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Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example,
analytic number theory is a subarea of
number theory devoted to the use of methods of
analysis for the study of
natural numbers .
This glossary is alphabetically sorted. This hides a large part of the relationships between areas. For the broadest areas of mathematics, see
Mathematics § Areas of mathematics . The
Mathematics Subject Classification is a hierarchical list of areas and subjects of study that has been elaborated by the community of mathematicians. It is used by most publishers for classifying mathematical articles and books.
A
Absolute differential calculus
An older name of
Ricci calculus
Absolute geometry
Also called
neutral geometry ,
[1] a
synthetic geometry similar to
Euclidean geometry but without the
parallel postulate .
[2]
Abstract algebra
The part of
algebra devoted to the study of
algebraic structures in themselves.
[3] Occasionally named
modern algebra in course titles.
Abstract analytic number theory
The study of
arithmetic semigroups as a means to extend notions from
classical analytic number theory .
[4]
Abstract differential geometry
A form of
differential geometry without the notion of
smoothness from
calculus . Instead it is built using
sheaf theory and
sheaf cohomology .
Abstract harmonic analysis
A modern branch of
harmonic analysis that extends upon the generalized
Fourier transforms that can be defined on
locally compact groups .
Abstract homotopy theory
A part of
topology that deals with homotopic functions, i.e. functions from one topological space to another which are homotopic (the functions can be deformed into one another).
Actuarial science
The discipline that applies
mathematical and
statistical methods to
assess risk in
insurance ,
finance and other industries and professions. More generally, actuaries apply rigorous mathematics to model matters of uncertainty.
Additive combinatorics
The part of
arithmetic combinatorics devoted to the operations of
addition and
subtraction .
Additive number theory
A part of
number theory that studies subsets of
integers and their behaviour under addition.
Affine geometry
A branch of
geometry that deals with properties that are independent from distances and angles, such as
alignment and
parallelism .
Affine geometry of curves
The study of
curve properties that are invariant under
affine transformations .
Affine differential geometry
A type of
differential geometry dedicated to differential
invariants under
volume -preserving
affine transformations .
Ahlfors theory
A part of
complex analysis being the geometric counterpart of
Nevanlinna theory . It was invented by
Lars Ahlfors .
Algebra
One of the major
areas of mathematics . Roughly speaking, it is the art of manipulating and computing with
operations acting on symbols called
variables that represent indeterminate
numbers or other
mathematical objects , such as
vectors ,
matrices , or elements of
algebraic structures .
Algebraic analysis
motivated by systems of
linear
partial differential equations , it is a branch of
algebraic geometry and
algebraic topology that uses methods from
sheaf theory and complex analysis, to study the properties and generalizations of
functions . It was started by
Mikio Sato .
Algebraic combinatorics
an area that employs methods of abstract algebra to problems of
combinatorics . It also refers to the application of methods from combinatorics to problems in abstract algebra.
Algebraic computation
An older name of
computer algebra .
Algebraic geometry
a branch that combines techniques from abstract algebra with the language and problems of geometry. Fundamentally, it studies
algebraic varieties .
Algebraic graph theory
a branch of
graph theory in which methods are taken from algebra and employed to problems about
graphs . The methods are commonly taken from
group theory and linear algebra.
Algebraic K-theory
an important part of
homological algebra concerned with defining and applying a certain sequence of
functors from
rings to
abelian groups .
Algebraic number theory
The part of
number theory devoted to the use of algebraic methods, mainly those of
commutative algebra , for the study of
number fields and their
rings of integers .
Algebraic statistics
the use of algebra to advance
statistics , although the term is sometimes restricted to label the use of algebraic geometry and
commutative algebra in
statistics .
Algebraic topology
a branch that uses tools from
abstract algebra for
topology to study
topological spaces .
Algorithmic number theory
also known as computational number theory , it is the study of
algorithms for performing
number theoretic
computations .
