|quantum theory]] is mathematically compatible with [[Physics:Special relativity|special relativity]].}}
{{term |[[Constructive set theory]]}}
{{defn | an approach to 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|differential geometry]] and [[Differential topology|topology]], closely related to and considered the odd-dimensional counterpart of [[Symplectic geometry|symplectic geometry]]. It is the study of a geometric structure called a contact structure on a [[Differentiable manifold|differentiable manifold]].}}
{{term |[[Convex analysis]]}}
{{defn | the study of properties of [[Convex function|convex function]]s and [[Convex set|convex set]]s.}}
{{term |[[Convex geometry]]}}
{{defn | part of geometry devoted to the study of [[Convex set|convex set]]s.}}
{{term |Coordinate geometry}}
{{defn | see ''analytic geometry''}}
{{term |CR geometry}}
{{defn | a branch of [[Differential geometry|differential geometry]], being the study of [[CR manifold]]s.}}
{{term |[[Cryptography]]}}
{{glossary end}}


==D==
 
{{glossary}}
{{term |[[Decision analysis]]}}
{{term |[[Decision theory]]}}
{{term |[[Noncommutative algebraic geometry|Derived noncommutative algebraic geometry]]}}
{{term |[[Descriptive set theory]]}}
{{defn |a part of [[Mathematical logic|mathematical logic]], more specifically a part of [[Set theory|set theory]] dedicated to the study of [[Polish space]]s.}}
{{term |[[Differential algebraic geometry]]}}
{{defn |the adaption of methods and concepts from algebraic geometry to systems of [[Algebraic differential equation|algebraic differential equation]]s.}}
{{term |[[Differential calculus]]}}
{{defn |A branch of [[Calculus|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|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>}}
{{term |[[Differential Galois theory]]}}
{{defn |the study of the [[Galois group]]s of differential fields.}}
{{term |[[Differential geometry]]}}
{{defn |a form of geometry that uses techniques from integral and [[Differential calculus|differential calculus]] as well as [[Linear algebra|linear]] and [[Multilinear algebra|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|differentiable manifold]]s. It is closely related to differential topology.}}
{{term |[[Differential geometry of curves]]}}
{{defn |the study of [[Curve|smooth curve]]s in [[Euclidean space]] by using techniques from [[Differential geometry|differential geometry]]}}
{{term |[[Differential geometry of surfaces]]}}
{{defn |the study of smooth [[Surface (topology)|surface]]s with various additional structures using the techniques of [[Differential geometry|differential geometry]].}}
{{term |[[Differential topology]]}}
{{defn |a branch of [[Topology|topology]] that deals with [[Differentiable function|differentiable function]]s on [[Differentiable manifold|differentiable manifold]]s.}}
{{term |[[Diffiety|Diffiety theory]]}}
{{term |[[Diophantine geometry]]}}
{{defn |in general the study of algebraic varieties over [[Field (mathematics)|field]]s that are finitely generated over their prime fields.}}
{{term |[[Discrepancy theory]]}}
{{term |[[Discrete differential geometry]]}}
{{term |[[Discrete exterior calculus]]}}
{{term |[[Discrete geometry]]}}
{{defn |a branch of [[Geometry|geometry]] that studies [[Combinatorics|combinatorial]] properties and constructive methods of [[Discrete mathematics|discrete]] geometric objects.}}
{{term |[[Discrete mathematics]]}}
{{defn |the study of [[Mathematical structure|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|partially ordered set]]s (posets) commonly called domains.}}
{{term |[[Donaldson theory]]}}
{{defn |the study of smooth [[4-manifold|4-manifold]]s using [[Physics:Gauge theory|gauge theory]].}}
{{term |[[Dyadics|Dyadic algebra]]}}
{{term |[[Dynamical systems theory]]}}
{{defn |an area used to describe the behavior of the complex [[Dynamical system|dynamical system]]s, usually by employing [[Differential equation|differential equation]]s or difference equations.}}
{{glossary end}}


