Mathematics is a broad subject that is commonly divided in manyareas orbranches that may be defined bytheir objects of study , by the used methods, or by both. For example,analytic number theory is a subarea ofnumber theory devoted to the use of methods ofanalysis for the study ofnatural numbers .
This glossary is alphabetically sorted. This hides a large part of the relationships between areas. For the broadest areas of mathematics, seeMathematics § Areas of mathematics . TheMathematics 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.
Absolute differential calculus An older name ofRicci calculus Absolute geometry Also calledneutral geometry [ 1] synthetic geometry similar toEuclidean geometry but without theparallel postulate .[ 2] Abstract algebra The part ofalgebra algebraic structures in themselves.[ 3] modern algebra Abstract analytic number theory The study ofarithmetic semigroups as a means to extend notions fromclassical analytic number theory .[ 4] Abstract differential geometry A form ofdifferential geometry without the notion ofsmoothness fromcalculus . Instead it is built usingsheaf theory andsheaf cohomology . Abstract harmonic analysis A modern branch ofharmonic analysis that extends upon the generalizedFourier transforms that can be defined onlocally compact groups . Abstract homotopy theory A part oftopology 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 appliesmathematical andstatistical methods toassess risk ininsurance ,finance and other industries and professions. More generally, actuaries apply rigorous mathematics to model matters of uncertainty. Additive combinatorics The part ofarithmetic combinatorics devoted to the operations ofaddition andsubtraction . Additive number theory A part ofnumber theory that studies subsets ofintegers and their behaviour under addition. Affine geometry A branch ofgeometry that deals with properties that are independent from distances and angles, such asalignment andparallelism . Affine geometry of curves The study ofcurve properties that are invariant underaffine transformations . Affine differential geometry A type ofdifferential geometry dedicated to differentialinvariants undervolume -preservingaffine transformations . Ahlfors theory A part ofcomplex analysis being the geometric counterpart ofNevanlinna theory . It was invented byLars Ahlfors . Algebra One of the majorareas of mathematics . Roughly speaking, it is the art of manipulating and computing withoperations acting on symbols calledvariables that represent indeterminatenumbers or othermathematical objects , such asvectors ,matrices , or elements ofalgebraic structures . Algebraic analysis motivated by systems oflinear partial differential equations , it is a branch ofalgebraic geometry andalgebraic topology that uses methods fromsheaf theory and complex analysis, to study the properties and generalizations offunctions . It was started byMikio Sato . Algebraic combinatorics an area that employs methods of abstract algebra to problems ofcombinatorics . It also refers to the application of methods from combinatorics to problems in abstract algebra. Algebraic computation An older name ofcomputer algebra . Algebraic geometry a branch that combines techniques from abstract algebra with the language and problems of geometry. Fundamentally, it studiesalgebraic varieties . Algebraic graph theory a branch ofgraph theory in which methods are taken from algebra and employed to problems aboutgraphs . The methods are commonly taken fromgroup theory and linear algebra. Algebraic K-theory an important part ofhomological algebra concerned with defining and applying a certain sequence offunctors fromrings toabelian groups . Algebraic number theory The part ofnumber theory commutative algebra number fields and theirrings of integers . Algebraic statistics the use of algebra to advancestatistics , although the term is sometimes restricted to label the use of algebraic geometry andcommutative algebra instatistics . Algebraic topology a branch that uses tools fromabstract algebra fortopology to studytopological spaces . Algorithmic number theory also known ascomputational number theory , it is the study ofalgorithms for performingnumber theoretic computations . Anabelian geometry an area of study based on the theory proposed byAlexander Grothendieck in the 1980s that describes the way a geometric object of analgebraic variety (such as analgebraic fundamental group ) can be mapped into another object, without it being anabelian group . Analysis A wide area of mathematics centered on the study ofcontinuous functions and including such topics asdifferentiation ,integration ,limits , andseries .[ 5] Analytic combinatorics part ofenumerative combinatorics where methods of complex analysis are applied togenerating functions . Analytic geometry 1. Also known asCartesian geometry Euclidean geometry Cartesian coordinates . 2. Analogue todifferential geometry differentiable functions are replaced withanalytic functions . It is a subarea of bothcomplex analysis algebraic geometry Analytic number theory An area ofnumber theory that applies methods frommathematical analysis to solve problems aboutintegers .[ 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 forscience ,engineering ,finance ,economics andlogistics . Approximation theory part ofanalysis that studies how well functions can be approximated by simpler ones (such aspolynomials ortrigonometric polynomials ) Arakelov geometry also known asArakelov theory Arakelov theory an approach toDiophantine geometry used to studyDiophantine equations in higher dimensions (using techniques from algebraic geometry). It is named afterSuren Arakelov . Arithmetic 1. Also known aselementary arithmetic addition ,subtraction ,multiplication anddivision of numbers. 