Anabelian geometry
an area of study based on the theory proposed by
Alexander Grothendieck in the 1980s that describes the way a geometric object of an
algebraic variety (such as an
algebraic fundamental group ) can be mapped into another object, without it being an
abelian group .
Analysis
A wide area of mathematics centered on the study of
continuous functions and including such topics as
differentiation ,
integration ,
limits , and
series .
[5]
Analytic combinatorics
part of
enumerative combinatorics where methods of complex analysis are applied to
generating functions .
Analytic geometry
1. Also known as
Cartesian geometry , the study of
Euclidean geometry using
Cartesian coordinates .
2. Analogue to
differential geometry , where
differentiable functions are replaced with
analytic functions . It is a subarea of both
complex analysis and
algebraic geometry .
Analytic number theory
An area of
number theory that applies methods from
mathematical analysis to solve problems about
integers .
[6]
Analytic theory of L-functions
Applied mathematics
a combination of various parts of mathematics that concern a variety of mathematical methods that can be applied to practical and theoretical problems. Typically the methods used are for
science ,
engineering ,
finance ,
economics and
logistics .
Approximation theory
part of
analysis that studies how well functions can be approximated by simpler ones (such as
polynomials or
trigonometric polynomials )
Arakelov geometry
also known as Arakelov theory
Arakelov theory
an approach to
Diophantine geometry used to study
Diophantine equations in higher dimensions (using techniques from algebraic geometry). It is named after
Suren Arakelov .
Arithmetic
1. Also known as
elementary arithmetic , the methods and rules for computing with
addition ,
subtraction ,
multiplication and
division of numbers.
2. Also known as
higher arithmetic , another name for
number theory .
Arithmetic algebraic geometry
See
arithmetic geometry .
Arithmetic combinatorics
the study of the estimates from
combinatorics that are associated with
arithmetic operations such as addition,
subtraction ,
multiplication and
division .
Arithmetic dynamics
Arithmetic dynamics is the study of the number-theoretic properties of
integer ,
rational , p -adic, and/or algebraic points under repeated application of a
polynomial or
rational function . A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.
Arithmetic geometry
The use of
algebraic geometry and more specially
scheme theory for solving problems of number theory.
Arithmetic topology
a combination of
algebraic number theory and
topology studying analogies between
prime ideals and
knots
Arithmetical algebraic geometry
Another name for
arithmetic algebraic geometry
Asymptotic combinatorics
It uses the internal structure of the objects to derive formulas for their
generating functions and then complex analysis techniques to get asymptotics.
Asymptotic theory
the study of
asymptotic expansions
Auslander–Reiten theory
the study of the
representation theory of
Artinian rings
Axiomatic geometry
also known as
synthetic geometry : it is a branch of geometry that uses
axioms and
logical arguments to draw conclusions as opposed to
analytic and algebraic methods.
Axiomatic set theory
the study of systems of
axioms in a context relevant to
set theory and
mathematical logic .
B
Bifurcation theory
the study of changes in the qualitative or topological structure of a given family. It is a part of
dynamical systems theory
Biostatistics
the development and application of
statistical methods to a wide range of topics in
biology .
Birational geometry
a part of
algebraic geometry that deals with the geometry (of an algebraic variety) that is dependent only on its
function field .
Bolyai–Lobachevskian geometry
see
hyperbolic geometry
C
C*-algebra theory
a
complex
algebra A of
continuous linear operators on a
complex
Hilbert space with two additional properties-(i) A is a topologically
closed set in the
norm topology of operators.(ii)A is closed under the operation of taking
adjoints of operators.
Cartesian geometry
see analytic geometry
Calculus
An area of mathematics connected by the
fundamental theorem of calculus .
[7]
Calculus of infinitesimals
Also called infinitesimal calculus
A foundation of
calculus , first developed in the 17th century,
[8] that makes use of
infinitesimal numbers.