==E==

{{glossary}}
{{term |[[Econometrics]]}}
{{defn |the application of mathematical and [[Statistics|statistical]] methods to [[Finance:Economics|economic]] [[Data|data]].}}
{{term |[[Effective descriptive set theory]]}}
{{defn |a branch of [[Descriptive set theory|descriptive set theory]] dealing with [[Set (mathematics)|set]] of [[Real number|real number]]s that have lightface definitions. It uses aspects of [[Computability theory|computability theory]].}}
{{term |[[Elementary algebra]]}}
{{defn |a fundamental form of [[Algebra|algebra]] extending on [[Elementary arithmetic|elementary arithmetic]] to include the concept of [[Variable (mathematics)|variable]]s.}}
{{term |[[Elementary arithmetic]]}}
{{defn |the simplified portion of arithmetic considered necessary for primary education. It includes the usage addition, [[Subtraction|subtraction]], [[Multiplication|multiplication]] and [[Division (mathematics)|division]] of the [[Natural number|natural number]]s. It also includes the concept of [[Fraction (mathematics)|fraction]]s and [[Negative number|negative number]]s.}}
{{term |[[Elementary mathematics]]}}
{{defn |parts of mathematics frequently taught at the [[Social:Primary school|primary]] and [[Social:Secondary school|secondary school]] levels. This includes [[Elementary arithmetic|elementary arithmetic]], geometry, [[Probability|probability]] and [[Statistics|statistics]], [[Elementary algebra|elementary algebra]] and [[Trigonometry|trigonometry]]. (calculus is not usually considered a part)}}
{{term |Elementary group theory}}
{{defn |the study of the basics of [[Group theory|group theory]]}}
{{term |[[Elimination theory]]}}
{{defn |the classical name for algorithmic approaches to eliminating between [[Polynomial|polynomial]]s of several variables. It is a part of [[Commutative algebra|commutative algebra]] and algebraic geometry.}}
{{term |[[Elliptic geometry]]}}
{{defn |a type of [[Non-Euclidean geometry|non-Euclidean geometry]] (it violates [[Biography:Euclid|Euclid]]'s [[Parallel postulate|parallel postulate]]) and is based on [[Spherical geometry|spherical geometry]]. It is constructed in elliptic space.}}
{{term |[[Enumerative combinatorics]]}}
{{defn |an area of combinatorics that deals with the number of ways that certain patterns can be formed.}}
{{term |[[Enumerative geometry]]}}
{{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|intersection theory]].}}
{{term |[[Social:Epidemiology|Epidemiology]]}}
{{term |[[Noncommutative algebraic geometry|Equivariant noncommutative algebraic geometry]]}}
{{term |[[Ergodic Ramsey theory]]}}
{{defn |a branch where problems are motivated by [[Additive combinatorics|additive combinatorics]] and solved using [[Ergodic theory|ergodic theory]].}}
{{term |[[Ergodic theory]]}}
{{defn |the study of [[Dynamical system|dynamical system]]s with an [[Invariant measure|invariant measure]], and related problems.}}
{{term |[[Euclidean geometry]]}}