2. Also known ashigher arithmetic number theory . Arithmetic algebraic geometry Seearithmetic geometry Arithmetic combinatorics the study of the estimates fromcombinatorics that are associated witharithmetic operations such as addition,subtraction ,multiplication anddivision . Arithmetic dynamics Arithmetic dynamics is the study of the number-theoretic properties ofinteger ,rational ,p -adic, and/or algebraic points under repeated application of apolynomial orrational function . A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures. Arithmetic geometry The use ofalgebraic geometry and more speciallyscheme theory for solving problems of number theory. Arithmetic topology a combination ofalgebraic number theory andtopology studying analogies betweenprime ideals andknots Arithmetical algebraic geometry Another name forarithmetic algebraic geometry Asymptotic combinatorics It uses the internal structure of the objects to derive formulas for theirgenerating functions and then complex analysis techniques to get asymptotics. Asymptotic theory the study ofasymptotic expansions Auslander–Reiten theory the study of therepresentation theory ofArtinian rings Axiomatic geometry also known assynthetic geometry axioms andlogical arguments to draw conclusions as opposed toanalytic and algebraic methods. Axiomatic set theory the study of systems ofaxioms in a context relevant toset theory andmathematical logic . Bifurcation theory the study of changes in the qualitative or topological structure of a given family. It is a part ofdynamical systems theory Biostatistics the development and application ofstatistical methods to a wide range of topics inbiology . Birational geometry a part ofalgebraic geometry that deals with the geometry (of an algebraic variety) that is dependent only on itsfunction field . Bolyai–Lobachevskian geometry seehyperbolic geometry C*-algebra theory acomplex algebra A ofcontinuous linear operators on acomplex Hilbert space with two additional properties-(i)A is a topologicallyclosed set in thenorm topology of operators.(ii)A is closed under the operation of takingadjoints of operators. Cartesian geometry seeanalytic geometry Calculus An area of mathematics connected by thefundamental theorem of calculus .[ 7] Calculus of infinitesimals Also calledinfinitesimal calculus
A foundation ofcalculus , first developed in the 17th century,[ 8] infinitesimal numbers. Calculus of moving surfaces an extension of the theory oftensor calculus to include deformingmanifolds . Calculus of variations the field dedicated to maximizing or minimizingfunctionals . It used to be calledfunctional calculus . Catastrophe theory a branch ofbifurcation theory fromdynamical systems theory , and also a special case of the more generalsingularity theory from geometry. It analyses thegerms of the catastrophe geometries. Categorical logic a branch ofcategory theory adjacent to themathematical logic . It is based ontype theory forintuitionistic 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 ofdynamical systems that are highly sensitive to their initial conditions. Character theory a branch ofgroup theory that studies the characters ofgroup representations ormodular representations . Class field theory a branch ofalgebraic number theory that studiesabelian extensions ofnumber fields . Classical differential geometry also known asEuclidean differential geometry . seeEuclidean differential geometry . Classical algebraic topology seealgebraic topology Classical analysis usually refers to the more traditional topics of analysis such asreal analysis and complex analysis. It includes any work that does not use techniques fromfunctional analysis and is sometimes calledhard analysis . However it may also refer to mathematical analysis done according to the principles ofclassical mathematics . Classical analytic number theory Classical differential calculus Classical Diophantine geometry Classical Euclidean geometry seeEuclidean geometry Classical geometry may refer tosolid geometry or classical Euclidean geometry. Seegeometry Classical invariant theory the form ofinvariant theory that deals with describingpolynomial functions that areinvariant under transformations from a givenlinear group . Classical mathematics the standard approach to mathematics based onclassical logic andZFC set theory . Classical projective geometry Classical tensor calculus Clifford algebra Clifford analysis the study ofDirac operators andDirac type operators from geometry and analysis usingclifford algebras . Clifford theory is a branch ofrepresentation theory spawned fromCliffords theorem . Cobordism theory Coding theory the study of the properties ofcodes and their respective fitness for specific applications. Cohomology theory Combinatorial analysis Combinatorial commutative algebra a discipline viewed as the intersection betweencommutative 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 ofsystems of finite sets whose intersections have certain properties. Combinatorial game theory Combinatorial geometry seediscrete geometry Combinatorial group theory the theory offree groups and thepresentation of a group . It is closely related togeometric group theory and is applied ingeometric topology . Combinatorial mathematics an area primarily concerned with counting, both as a means and an end in obtaining results, and certain properties offinite structures . Combinatorial