Calculus of moving surfaces
an extension of the theory of
tensor calculus to include deforming
manifolds .
Calculus of variations
the field dedicated to maximizing or minimizing
functionals . It used to be called functional calculus .
Catastrophe theory
a branch of
bifurcation theory from
dynamical systems theory , and also a special case of the more general
singularity theory from geometry. It analyses the
germs of the catastrophe geometries.
Categorical logic
a branch of
category theory adjacent to the
mathematical logic . It is based on
type theory for
intuitionistic logics .
Category theory
the study of the properties of particular mathematical concepts by formalising them as collections of objects and arrows.
Chaos theory
the study of the behaviour of
dynamical systems that are highly sensitive to their initial conditions.
Character theory
a branch of
group theory that studies the characters of
group representations or
modular representations .
Class field theory
a branch of
algebraic number theory that studies
abelian extensions of
number fields .
Classical differential geometry
also known as
Euclidean differential geometry . see Euclidean differential geometry .
Classical algebraic topology
see algebraic topology
Classical analysis
usually refers to the more traditional topics of analysis such as
real analysis and complex analysis. It includes any work that does not use techniques from
functional analysis and is sometimes called hard analysis . However it may also refer to mathematical analysis done according to the principles of
classical mathematics .
Classical analytic number theory
Classical differential calculus
Classical Diophantine geometry
Classical Euclidean geometry
see Euclidean geometry
Classical geometry
may refer to
solid geometry or classical Euclidean geometry. See geometry
Classical invariant theory
the form of
invariant theory that deals with describing
polynomial functions that are
invariant under transformations from a given
linear group .
Classical mathematics
the standard approach to mathematics based on
classical logic and
ZFC set theory .
Classical projective geometry
Classical tensor calculus
Clifford algebra
Clifford analysis
the study of
Dirac operators and
Dirac type operators from geometry and analysis using
clifford algebras .
Clifford theory
is a branch of
representation theory spawned from
Cliffords theorem .
Cobordism theory
Coding theory
the study of the properties of
codes and their respective fitness for specific applications.
Cohomology theory
Combinatorial analysis
Combinatorial commutative algebra
a discipline viewed as the intersection between
commutative algebra and combinatorics. It frequently employs methods from one to address problems arising in the other.
Polyhedral geometry also plays a significant role.
Combinatorial design theory
a part of combinatorial mathematics that deals with the existence and construction of
systems of finite sets whose intersections have certain properties.
Combinatorial game theory
Combinatorial geometry
see discrete geometry
Combinatorial group theory
the theory of
free groups and the
presentation of a group . It is closely related to
geometric group theory and is applied in
geometric topology .
Combinatorial mathematics
an area primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of
finite
structures .
Combinatorial number theory
Combinatorial optimization
Combinatorial set theory
also known as
Infinitary combinatorics . see infinitary combinatorics
Combinatorial theory
Combinatorial topology
an old name for algebraic topology, when
topological invariants of spaces were regarded as derived from combinatorial decompositions.
Combinatorics
a branch of
discrete mathematics concerned with
countable
structures . Branches of it include
enumerative combinatorics ,
combinatorial design theory ,
matroid theory ,
extremal combinatorics and
algebraic combinatorics , as well as many more.
Commutative algebra
a branch of abstract algebra studying
commutative rings .
Complex algebraic geometry
the mainstream of algebraic geometry devoted to the study of the
complex points of
algebraic varieties .
Complex analysis
a part of
analysis that deals with functions of a
complex variable.
Complex analytic dynamics
a subdivision of
complex dynamics being the study of the
dynamic systems defined by
analytic functions .
Complex analytic geometry
the application of complex numbers to
plane geometry .
Complex differential geometry
a branch of
differential geometry that studies
complex manifolds .
Complex dynamics
the study of
dynamical systems defined by
iterated functions on complex
number spaces .
Complex geometry
the study of
complex manifolds and functions of
complex variables. It includes
complex algebraic geometry and
complex analytic geometry .