{{defn | An area of [[Geometry|geometry]] based on the axiom system and [[Synthetic geometry|synthetic method]]s of the ancient Greek mathematician [[Biography:Euclid|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|algebraic topology]] and [[Integral geometry|integral geometry]] that integrates [[Constructible function|constructible function]]s and more recently definable functions by integrating with respect to the [[Euler characteristic]] as a finitely-additive 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}}


==F==

{{glossary}}
{{term |[[Field theory (mathematics)|Field theory]]}}
{{defn | The branch of [[Algebra|algebra]] dedicated to [[Field (mathematics)|field]]s, a type of [[Algebraic structure|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>}}
{{term |[[Finite geometry]]}}
{{term |[[Finite model theory]]}}
{{defn |a restriction of [[Model theory|model theory]] to [[Philosophy:Interpretation (logic)|interpretation]]s on finite [[Structure (mathematical logic)|structure]]s, which have a finite universe.}}
{{term |Finsler geometry}}
{{defn |a branch of [[Differential geometry|differential geometry]] whose main object of study is [[Finsler manifold]]s, a generalisation of a [[Riemannian manifold]]s.}}
{{term |[[Arithmetic|First order arithmetic]]}}
{{term |[[Fourier analysis]]}}
{{defn |the study of the way general [[Function (mathematics)|function]]s may be represented or approximated by sums of trigonometric functions.}}
{{term |Fractal geometry}}
{{term |[[Fractional calculus]]}}
{{defn |a branch of analysis that studies the possibility of taking [[Real number|real]] or complex powers of the differentiation operator.}}
{{term |Fractional dynamics}}
{{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|fractional calculus]].}}
{{term |[[Fredholm theory]]}}
{{defn |part of [[Spectral theory|spectral theory]] studying [[Integral equation|integral equation]]s.}}
{{term |Function theory}}
{{defn |an ambiguous term that generally refers to [[Mathematical analysis|mathematical analysis]].}}
{{term |[[Functional analysis]]}}
{{defn |a branch of [[Mathematical analysis|mathematical analysis]], the core of which is formed by the study of [[Function space|function space]]s, which are some sort of [[Topological vector space|topological vector space]]s.}}
{{term |[[Functional calculus]]}}
{{defn |historically the term was used synonymously with [[Calculus of variations|calculus of variations]], but now refers to a branch of [[Functional analysis|functional analysis]] connected with [[Spectral theory|spectral theory]]}}
{{term |[[Fuzzy mathematics]]}}
{{defn |a branch of mathematics based on fuzzy set theory and [[Fuzzy logic|fuzzy logic]].}}
{{term |[[Fuzzy measure theory]]}}
{{term |Fuzzy set theory}}
{{defn |a form of [[Set theory|set theory]] that studies [[Fuzzy set|fuzzy set]]s, that is [[Set (mathematics)|set]]s that have degrees of membership.}}
{{glossary end}}