number theory Combinatorial optimization Combinatorial set theory also known asInfinitary combinatorics . seeinfinitary combinatorics Combinatorial theory Combinatorial topology an old name for algebraic topology, whentopological invariants of spaces were regarded as derived from combinatorial decompositions. Combinatorics a branch ofdiscrete mathematics concerned withcountable structures . Branches of it includeenumerative combinatorics ,combinatorial design theory ,matroid theory ,extremal combinatorics andalgebraic combinatorics , as well as many more. Commutative algebra a branch of abstract algebra studyingcommutative rings . Complex algebraic geometry the mainstream of algebraic geometry devoted to the study of thecomplex points ofalgebraic varieties . Complex analysis a part ofanalysis that deals with functions of acomplex variable. Complex analytic dynamics a subdivision ofcomplex dynamics being the study of thedynamic systems defined byanalytic functions . Complex analytic geometry the application of complex numbers toplane geometry . Complex differential geometry a branch ofdifferential geometry that studiescomplex manifolds . Complex dynamics the study ofdynamical systems defined byiterated functions on complexnumber spaces . Complex geometry the study ofcomplex manifolds and functions ofcomplex variables. It includescomplex algebraic geometry andcomplex analytic geometry . Complexity theory the study ofcomplex systems with the inclusion of the theory ofcomplex systems . Computable analysis the study of which parts ofreal analysis andfunctional analysis can be carried out in acomputable manner. It is closely related toconstructive analysis . Computable model theory a branch ofmodel theory dealing with the relevant questionscomputability . Computability theory a branch ofmathematical logic originating in the 1930s with the study ofcomputable functions andTuring degrees , but now includes the study of generalized computability and definability. It overlaps withproof theory andeffective descriptive set theory . Computational algebraic geometry Computational complexity theory a branch of mathematics andtheoretical computer science that focuses on classifyingcomputational problems according to their inherent difficulty, and relating thoseclasses to each other. Computational geometry a branch ofcomputer science devoted to the study of algorithms which can be stated in terms ofgeometry . Computational group theory the study ofgroups by means of computers. Computational mathematics the mathematical research in areas ofscience wherecomputing plays an essential role. Computational number theory also known asalgorithmic number theory , it is the study ofalgorithms for performingnumber theoretic computations . Computational statistics Computational synthetic geometry Computational topology Computer algebra seesymbolic computation Conformal geometry the study ofconformal transformations on a space. Constructive analysis mathematical analysis done according to the principles ofconstructive mathematics . This differs fromclassical analysis . Constructive function theory a branch of analysis that is closely related toapproximation theory , studying the connection between thesmoothness of a function and itsdegree of approximation Constructive mathematics mathematics which tends to useintuitionistic logic . Essentially that is classical logic but without the assumption that thelaw of the excluded middle is anaxiom . Constructive quantum field theory a branch ofmathematical physics that is devoted to showing thatquantum theory is mathematically compatible withspecial relativity . Constructive set theory an approach tomathematical constructivism following the program ofaxiomatic set theory , using the usualfirst-order language of classical set theory. Contact geometry a branch ofdifferential geometry andtopology , closely related to and considered the odd-dimensional counterpart ofsymplectic geometry . It is the study of a geometric structure called a contact structure on adifferentiable manifold . Convex analysis the study of properties ofconvex functions andconvex sets . Convex geometry part of geometry devoted to the study ofconvex sets . Coordinate geometry seeanalytic geometry CR geometry a branch ofdifferential geometry , being the study ofCR manifolds . Cryptography Decision analysis Decision theory Derived noncommutative algebraic geometry Descriptive set theory a part ofmathematical logic , more specifically a part ofset theory dedicated to the study ofPolish spaces . Differential algebraic geometry the adaption of methods and concepts from algebraic geometry to systems ofalgebraic differential equations . Differential calculus A branch ofcalculus that's contrasted tointegral calculus ,[ 9] derivatives .