Complexity theory
the study of
complex systems with the inclusion of the theory of
complex systems .
Computable analysis
the study of which parts of
real analysis and
functional analysis can be carried out in a
computable manner. It is closely related to
constructive analysis .
Computable model theory
a branch of
model theory dealing with the relevant questions
computability .
Computability theory
a branch of
mathematical logic originating in the 1930s with the study of
computable functions and
Turing degrees , but now includes the study of generalized computability and definability. It overlaps with
proof theory and
effective descriptive set theory .
Computational algebraic geometry
Computational complexity theory
a branch of mathematics and
theoretical computer science that focuses on classifying
computational problems according to their inherent difficulty, and relating those
classes to each other.
Computational geometry
a branch of
computer science devoted to the study of algorithms which can be stated in terms of
geometry .
Computational group theory
the study of
groups by means of computers.
Computational mathematics
the mathematical research in areas of
science where
computing plays an essential role.
Computational number theory
also known as algorithmic number theory , it is the study of
algorithms for performing
number theoretic
computations .
Computational statistics
Computational synthetic geometry
Computational topology
Computer algebra
see symbolic computation
Conformal geometry
the study of
conformal transformations on a space.
Constructive analysis
mathematical analysis done according to the principles of
constructive mathematics . This differs from classical analysis .
Constructive function theory
a branch of analysis that is closely related to
approximation theory , studying the connection between the
smoothness of a function and its
degree of approximation
Constructive mathematics
mathematics which tends to use
intuitionistic logic . Essentially that is classical logic but without the assumption that the
law of the excluded middle is an
axiom .
Constructive quantum field theory
a branch of
mathematical physics that is devoted to showing that
quantum theory is mathematically compatible with
special relativity .
Constructive set theory
an approach to
mathematical constructivism following the program of
axiomatic set theory , using the usual
first-order language of classical set theory.
Contact geometry
a branch of
differential geometry and
topology , closely related to and considered the odd-dimensional counterpart of
symplectic geometry . It is the study of a geometric structure called a contact structure on a
differentiable manifold .
Convex analysis
the study of properties of
convex functions and
convex sets .
Convex geometry
part of geometry devoted to the study of
convex sets .
Coordinate geometry
see analytic geometry
CR geometry
a branch of
differential geometry , being the study of
CR manifolds .
Cryptography
D
Decision analysis
Decision theory
Derived noncommutative algebraic geometry
Descriptive set theory
a part of
mathematical logic , more specifically a part of
set theory dedicated to the study of
Polish spaces .
Differential algebraic geometry
the adaption of methods and concepts from algebraic geometry to systems of
algebraic differential equations .
Differential calculus
A branch of
calculus that's contrasted to
integral calculus ,
[9] and concerned with
derivatives .
[10]
Differential Galois theory
the study of the
Galois groups of
differential fields .
Differential geometry
a form of geometry that uses techniques from
integral and
differential calculus as well as
linear and
multilinear algebra to study problems in geometry. Classically, these were problems of Euclidean geometry, although now it has been expanded. It is generally concerned with geometric structures on
differentiable manifolds . It is closely related to differential topology.
Differential geometry of curves
the study of
smooth curves in
Euclidean space by using techniques from
differential geometry
Differential geometry of surfaces
the study of
smooth
surfaces with various additional structures using the techniques of
differential geometry .
Differential topology
a branch of
topology that deals with
differentiable functions on
differentiable manifolds .
Diffiety theory
Diophantine geometry
in general the study of algebraic varieties over
fields that are finitely generated over their
prime fields .
Discrepancy theory
Discrete differential geometry
Discrete exterior calculus
Discrete geometry
a branch of
geometry that studies
combinatorial properties and constructive methods of
discrete geometric objects.
Discrete mathematics
the study of
mathematical structures that are fundamentally
discrete rather than
continuous .