==G==

{{glossary}}
{{term |[[Galois cohomology]]}}
{{defn |an application of [[Homological algebra|homological algebra]], it is the study of [[Group cohomology|group cohomology]] of [[Galois module]]s.}}
{{term |[[Galois theory]]}}
{{defn |named after [[Biography:Évariste Galois|Évariste Galois]], it is a branch of abstract algebra providing a connection between [[Field theory (mathematics)|field theory]] and [[Group theory|group theory]].}}
{{term |[[Galois geometry]]}}
{{defn |a branch of [[Finite geometry|finite geometry]] concerned with algebraic and [[Analytic geometry|analytic]] geometry over a Galois field.}}
{{term |[[Game theory]]}}
{{defn |the study of [[Mathematical model|mathematical model]]s of strategic interaction among rational decision-makers.}}
{{term |[[Physics:Gauge theory|Gauge theory]]}}
{{term |[[General topology]]}}
{{defn |also known as ''point-set topology'', it is a branch of [[Topology|topology]] studying the properties of [[Topological space|topological space]]s and structures defined on them. It differs from other branches of [[Topology|topology]] as the [[Topological space|topological space]]s do not have to be similar to manifolds.}}
{{term |[[Generalized trigonometry]]}}
{{defn |developments of [[Trigonometry|trigonometric]] methods from the application to [[Real number|real number]]s of Euclidean geometry to any geometry or [[Space|space]]. This includes [[Spherical trigonometry|spherical trigonometry]], [[Hyperbolic geometry|hyperbolic trigonometry]], gyrotrigonometry, and [[Rational trigonometry|universal hyperbolic trigonometry]].}}
{{term |[[Geometric algebra]]}}
{{defn |an alternative approach to classical, [[Computational geometry|computational]] and [[Physics:Special relativity|relativistic geometry]]. It shows a natural correspondence between geometric entities and elements of algebra.}}
{{term |[[Geometric analysis]]}}
{{defn |a discipline that uses methods from [[Differential geometry|differential geometry]] to study [[Partial differential equation|partial differential equation]]s as well as the applications to geometry.}}
{{term |[[Geometric calculus]]}}
{{defn |extends the [[Geometric algebra|geometric algebra]] to include [[Derivative|differentiation]] and [[Integral|integration]].}}
{{term |[[Geometric combinatorics]]}}
{{defn |a branch of [[Combinatorics|combinatorics]].  It includes a number of subareas such as [[Polyhedral combinatorics|polyhedral combinatorics]] (the study of [[Face (geometry)|face]]s of convex polyhedra), [[Convex geometry|convex geometry]] (the study of [[Convex set|convex set]]s, in particular combinatorics of their intersections), and [[Discrete geometry|discrete geometry]], which in turn has many applications to [[Computational geometry|computational geometry]].}}
{{term |[[Geometric function theory]]}}
{{defn |the study of geometric properties of [[Analytic function|analytic function]]s.}}
{{term |[[Geometric invariant theory]]}}
{{defn |a method for constructing quotients by [[Group action (mathematics)|group action]]s in [[Algebraic geometry|algebraic geometry]], used to construct [[Moduli space|moduli space]]s.}}
{{term |[[Geometric graph theory]]}}
{{defn |a large and amorphous subfield of [[Graph theory|graph theory]], concerned with graphs defined by geometric means.}}
{{term |[[Geometric group theory]]}}
{{defn |the study of [[Finitely generated group|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|topology]] studying manifolds and mappings between them; in particular the [[Embedding|embedding]] of one manifold into another.}}
{{term |[[Geometry]]}}
{{defn |a branch of mathematics concerned with [[Shape|shape]] and the properties of [[Space (mathematics)|space]]. Classically it arose as what is now known as [[Solid geometry|solid geometry]]; this was concerning practical knowledge of [[Physics:Length|length]], [[Area|area]] and [[Physics:Volume|volume]]. It was then put into an [[Axiomatic system|axiomatic form]] by [[Biography:Euclid|Euclid]], giving rise to what is now known as classical Euclidean geometry. The use of coordinates by [[Biography:René Descartes|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|projective geometry]], [[Differential geometry|differential geometry]], [[Non-Euclidean geometry|non-Euclidean geometry]], Fractal geometry and algebraic geometry. Geometry also gave rise to the modern discipline of [[Topology|topology]].}}
{{term |Geometry of numbers}}
{{defn |initiated by [[Biography:Hermann Minkowski|Hermann Minkowski]], it is a branch of [[Number theory|number theory]] studying convex bodies and [[Integer|integer]] [[Euclidean vector|vector]]s.}}
{{term |[[Global analysis]]}}
{{defn |the study of [[Differential equation|differential equation]]s on manifolds and the relationship between [[Differential equation|differential equation]]s and [[Topology|topology]].}}
{{term |[[Arithmetic dynamics|Global arithmetic dynamics]]}}
{{term |[[Graph theory]]}}
{{defn |a branch of [[Discrete mathematics|discrete mathematics]] devoted to the study of [[Graph (discrete mathematics)|graph]]s. It has many applications in physical, biological and [[Social: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|group representation]]s.}}
{{term |Group representation theory}}
{{term |[[Group theory]]}}
{{defn |the study of [[Algebraic structure|algebraic structure]]s known as [[Group (mathematics)|group]]s.}}
{{term |Gyrotrigonometry}}
{{defn |a form of [[Trigonometry|trigonometry]] used in [[Physics:Gyrovector space|gyrovector space]] for [[Hyperbolic geometry|hyperbolic geometry]]. (An analogy of the [[Vector space|vector space]] in Euclidean geometry.)}}
{{glossary end}}