[ 10] Differential Galois theory the study of theGalois groups ofdifferential fields . Differential geometry a form of geometry that uses techniques fromintegral anddifferential calculus as well aslinear andmultilinear 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 ondifferentiable manifolds . It is closely related to differential topology. Differential geometry of curves the study ofsmooth curves inEuclidean space by using techniques fromdifferential geometry Differential geometry of surfaces the study ofsmooth surfaces with various additional structures using the techniques ofdifferential geometry . Differential topology a branch oftopology that deals withdifferentiable functions ondifferentiable manifolds . Diffiety theory Diophantine geometry in general the study of algebraic varieties overfields that are finitely generated over theirprime fields . Discrepancy theory Discrete differential geometry Discrete exterior calculus Discrete geometry a branch ofgeometry that studiescombinatorial properties and constructive methods ofdiscrete geometric objects. Discrete mathematics the study ofmathematical structures that are fundamentallydiscrete rather thancontinuous . Discrete Morse theory acombinatorial adaption ofMorse theory . Distance geometry Domain theory a branch that studies special kinds ofpartially ordered sets (posets) commonly called domains. Donaldson theory the study of smooth4-manifolds usinggauge theory . Dyadic algebra Dynamical systems theory an area used to describe the behavior of thecomplex dynamical systems , usually by employingdifferential equations ordifference equations . Econometrics the application of mathematical andstatistical methods toeconomic data . Effective descriptive set theory a branch ofdescriptive set theory dealing withset ofreal numbers that havelightface definitions. It uses aspects ofcomputability theory . Elementary algebra a fundamental form ofalgebra extending onelementary arithmetic to include the concept ofvariables . Elementary arithmetic the simplified portion of arithmetic considered necessary forprimary education . It includes the usage addition,subtraction ,multiplication anddivision of thenatural numbers . It also includes the concept offractions andnegative numbers . Elementary mathematics parts of mathematics frequently taught at theprimary andsecondary school levels. This includeselementary arithmetic , geometry,probability andstatistics ,elementary algebra andtrigonometry . (calculus is not usually considered a part) Elementary group theory the study of the basics ofgroup theory Elimination theory the classical name for algorithmic approaches to eliminating betweenpolynomials of several variables. It is a part ofcommutative algebra and algebraic geometry. Elliptic geometry a type ofnon-Euclidean geometry (it violatesEuclid 'sparallel postulate ) and is based onspherical geometry . It is constructed inelliptic 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 ofintersection theory . Epidemiology Equivariant noncommutative algebraic geometry Ergodic Ramsey theory a branch where problems are motivated byadditive combinatorics and solved usingergodic theory . Ergodic theory the study ofdynamical systems with aninvariant measure , and related problems. Euclidean geometry An area ofgeometry based on theaxiom system andsynthetic methods of the ancient Greek mathematicianEuclid .[ 11] Euclidean differential geometry also known asclassical differential geometry . Seedifferential geometry . Euler calculus a methodology from appliedalgebraic topology andintegral geometry that integratesconstructible functions and more recentlydefinable functions by integrating with respect to theEuler characteristic as a finitely-additivemeasure . 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. Field theory The branch ofalgebra dedicated tofields , a type ofalgebraic structure .[ 12] Finite geometry Finite model theory a restriction ofmodel theory tointerpretations on finitestructures , which have a finite universe. Finsler geometry a branch ofdifferential geometry whose main object of study isFinsler manifolds , a generalisation of aRiemannian manifolds . First order arithmetic Fourier analysis the study of the way generalfunctions may be represented or approximated by sums oftrigonometric functions . Fractal geometry Fractional calculus a branch of analysis that studies the possibility of takingreal or complex powers of thedifferentiation operator . Fractional dynamics investigates the behaviour of objects and systems that are described bydifferentiation andintegration offractional orders using methods offractional calculus . Fredholm theory part ofspectral theory studyingintegral equations . Function theory an ambiguous term that generally refers tomathematical analysis . Functional analysis a branch ofmathematical analysis , the core of which is formed by the study offunction spaces , which are some sort oftopological vector spaces . Functional calculus historically the term was used synonymously withcalculus of variations , but now refers to a branch offunctional analysis connected withspectral theory Fuzzy mathematics a branch of mathematics based onfuzzy set theory andfuzzy logic . Fuzzy measure theory Fuzzy set theory a form ofset theory that studiesfuzzy sets , that issets that have degrees of membership. Galois cohomology an application ofhomological algebra , it is the study ofgroup cohomology ofGalois modules . Galois theory named afterÉvariste Galois , it is a branch of abstract algebra providing a connection betweenfield theory andgroup theory . Galois geometry a branch offinite geometry concerned with algebraic andanalytic geometry over aGalois field . Game theory the study ofmathematical models of strategic interaction among rational decision-makers. Gauge theory General topology also known aspoint-set topology , it is a branch oftopology studying the properties oftopological spaces and structures defined on them. It differs from other branches oftopology as thetopological spaces do not have to be similar to manifolds. Generalized trigonometry developments oftrigonometric methods from the application toreal numbers of Euclidean geometry to any geometry orspace . This includesspherical trigonometry ,hyperbolic trigonometry ,gyrotrigonometry , anduniversal hyperbolic trigonometry . Geometric algebra an alternative approach to classical,computational andrelativistic