Discrete Morse theory
a
combinatorial adaption of
Morse theory .
Distance geometry
Domain theory
a branch that studies special kinds of
partially ordered sets (posets) commonly called domains.
Donaldson theory
the study of smooth
4-manifolds using
gauge theory .
Dyadic algebra
Dynamical systems theory
an area used to describe the behavior of the
complex
dynamical systems , usually by employing
differential equations or
difference equations .
E
Econometrics
the application of mathematical and
statistical methods to
economic
data .
Effective descriptive set theory
a branch of
descriptive set theory dealing with
set of
real numbers that have
lightface definitions. It uses aspects of
computability theory .
Elementary algebra
a fundamental form of
algebra extending on
elementary arithmetic to include the concept of
variables .
Elementary arithmetic
the simplified portion of arithmetic considered necessary for
primary education . It includes the usage addition,
subtraction ,
multiplication and
division of the
natural numbers . It also includes the concept of
fractions and
negative numbers .
Elementary mathematics
parts of mathematics frequently taught at the
primary and
secondary school levels. This includes
elementary arithmetic , geometry,
probability and
statistics ,
elementary algebra and
trigonometry . (calculus is not usually considered a part)
Elementary group theory
the study of the basics of
group theory
Elimination theory
the classical name for algorithmic approaches to eliminating between
polynomials of several variables. It is a part of
commutative algebra and algebraic geometry.
Elliptic geometry
a type of
non-Euclidean geometry (it violates
Euclid 's
parallel postulate ) and is based on
spherical geometry . It is constructed in
elliptic space .
Enumerative combinatorics
an area of combinatorics that deals with the number of ways that certain patterns can be formed.
Enumerative geometry
a branch of algebraic geometry concerned with counting the number of solutions to geometric questions. This is usually done by means of
intersection theory .
Epidemiology
Equivariant noncommutative algebraic geometry
Ergodic Ramsey theory
a branch where problems are motivated by
additive combinatorics and solved using
ergodic theory .
Ergodic theory
the study of
dynamical systems with an
invariant measure , and related problems.
Euclidean geometry
An area of
geometry based on the
axiom system and
synthetic methods of the ancient Greek mathematician
Euclid .
[11]
Euclidean differential geometry
also known as classical differential geometry . See differential geometry .
Euler calculus
a methodology from applied
algebraic topology and
integral geometry that integrates
constructible functions and more recently
definable functions by integrating with respect to the
Euler characteristic as a finitely-additive
measure .
Experimental mathematics
an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns.
Exterior algebra
Exterior calculus
Extremal combinatorics
a branch of combinatorics, it is the study of the possible sizes of a collection of finite objects given certain restrictions.
Extremal graph theory
a branch of mathematics that studies how global properties of a graph influence local substructure.
F
Field theory
The branch of
algebra dedicated to
fields , a type of
algebraic structure .
[12]
Finite geometry
Finite model theory
a restriction of
model theory to
interpretations on finite
structures , which have a finite universe.
Finsler geometry
a branch of
differential geometry whose main object of study is
Finsler manifolds , a generalisation of a
Riemannian manifolds .
First order arithmetic
Fourier analysis
the study of the way general
functions may be represented or approximated by sums of
trigonometric functions .
Fractal geometry
Fractional calculus
a branch of analysis that studies the possibility of taking
real or complex powers of the
differentiation operator .
Fractional dynamics
investigates the behaviour of objects and systems that are described by
differentiation and
integration of
fractional orders using methods of
fractional calculus .
Fredholm theory
part of
spectral theory studying
integral equations .
Function theory
an ambiguous term that generally refers to
mathematical analysis .
Functional analysis
a branch of
mathematical analysis , the core of which is formed by the study of
function spaces , which are some sort of
topological vector spaces .
Functional calculus
historically the term was used synonymously with
calculus of variations , but now refers to a branch of
functional analysis connected with
spectral theory
Fuzzy mathematics
a branch of mathematics based on
fuzzy set theory and
fuzzy logic .