==H==

{{glossary}}
{{term |Hard analysis}}
{{defn |see ''classical analysis''}}
{{term |[[Physics:Harmonic analysis|Harmonic analysis]]}}
{{defn |part of analysis concerned with the representations of [[Function (mathematics)|function]]s in terms of [[Wave|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|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 groups of a smooth manifold ''M'' using [[Partial differential equation|partial differential equation]]s.}}
{{term |Hodge-Arakelov theory}}
{{term|[[Holomorphic functional calculus]]}}
{{defn |a branch of [[Functional calculus|functional calculus]] starting with [[Holomorphic function|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|non-Euclidean geometry]] looking at [[Hyperbolic space|hyperbolic space]].}}
{{term|[[Hyperbolic geometry|hyperbolic trigonometry]]}}
{{defn |the study of [[Hyperbolic triangle|hyperbolic triangle]]s in [[Hyperbolic geometry|hyperbolic geometry]], or [[Hyperbolic function|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|real analysis]] and [[Complex analysis|complex analysis]] to the study of functions where the [[Argument of a function|argument]] is a [[Hypercomplex number|hypercomplex number]].}}
{{term|[[Hyperfunction|Hyperfunction theory]]}}
{{glossary end}}


==I==

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


==J==

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


==K==
{{glossary}}
{{term |[[K-theory]]}}
{{defn |originated as the study of a [[Ring (mathematics)|ring]] generated by [[Vector bundle|vector bundle]]s over a [[Topological space|topological space]] or [[Scheme (mathematics)|scheme]]. In algebraic topology it is an extraordinary cohomology theory known as [[Topological K-theory|topological K-theory]]. In algebra and algebraic geometry it is referred to as [[Algebraic K-theory|algebraic K-theory]]. In [[Physics:Mathematical physics|physics]], [[Physics:K-theory|K-theory]] has appeared in [[Physics:Type II string theory|type II string theory]]. (In particular [[Twisted K-theory|twisted K-theory]].)}}
{{term |[[K-homology]]}}
{{defn |a [[Homology (mathematics)|homology]] theory on the [[Category (mathematics)|category]] of locally [[Compact space|compact]] [[Hausdorff space]]s. }}
{{term |[[Kähler manifold|Kähler geometry]]}}
{{defn |a branch of [[Differential geometry|differential geometry]], more specifically a union of [[Riemannian geometry]], [[Differential geometry|complex differential geometry]] and [[Symplectic geometry|symplectic geometry]]. It is the study of [[Kähler manifold]]s. (named after [[Biography:Erich Kähler|Erich Kähler]])}}
{{term |[[KK-theory]]}}
{{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|homogeneous space]] ''X'' together with a [[Group action (mathematics)|transitive action]] on ''X'' by a [[Lie group]] ''G'', which acts as the [[Symmetry group|symmetry group]] of the geometry.}}
{{term |[[Knot theory]]}}
{{defn |part of [[Topology|topology]] dealing with knots}}
{{term |[[Kummer theory]]}}
{{defn |provides a description of certain types of field extensions involving the adjunction of ''n''th roots of elements of the base [[Field (mathematics)|field]]}}
{{glossary end}}

==L==

{{glossary}}
{{term |[[L-theory]]}}
{{defn | the [[K-theory]] of [[Quadratic form|quadratic form]]s.}}
{{term |[[Large deviations theory]]}}
{{defn | part of [[Probability theory|probability theory]] studying [[Event (probability theory)|event]]s of small probability ([[Event (probability theory)|tail event]]s).}}
{{term |Large sample theory}}
{{defn | also known as ''asymptotic theory''}}
{{term |Lattice theory}}
{{defn | the study of [[Lattice (order)|lattice]]s, being important in [[Order theory|order theory]] and [[Universal algebra|universal algebra]]}}
{{term |[[Lie algebra|Lie algebra theory]]}}
{{term |Lie group theory}}
{{term |[[Lie sphere geometry]]}}
{{defn |[[Geometry|geometrical]] theory of planar or spatial geometry in which the fundamental concept is the [[Circle|circle]] or [[Sphere|sphere]].}}
{{term |[[Lie theory]]}}
{{term |Line geometry}}
{{term |[[Linear algebra]]}}
{{defn |a branch of algebra studying linear spaces and [[Linear map|linear map]]s. It has applications in fields such as abstract algebra and [[Functional analysis|functional analysis]]; it can be represented in analytic geometry and it is generalized in [[Operator theory|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|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|mathematical model]] whose requirements are represented by [[Linear function#As a polynomial function|linear relationship]]s.}}
{{term |List of graphical methods}}
{{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|local ring]]s.}}
{{term |[[Local class field theory]]}}
{{defn |the study of [[Abelian extension|abelian extension]]s of [[Local field|local field]]s.}}
{{term |[[Low-dimensional topology]]}}
{{defn |the branch of [[Topology|topology]] that studies [[Manifold|manifold]]s, or more generally topological spaces, of four or fewer [[Physics:Dimension|dimension]]s.}}
{{glossary end}}