geometry . It shows a natural correspondence between geometric entities and elements of algebra. Geometric analysis a discipline that uses methods fromdifferential geometry to studypartial differential equations as well as the applications to geometry. Geometric calculus extends thegeometric algebra to includedifferentiation andintegration . Geometric combinatorics a branch ofcombinatorics . It includes a number of subareas such aspolyhedral combinatorics (the study offaces ofconvex polyhedra ),convex geometry (the study ofconvex sets , in particular combinatorics of their intersections), anddiscrete geometry , which in turn has many applications tocomputational geometry . Geometric function theory the study of geometric properties ofanalytic functions . Geometric invariant theory a method for constructing quotients bygroup actions inalgebraic geometry , used to constructmoduli spaces . Geometric graph theory a large and amorphous subfield ofgraph theory , concerned with graphs defined by geometric means. Geometric group theory the study offinitely generated groups via exploring the connections between algebraic properties of such groups andtopological andgeometric properties of spaces on which these groupsact (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces). Geometric measure theory the study ofgeometric properties ofsets (typically inEuclidean space ) throughmeasure theory . Geometric number theory Geometric topology a branch oftopology studying manifolds and mappings between them; in particular theembedding of one manifold into another. Geometry a branch of mathematics concerned withshape and the properties ofspace . Classically it arose as what is now known assolid geometry ; this was concerning practical knowledge oflength ,area andvolume . It was then put into anaxiomatic form byEuclid , giving rise to what is now known as classical Euclidean geometry. The use ofcoordinates byRené Descartes gave rise toCartesian geometry enabling a more analytical approach to geometric entities. Since then many other branches have appeared includingprojective geometry ,differential geometry ,non-Euclidean geometry ,Fractal geometry and algebraic geometry. Geometry also gave rise to the modern discipline oftopology . Geometry of numbers initiated byHermann Minkowski , it is a branch ofnumber theory studyingconvex bodies andinteger vectors . Global analysis the study ofdifferential equations on manifolds and the relationship betweendifferential equations andtopology . Global arithmetic dynamics Graph theory a branch ofdiscrete mathematics devoted to the study ofgraphs . It has many applications inphysical ,biological andsocial systems. Group-character theory the part of character theory dedicated to the study of characters ofgroup representations . Group representation theory Group theory the study ofalgebraic structures known asgroups . Gyrotrigonometry a form oftrigonometry used ingyrovector space forhyperbolic geometry . (An analogy of thevector space in Euclidean geometry.) Hard analysis seeclassical analysis Harmonic analysis part of analysis concerned with the representations offunctions in terms ofwaves . It generalizes the notions ofFourier series andFourier transforms from theFourier analysis . Higher arithmetic Higher category theory the part ofcategory theory at ahigher order , which means that some equalities are replaced by explicitarrows in order to be able to explicitly study the structure behind those equalities. Higher-dimensional algebra the study ofcategorified structures. Hodge theory a method for studying thecohomology groups of asmooth manifold M usingpartial differential equations . Hodge–Arakelov theory Holomorphic functional calculus a branch offunctional calculus starting withholomorphic functions . Homological algebra the study ofhomology in general algebraic settings. Homology theory Homotopy theory Hyperbolic geometry also known asLobachevskian geometry orBolyai-Lobachevskian geometry . It is anon-Euclidean geometry looking athyperbolic space . hyperbolic trigonometry the study ofhyperbolic triangles inhyperbolic geometry , orhyperbolic functions in Euclidean geometry. Other forms includegyrotrigonometry anduniversal hyperbolic trigonometry . Hypercomplex analysis the extension ofreal analysis andcomplex analysis to the study of functions where theargument is ahypercomplex number . Hyperfunction theory Ideal theory once the precursor name for what is now known ascommutative algebra ; it is the theory ofideals incommutative rings . Idempotent analysis the study ofidempotent semirings , such as thetropical semiring . Incidence geometry the study of relations ofincidence between various geometric objects, likecurves andlines . Inconsistent mathematics seeparaconsistent mathematics . Infinitary combinatorics an expansion of ideas in combinatorics to account forinfinite sets . Infinitesimal analysis once a synonym forinfinitesimal calculus Infinitesimal calculus Seecalculus of infinitesimals Information geometry an interdisciplinary field that applies the techniques ofdifferential geometry to studyprobability theory andstatistics . It studiesstatistical manifolds , which areRiemannian manifolds whose points correspond toprobability distributions . Integral calculus Integral geometry the theory ofmeasures on a geometrical space invariant under thesymmetry group of that space. Intersection theory a branch of algebraic geometry and algebraic topology Intuitionistic type theory atype theory and an alternativefoundation of mathematics . Invariant theory studies howgroup 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 