Fuzzy measure theory
Fuzzy set theory
a form of
set theory that studies
fuzzy sets , that is
sets that have degrees of membership.
G
Galois cohomology
an application of
homological algebra , it is the study of
group cohomology of
Galois modules .
Galois theory
named after
Évariste Galois , it is a branch of abstract algebra providing a connection between
field theory and
group theory .
Galois geometry
a branch of
finite geometry concerned with algebraic and
analytic geometry over a
Galois field .
Game theory
the study of
mathematical models of strategic interaction among rational decision-makers.
Gauge theory
General topology
also known as point-set topology , it is a branch of
topology studying the properties of
topological spaces and structures defined on them. It differs from other branches of
topology as the
topological spaces do not have to be similar to manifolds.
Generalized trigonometry
developments of
trigonometric methods from the application to
real numbers of Euclidean geometry to any geometry or
space . This includes
spherical trigonometry ,
hyperbolic trigonometry ,
gyrotrigonometry , and
universal hyperbolic trigonometry .
Geometric algebra
an alternative approach to classical,
computational and
relativistic geometry . It shows a natural correspondence between geometric entities and elements of algebra.
Geometric analysis
a discipline that uses methods from
differential geometry to study
partial differential equations as well as the applications to geometry.
Geometric calculus
extends the
geometric algebra to include
differentiation and
integration .
Geometric combinatorics
a branch of
combinatorics . It includes a number of subareas such as
polyhedral combinatorics (the study of
faces of
convex polyhedra ),
convex geometry (the study of
convex sets , in particular combinatorics of their intersections), and
discrete geometry , which in turn has many applications to
computational geometry .
Geometric function theory
the study of geometric properties of
analytic functions .
Geometric invariant theory
a method for constructing quotients by
group actions in
algebraic geometry , used to construct
moduli spaces .
Geometric graph theory
a large and amorphous subfield of
graph theory , concerned with graphs defined by geometric means.
Geometric group theory
the study of
finitely generated groups via exploring the connections between algebraic properties of such groups and
topological and
geometric properties of spaces on which these groups
act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).
Geometric measure theory
the study of
geometric properties of
sets (typically in
Euclidean space ) through
measure theory .
Geometric number theory
Geometric topology
a branch of
topology studying manifolds and mappings between them; in particular the
embedding of one manifold into another.
Geometry
a branch of mathematics concerned with
shape and the properties of
space . Classically it arose as what is now known as
solid geometry ; this was concerning practical knowledge of
length ,
area and
volume . It was then put into an
axiomatic form by
Euclid , giving rise to what is now known as classical Euclidean geometry. The use of
coordinates by
René Descartes gave rise to
Cartesian geometry enabling a more analytical approach to geometric entities. Since then many other branches have appeared including
projective geometry ,
differential geometry ,
non-Euclidean geometry ,
Fractal geometry and algebraic geometry. Geometry also gave rise to the modern discipline of
topology .
Geometry of numbers
initiated by
Hermann Minkowski , it is a branch of
number theory studying
convex bodies and
integer
vectors .
Global analysis
the study of
differential equations on manifolds and the relationship between
differential equations and
topology .
Global arithmetic dynamics
Graph theory
a branch of
discrete mathematics devoted to the study of
graphs . It has many applications in
physical ,
biological and
social systems.
Group-character theory
the part of character theory dedicated to the study of characters of
group representations .
Group representation theory
Group theory
the study of
algebraic structures known as
groups .
Gyrotrigonometry
a form of
trigonometry used in
gyrovector space for
hyperbolic geometry . (An analogy of the
vector space in Euclidean geometry.)
H
Hard analysis
see classical analysis
Harmonic analysis
part of analysis concerned with the representations of
functions in terms of
waves . It generalizes the notions of
Fourier series and
Fourier transforms from the
Fourier analysis .