==M==
{{glossary}}
{{term |[[Malliavin calculus]]}}
{{defn |a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from  deterministic functions to stochastic processes.}}
{{term |Mathematical biology}}
{{defn |the mathematical modeling of biological phenomena.}}
{{term |[[Chemistry:Mathematical chemistry|Mathematical chemistry]]}}
{{defn |the mathematical modeling of chemical phenomena.}}
{{term |[[Finance:Mathematical economics|Mathematical economics]]}}
{{defn |the application of mathematical methods to represent theories and analyze problems in [[Finance:Economics|economics]].}}
{{term |[[Finance:Mathematical finance|Mathematical finance]]}}
{{defn |a field of [[Applied mathematics|applied mathematics]], concerned with mathematical modeling of [[Finance:Financial market|financial market]]s. }}
{{term |[[Mathematical logic]]}}
{{defn |a subfield of [[Mathematics|mathematics]] exploring the applications of formal [[Logic|logic]] to mathematics.}}
{{term |[[Mathematical optimization]]}}
{{term |[[Physics:Mathematical physics|Mathematical physics]]}}
{{defn | The development of mathematical methods suitable for application to problems in [[Physics:Physics|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 [[Social: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 sciences}}
{{defn |refers to [[Academic discipline|academic discipline]]s that are mathematical in nature, but are not considered proper subfields of mathematics. Examples include [[Statistics|statistics]], [[Cryptography|cryptography]], [[Game theory|game theory]] and [[Actuarial science|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|probability theory]], a branch of [[Mathematics|mathematics]], to [[Statistics|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|set theory]]) from the perspective of [[Mathematical logic|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 [[Biography:Alexander Grothendieck|Alexander Grothendieck]] and [[Biography:Jean-Pierre Serre|Jean-Pierre Serre]] drawing on sheaf theory.}}
{{term |[[Invariant theory|Modern invariant theory]]}}
{{defn |the form of [[Invariant theory|invariant theory]] that analyses the decomposition of [[Representation (mathematics)|representation]]s into irreducibles.}}
{{term |[[Modular representation theory]]}}
{{defn |a part of [[Representation theory|representation theory]] that studies linear representations of [[Finite group|finite group]]s over a [[Field (mathematics)|field]] ''K'' of positive [[Characteristic (algebra)|characteristic]] ''p'', necessarily a prime number.}}
{{term |Module theory}}
{{term |[[Physics:Molecular geometry|Molecular geometry]]}}
{{term |[[Morse theory]]}}
{{defn |a part of differential topology, it analyzes the [[Topological space|topological space]] of a manifold by studying [[Differentiable function|differentiable function]]s on that manifold.}}
{{term |[[Motivic cohomology]]}}
{{term |[[Multilinear algebra]]}}
{{defn |an extension of linear algebra building upon concepts of p-vectors and [[Multivector|multivector]]s with Grassmann algebra.}}
{{term |[[Multiplicative number theory]]}}
{{defn |a subfield of analytic number theory that deals with [[Prime number|prime number]]s, [[Factorization|factorization]] and [[Divisor|divisor]]s.}}
{{term |[[Multivariable calculus]]}}
{{defn |the extension of [[Calculus|calculus]] in one [[Variable (mathematics)|variable]] to calculus with functions of several variables: the [[Differential calculus|differentiation]] and [[Integral|integration]] of functions involving several variables, rather than just one.}}
{{term |[[Multiple-scale analysis]]}}
{{glossary end}}


==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|meromorphic function]]s. It is named after [[Biography:Rolf Nevanlinna|Rolf Nevanlinna]]}}
{{term |[[Nielsen theory]]}}
{{defn |an area of mathematical research with its origins in fixed point topology, developed by [[Biography: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|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|pure mathematics]] primarily devoted to the study of the [[Integer|integer]]s. Originally it was known as ''arithmetic'' or ''higher arithmetic''.}}
{{term |[[Numerical analysis]]}}
{{term |[[Numerical linear algebra]]}}
{{glossary end}}


==O==
{{glossary}}
{{term| [[Operad theory]]}} 
{{defn| a type of abstract algebra concerned with prototypical [[Algebra over a field|algebra]]s.}}
{{term|Operation research}}
{{term|[[Operator K-theory]]}}
{{term|[[Operator theory]]}}
{{defn|part of [[Functional analysis|functional analysis]] studying [[Operator (mathematics)|operator]]s.}}
{{term|Optimal control theory}}
{{defn|a generalization of the calculus of variations.}}
{{term|[[Optimal maintenance]]}}
{{term|[[Orbifold|Orbifold theory]]}}
{{term|[[Order theory]]}}
{{defn|a branch that investigates the intuitive notion of order using [[Binary relation|binary relation]]s.}}
{{term|[[Ordered geometry]]}}
{{defn|a form of geometry  omitting the notion of [[Measurement|measurement]] but featuring the concept of intermediacy. It is a fundamental geometry forming a common framework for [[Affine geometry|affine geometry]], Euclidean geometry, [[Absolute geometry|absolute geometry]] and [[Hyperbolic geometry|hyperbolic geometry]].}}
{{term|[[Oscillation theory]]}}
{{glossary end}}