tostochastic processes such asBrownian motion (seeWiener process ). It has important applications inmathematical finance andstochastic differential equations . Iwasawa theory the study of objects of arithmetic interest over infinitetowers ofnumber fields . Iwasawa-Tate theory Job shop scheduling K-theory originated as the study of aring generated byvector bundles over atopological space orscheme . In algebraic topology it is anextraordinary cohomology theory known astopological K-theory . In algebra and algebraic geometry it is referred to asalgebraic K-theory . Inphysics ,K-theory has appeared intype II string theory . (In particulartwisted K-theory .) K-homology ahomology theory on thecategory of locallycompact Hausdorff spaces . Kähler geometry a branch ofdifferential geometry , more specifically a union ofRiemannian geometry ,complex differential geometry andsymplectic geometry . It is the study ofKähler manifolds . (named afterErich Kähler ) KK-theory a common generalization both ofK-homology andK-theory as an additivebivariant functor onseparable C*-algebras . Klein geometry More specifically, it is ahomogeneous space X together with atransitive action onX by aLie group G , which acts as thesymmetry group of the geometry. Knot theory part oftopology dealing withknots Kummer theory provides a description of certain types offield extensions involving theadjunction ofn th roots of elements of the basefield L-theory theK-theory ofquadratic forms . Large deviations theory part ofprobability theory studyingevents of small probability (tail events ). Large sample theory also known asasymptotic theory Lattice theory the study oflattices , being important inorder theory anduniversal algebra Lie algebra theory Lie group theory Lie sphere geometry geometrical theory ofplanar orspatial geometry in which the fundamental concept is thecircle orsphere .Lie theory Line geometry Linear algebra a branch of algebra studyinglinear spaces andlinear maps . It has applications in fields such as abstract algebra andfunctional analysis ; it can be represented in analytic geometry and it is generalized inoperator theory and inmodule theory . Sometimesmatrix theory is considered a branch, although linear algebra is restricted to only finite dimensions. Extensions of the methods used belong tomultilinear algebra . Linear functional analysis Linear programming a method to achieve the best outcome (such as maximum profit or lowest cost) in amathematical model whose requirements are represented bylinear 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 oflocal rings . Local class field theory the study ofabelian extensions oflocal fields . Low-dimensional topology the branch oftopology that studiesmanifolds , or more generally topological spaces, of four or fewerdimensions . Malliavin calculus a set of mathematical techniques and ideas that extend the mathematical field ofcalculus of variations from deterministic functions tostochastic processes . Mathematical biology themathematical modeling of biological phenomena. Mathematical chemistry themathematical modeling of chemical phenomena. Mathematical economics the application of mathematical methods to represent theories and analyze problems ineconomics . Mathematical finance a field ofapplied mathematics , concerned with mathematical modeling offinancial markets . Mathematical logic a subfield ofmathematics exploring the applications of formallogic to mathematics. Mathematical optimization Mathematical physics The development of mathematical methods suitable for application to problems inphysics .[ 13] Mathematical psychology an approach topsychological research that is based onmathematical 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 toacademic disciplines that are mathematical in nature, but are not considered proper subfields of mathematics. Examples includestatistics ,cryptography ,game theory andactuarial science . Mathematical sociology the area of sociology that uses mathematics to construct social theories. Mathematical statistics the application ofprobability theory , a branch ofmathematics , tostatistics , 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 mathematicalstructures (e.g.groups ,fields ,graphs , universes ofset theory ) from the perspective ofmathematical logic . Modern algebra Occasionally used forabstract algebra van der Waerden as the title of his bookModerne Algebra Algebra in the latest editions. Modern algebraic geometry the form of algebraic geometry given byAlexander Grothendieck andJean-Pierre Serre drawing onsheaf theory . Modern invariant theory the form ofinvariant theory that analyses the decomposition ofrepresentations into irreducibles. Modular representation theory a part ofrepresentation theory that studieslinear representations offinite groups over afield K of positivecharacteristic p , necessarily a prime number. Module theory Molecular geometry Morse theory a part of differential topology, it analyzes thetopological space of a manifold by studyingdifferentiable functions on that manifold. Motivic cohomology Multilinear algebra an extension of linear algebra building upon concepts ofp-vectors andmultivectors withGrassmann algebra . Multiplicative number theory a subfield of analytic number theory that deals withprime numbers ,factorization anddivisors . Multivariable calculus the extension ofcalculus in onevariable to calculus withfunctions of several variables : thedifferentiation andintegration of functions involving several variables, rather than just one. Multiple-scale analysis Neutral geometry Seeabsolute