==P==
{{glossary}}
{{term|[[P-adic analysis|p-adic analysis]]}}
{{defn|a branch of [[Number theory|number theory]] that deals with the analysis of functions of [[P-adic number|p-adic number]]s.}}
{{term|[[P-adic analysis|p-adic dynamics]]}}
{{defn|an application of [[P-adic analysis|p-adic analysis]] looking at [[P-adic number|p-adic]] [[Differential equation|differential equation]]s.}}
{{term|[[P-adic Hodge theory|p-adic Hodge theory]]}}
{{term|Parabolic geometry}}
{{term|[[Philosophy:Paraconsistent mathematics|Paraconsistent mathematics]]}}
{{defn|sometimes called ''inconsistent mathematics'', it is an attempt to develop the classical infrastructure of mathematics based on a foundation of [[Philosophy:Paraconsistent logic|paraconsistent logic]] instead of [[Philosophy:Classical logic|classical logic]].}}
{{term|Partition theory}}
{{term|[[Physics:Perturbation theory|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|discrete geometry]] that studies the problems of  describing [[Convex polytope|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|projective transformation]].}}
{{term|[[Projective differential geometry]]}}
{{term|[[Proof theory]]}}
{{term|Pseudo-Riemannian geometry}}
{{defn|generalizes [[Riemannian geometry]] to the study of [[Pseudo-Riemannian manifold|pseudo-Riemannian manifold]]s.}}
{{term|[[Pure mathematics]]}}
{{defn|the part of mathematics that studies entirely abstract concepts.}}
{{glossary end}}


==Q==
{{glossary}}
{{term|[[Physics:Quantum calculus|Quantum calculus]]}}
{{defn|a form of calculus without the notion of [[Limit (mathematics)|limit]]s.}}
{{term|[[Physics:Quantum geometry|Quantum geometry]]}}
{{defn|the generalization of concepts of geometry used to describe the [[Physics:Physics|physical]] phenomena of quantum physics}}
{{term|[[Quaternionic analysis]]}}
{{glossary end}}
==R==
{{glossary}}
{{term|[[Ramsey theory]]}}
{{defn|the study of the conditions in which order must appear. It is named after [[Biography:Frank P. Ramsey|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|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|real number]]s and [[Function (mathematics)|function]]s of [[Real number|Real]] values. It provides a rigorous formulation of the calculus of [[Real number|real number]]s in terms of [[Continuous function|continuity]] and smoothness, whilst the theory is extended to the [[Complex number|complex number]]s in [[Complex analysis|complex analysis]].}}
{{term | Real Clifford algebra}}
{{term|Real K-theory}}
{{term|Recreational mathematics}}
{{defn|the area dedicated to mathematical puzzles and mathematical games.}}
{{term|Recursion theory}}
{{defn|see ''computability theory''}}
{{term|[[Representation theory]]}}
{{defn|a subfield of abstract algebra; it studies [[Algebraic structure|algebraic structure]]s by representing their elements as linear transformations of [[Vector space|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|[[Physics:Representation theory of the Galilean group|Representation theory of the Galilean group]]}}
{{term|[[Physics:Representation theory of the Lorentz group|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|topology]] studying [[Ribbon (mathematics)|ribbon]]s.}}
{{term|[[Ricci calculus]]}}
{{defn| {{ghat|Also called '''absolute differential calculus'''.}}
A foundation of [[Tensor calculus|tensor calculus]], developed by [[Biography:Gregorio Ricci-Curbastro|Gregorio Ricci-Curbastro]] in 1887–1896,<ref>{{Citation |last1=Ricci |first1=Gregorio |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  [[Physics:General relativity|general relativity]] and [[Differential geometry|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|differential geometry]] that is more specifically, the study of [[Riemannian manifold]]s. It is named after [[Biography:Bernhard Riemann|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|set theory]] based on [[Rough set|rough set]]s.}}
{{glossary end}}