geometry Nevanlinna theory part of complex analysis studying the value distribution ofmeromorphic functions . It is named afterRolf Nevanlinna Nielsen theory an area of mathematical research with its origins infixed point topology , developed byJakob Nielsen Non-abelian class field theory Non-classical analysis Non-Euclidean geometry Non-standard analysis Non-standard calculus Nonarchimedean dynamics also known asp-adic analysis orlocal arithmetic dynamics Noncommutative algebra Noncommutative algebraic geometry a direction innoncommutative geometry studying the geometric properties of formal duals of non-commutative algebraic objects. Noncommutative geometry Noncommutative harmonic analysis seerepresentation theory Noncommutative topology Nonlinear analysis Nonlinear functional analysis Number theory a branch ofpure mathematics primarily devoted to the study of theintegers . Originally it was known asarithmetic orhigher arithmetic . Numerical analysis Numerical linear algebra Operad theory a type of abstract algebra concerned with prototypicalalgebras . Operation research Operator K-theory Operator theory part offunctional analysis studyingoperators . Optimal control theory a generalization of thecalculus of variations . Optimal maintenance Orbifold theory Order theory a branch that investigates the intuitive notion oforder usingbinary relations . Ordered geometry a form of geometry omitting the notion ofmeasurement but featuring the concept ofintermediacy . It is a fundamental geometry forming a common framework foraffine geometry , Euclidean geometry,absolute geometry andhyperbolic geometry . Oscillation theory p-adic analysis a branch ofnumber theory that deals with the analysis of functions ofp-adic numbers . p-adic dynamics an application ofp-adic analysis looking atp-adic differential equations . p-adic Hodge theory Parabolic geometry Paraconsistent mathematics sometimes calledinconsistent mathematics , it is an attempt to develop the classical infrastructure of mathematics based on a foundation ofparaconsistent logic instead ofclassical logic . Partition theory Perturbation theory Picard–Vessiot theory Plane geometry Point-set topology seegeneral topology Pointless topology Poisson geometry Polyhedral combinatorics a branch within combinatorics anddiscrete geometry that studies the problems of describingconvex 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 areinvariant under aprojective transformation . Projective differential geometry Proof theory Pseudo-Riemannian geometry generalizesRiemannian geometry to the study ofpseudo-Riemannian manifolds . Pure mathematics the part of mathematics that studies entirely abstract concepts. Quantum calculus a form of calculus without the notion oflimits . Quantum geometry the generalization of concepts of geometry used to describe thephysical phenomena ofquantum physics Quaternionic analysis Ramsey theory the study of the conditions in which order must appear. It is named afterFrank P. Ramsey . Rational geometry Real algebra the study of the part of algebra relevant toreal algebraic geometry . Real algebraic geometry the part of algebraic geometry that studiesreal points of the algebraic varieties. Real analysis a branch of mathematical analysis; in particularhard analysis , that is the study ofreal numbers andfunctions ofReal values. It provides a rigorous formulation of the calculus ofreal numbers in terms ofcontinuity andsmoothness , whilst the theory is extended to thecomplex numbers incomplex analysis . Real Clifford algebra Real K-theory Recreational mathematics the area dedicated tomathematical puzzles andmathematical games . Recursion theory seecomputability theory Representation theory a subfield of abstract algebra; it studiesalgebraic structures by representing their elements aslinear transformations ofvector spaces . It also studiesmodules 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 oftopology studyingribbons . Ricci calculus Also calledabsolute differential calculus .
A foundation oftensor calculus , developed byGregorio Ricci-Curbastro in 1887–1896,[ 14] general relativity anddifferential geometry .[ 15] Ring theory Riemannian geometry a branch ofdifferential geometry that is more specifically, the study ofRiemannian manifolds . It is named afterBernhard Riemann and it features many generalizations of concepts from Euclidean geometry, analysis and calculus. Rough set theory the a form ofset theory based onrough sets . Sampling theory Scheme theory the study ofschemes introduced byAlexander Grothendieck . It allows the use ofsheaf theory to study algebraic varieties and is considered the central part ofmodern algebraic geometry . Secondary calculus Semialgebraic geometry a part of algebraic geometry; more specifically a branch ofreal algebraic geometry that studiessemialgebraic sets . Set-theoretic topology Set theory Sheaf theory The study ofsheaves , which connect local and global properties ofgeometric objects .[ 16] Sheaf cohomology Sieve theory Single operator theory deals with the properties and classifications of singleoperators . Singularity theory a branch, notably of geometry; that studies the failure of manifold structure. Smooth infinitesimal analysis a rigorous reformation ofinfinitesimal calculus employing methods ofcategory theory . As a theory, it is a subset ofsynthetic differential geometry . Solid geometry Spatial geometry Spectral geometry a field that concerns the relationships between geometric structures of manifolds andspectra of canonically defineddifferential operators . Spectral graph theory the study of properties of agraph using methods frommatrix theory . Spectral theory part ofoperator theory extending the concepts ofeigenvalues andeigenvectors from linear algebra andmatrix theory . Spectral theory of ordinary differential equations part ofspectral theory concerned with thespectrum andeigenfunction expansion associated withlinear ordinary differential equations . Spectrum continuation analysis generalizes the concept of aFourier series to non-periodicfunctions . Spherical geometry a branch ofnon-Euclidean geometry , studying the 2-dimensional surface of asphere . Spherical trigonometry a branch ofspherical geometry that studiespolygons on the surface of asphere . Usually thepolygons aretriangles . Statistical mechanics Statistical modelling Statistical theory Statistics although the term may refer to the more general study ofstatistics , the term is used in mathematics to refer to themathematical study of statistics and related fields . This includesprobability 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 ofgeometric topology referring to methods used to produce one manifold from another (in a controlled way.) Survey sampling Survey methodology Symbolic computation also known asalgebraic computation andcomputer algebra . It refers to the techniques used to manipulatemathematical expressions andequations insymbolic form as opposed to manipulating them by the numerical quantities represented by them. Symbolic dynamics Symplectic geometry a branch ofdifferential geometry and topology whose main object of study is thesymplectic manifold . Symplectic topology Synthetic differential geometry a reformulation ofdifferential geometry in the language oftopos theory and in the context of anintuitionistic logic . Synthetic geometry also known asaxiomatic geometry , it is a branch of geometry that usesaxioms andlogical arguments to draw conclusions as opposed toanalytic and algebraic methods. Systolic geometry a branch ofdifferential geometry studying systolicinvariants ofmanifolds andpolyhedra . Systolic hyperbolic geometry the study ofsystoles inhyperbolic geometry . Tensor algebra, Tensor analysis, Tensor calculus, Tensor theory the study and use oftensors , which are generalizations ofvectors . Atensor algebra is also analgebraic structure that is used in theformal definition of tensors. Tessellation when periodic tiling has a repeating pattern. Theoretical physics a branch primarily of thescience physics that usesmathematical models andabstraction ofphysics to rationalize and predictphenomena . 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 ofnumber theory that revolves around thetranscendental numbers . Transformation geometry Trigonometry the study oftriangles and the relationships between thelength of their sides, and theangles between them. It is essential to many parts ofapplied mathematics . Tropical analysis seeidempotent analysis Tropical geometry Twisted K-theory a variation onK-theory , spanning abstract algebra, algebraic topology andoperator theory . Type theory Umbral calculus the study ofSheffer sequences Uncertainty theory a new branch ofmathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measureaxioms . Universal algebra a field studying the formalization of algebraic structures itself. Universal hyperbolic trigonometry an approach tohyperbolic trigonometry based onrational geometry . Valuation theory Variational analysis Vector algebra a part of linear algebra concerned with theoperations ofvector addition andscalar multiplication , although it may also refer tovector operations ofvector calculus , including thedot andcross product . In this case it can be contrasted withgeometric algebra which generalizes into higher dimensions. Vector analysis also known asvector calculus , seevector calculus . Vector calculus a branch ofmultivariable calculus concerned withdifferentiation andintegration ofvector fields . Primarily it is concerned with 3-dimensionalEuclidean space . Wavelets ^ Greenberg, Marvin Jay (2007),Euclidean and Non-Euclidean Geometries: Development and History (4th ed.), New York: W. H. Freeman,ISBN 978-0-7167-9948-1 ^ Faber, Richard L. (1983),Foundations of Euclidean and Non-Euclidean Geometry , New York: Marcel Dekker,ISBN 0-8247-1748-1 ^ Whitehead, C. (2002),Guide to Abstract Algebra (2nd ed.), Houndmills: Palgrave,ISBN 978-0-333-79447-0 ^ Knopfmacher, John (1990) [1975].Abstract Analytic Number Theory (2nd ed.). New York, NY: Dover Publishing.ISBN 0-486-66344-2 Zbl 0743.11002 . ^ Apostol, Tom M. Mathematical Analysis: A Modern Approach to Advanced Calculus (2 ed.). Addison-Wesley.^ 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 ^ DeBaggis, Henry F.; Miller, Kenneth S. (1966).Foundations of the Calculus . Philadelphia: Saunders.OCLC 527896 . ^ Boyer, Carl B. (1959).The History of the Calculus and its Conceptual Development OCLC 643872 .^ 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 ^ "Definition of DIFFERENTIAL CALCULUS" .www.merriam-webster.com . Retrieved2020-05-09 .^ Eves, Howard (1963).A Survey of Geometry (Volume One) . Allyn and Bacon. ^ Adamson, I. T. (2007),Introduction to Field Theory , Dover Publications,ISBN 978-0-486-46266-0 ^ Definition from theJournal of Mathematical Physics ."About the Journal" . Archived fromthe original on 2006-10-03. Retrieved2006-10-03 . ^ 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 54 (1– 2), Springer:125– 201,doi :10.1007/BF01454201 ,S2CID 120009332 , retrieved19 October 2019 ^ 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) ^ Tennison, Barry R. 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