==S==
{{glossary}}
{{term |[[Sampling (statistics)|Sampling theory]]}}
{{term |[[Scheme (mathematics)|Scheme theory]]}}
{{defn |the study of [[Scheme (mathematics)|scheme]]s introduced by [[Biography:Alexander Grothendieck|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|real algebraic geometry]] that studies [[Semialgebraic set|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={{wipe|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|category theory]]. As a theory, it is a subset of [[Synthetic differential geometry|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 spectra of canonically defined [[Differential operator|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 eigenvalues and eigenvectors from linear algebra and [[Matrix (mathematics)|matrix theory]].}}
{{term |Spectral theory of ordinary differential equations}}
{{defn |part of [[Spectral theory|spectral theory]] concerned with the spectrum and [[Eigenfunction|eigenfunction]] expansion associated with [[Linear differential equation|linear]] [[Ordinary differential equation|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|non-Euclidean geometry]], studying the 2-dimensional surface of a [[Sphere|sphere]].}}
{{term |[[Spherical trigonometry]]}}
{{defn |a branch of [[Spherical geometry|spherical geometry]] that studies [[Polygon|polygon]]s on the surface of a [[Sphere|sphere]]. Usually the [[Polygon|polygon]]s are [[Triangle|triangle]]s.}}
{{term |[[Physics:Statistical mechanics|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|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|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|geometric topology]] referring to methods used to produce one manifold from another (in a controlled way.)}}
{{term |[[Survey sampling]]}}
{{term |[[Engineering:Survey methodology|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|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|differential geometry]] and topology whose main object of study is the [[Physics:Symplectic manifold|symplectic manifold]].}}
{{term |Symplectic topology}}
{{term |[[Synthetic differential geometry]]}}
{{defn |a reformulation of [[Differential geometry|differential geometry]] in the language of topos theory and in the context of an [[Intuitionistic logic|intuitionistic logic]].}}
{{term |[[Synthetic geometry]]}}
{{defn |also known as '''axiomatic geometry''', it is a branch of geometry that uses [[Axiom|axiom]]s and logical arguments to draw conclusions as opposed to [[Analytic geometry|analytic]] and algebraic methods.}}
{{term |[[Systolic geometry]]}}
{{defn |a branch of [[Differential geometry|differential geometry]] studying systolic [[Invariant (mathematics)|invariant]]s of [[Manifold|manifold]]s and polyhedra.}}
{{term |[[Systolic geometry|Systolic hyperbolic geometry]]}}
{{defn |the study of [[Systolic geometry|systole]]s in [[Hyperbolic geometry|hyperbolic geometry]].}}
{{glossary end}}


==T==
{{glossary}}
{{term |tensor |[[Tensor|Tensor algebra, Tensor analysis, Tensor calculus, Tensor theory]]}}
{{defn |the study and use of [[Tensor|tensor]]s, which are generalizations of [[Vector (mathematics and physics)|vectors]]. A [[Tensor algebra|tensor algebra]] is also an [[Algebraic structure|algebraic structure]] that is used in the formal definition of tensors.}}
{{term |[[Tessellation]]}}
{{defn |when periodic tiling has a repeating pattern.}}
{{term |[[Physics:Theoretical physics|Theoretical physics]]}}
{{defn |a branch primarily of the [[Science|science]] [[Physics:Physics|physics]] that uses [[Mathematical model|mathematical model]]s and [[Philosophy:Abstraction|abstraction]] of [[Physics: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|number theory]] that revolves around the [[Transcendental number|transcendental number]]s.}}
{{term |[[Transformation geometry]]}}
{{term |[[Trigonometry]]}}
{{defn |the study of [[Triangle|triangle]]s and the relationships between the [[Physics:Length|length]] of their sides, and the [[Angle|angle]]s between them. It is essential to many parts of [[Applied mathematics|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|operator theory]].}}
{{term |[[Type theory]]}}
{{glossary end}}


==U==
{{glossary}}
{{term |[[Umbral calculus]]}}
{{defn |the study of [[Sheffer sequence]]s}}
{{term |[[Uncertainty theory]]}}
{{defn |a new branch of [[Mathematics|mathematics]] based on normality, monotonicity, self-duality, countable subadditivity, and product measure [[Axiom|axiom]]s.}}
{{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==
{{glossary}}
{{term |Valuation theory}}
{{term |[[Variational analysis]]}}
{{term |Vector algebra}}
{{defn |a part of linear algebra concerned with the [[Operation (mathematics)|operation]]s of vector addition and scalar [[Multiplication|multiplication]], although it may also refer to vector [[Operation (mathematics)|operation]]s of [[Vector calculus|vector calculus]], including the [[Dot product|dot]] and [[Cross product|cross product]]. In this case it can be contrasted with [[Geometric algebra|geometric algebra]] which generalizes into higher dimensions.}}
{{term |Vector analysis}}
{{defn |also known as [[Vector calculus|vector calculus]], see ''vector calculus''.}}
{{term |[[Vector calculus]]}}
{{defn |a branch of [[Multivariable calculus|multivariable calculus]] concerned with [[Derivative|differentiation]] and [[Integral|integration]] of [[Vector field|vector field]]s. Primarily it is concerned with 3-dimensional [[Euclidean space]].}}
{{glossary end}}

==W==
{{glossary}}
{{term |[[Wavelet]]s}}
{{glossary end}}

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

==References== 
{{reflist}}

{{Glossaries of science and engineering}}

[[Category:Fields of mathematics|*]]
[[Category:Glossaries of mathematics|Areas of mathematics]]

{{Sourceattribution|Glossary of areas of mathematics}}