Higher arithmetic
Higher category theory
the part of
category theory at a higher order , which means that some equalities are replaced by explicit
arrows in order to be able to explicitly study the structure behind those equalities.
Higher-dimensional algebra
the study of
categorified structures.
Hodge theory
a method for studying the
cohomology groups of a
smooth manifold M using
partial differential equations .
Hodge-Arakelov theory
Holomorphic functional calculus
a branch of
functional calculus starting with
holomorphic functions .
Homological algebra
the study of
homology in general algebraic settings.
Homology theory
Homotopy theory
Hyperbolic geometry
also known as Lobachevskian geometry or Bolyai-Lobachevskian geometry . It is a
non-Euclidean geometry looking at
hyperbolic space .
hyperbolic trigonometry
the study of
hyperbolic triangles in
hyperbolic geometry , or
hyperbolic functions in Euclidean geometry. Other forms include
gyrotrigonometry and
universal hyperbolic trigonometry .
Hypercomplex analysis
the extension of
real analysis and
complex analysis to the study of functions where the
argument is a
hypercomplex number .
Hyperfunction theory
I
Ideal theory
once the precursor name for what is now known as
commutative algebra ; it is the theory of
ideals in
commutative rings .
Idempotent analysis
the study of
idempotent semirings , such as the
tropical semiring .
Incidence geometry
the study of relations of
incidence between various geometric objects, like
curves and
lines .
Inconsistent mathematics
see paraconsistent mathematics .
Infinitary combinatorics
an expansion of ideas in combinatorics to account for
infinite sets .
Infinitesimal analysis
once a synonym for infinitesimal calculus
Infinitesimal calculus
See
calculus of infinitesimals
Information geometry
an interdisciplinary field that applies the techniques of
differential geometry to study
probability theory and
statistics . It studies
statistical manifolds , which are
Riemannian manifolds whose points correspond to
probability distributions .
Integral calculus
Integral geometry
the theory of
measures on a geometrical space invariant under the
symmetry group of that space.
Intersection theory
a branch of algebraic geometry and algebraic topology
Intuitionistic type theory
a
type theory and an alternative
foundation of mathematics .
Invariant theory
studies how
group actions on algebraic varieties affect functions.
Inventory theory
Inversive geometry
the study of invariants preserved by a type of transformation known as inversion
Inversive plane geometry
inversive geometry that is limited to two dimensions
Inversive ring geometry
Itô calculus
extends the methods of calculus to
stochastic processes such as
Brownian motion (see
Wiener process ). It has important applications in
mathematical finance and
stochastic differential equations .
Iwasawa theory
the study of objects of arithmetic interest over infinite
towers of
number fields .
Iwasawa-Tate theory
J
Job shop scheduling
K
K-theory
originated as the study of a
ring generated by
vector bundles over a
topological space or
scheme . In algebraic topology it is an
extraordinary cohomology theory known as
topological K-theory . In algebra and algebraic geometry it is referred to as
algebraic K-theory . In
physics ,
K-theory has appeared in
type II string theory . (In particular
twisted K-theory .)
K-homology
a
homology theory on the
category of locally
compact
Hausdorff spaces .
Kähler geometry
a branch of
differential geometry , more specifically a union of
Riemannian geometry ,
complex differential geometry and
symplectic geometry . It is the study of
Kähler manifolds . (named after
Erich Kähler )
KK-theory
a common generalization both of
K-homology and
K-theory as an additive
bivariant functor on
separable
C*-algebras .
Klein geometry
More specifically, it is a
homogeneous space X together with a
transitive action on X by a
Lie group G , which acts as the
symmetry group of the geometry.
Knot theory
part of
topology dealing with
knots
Kummer theory
provides a description of certain types of
field extensions involving the
adjunction of n th roots of elements of the base
field
L
L-theory
the
K-theory of
quadratic forms .
Large deviations theory
part of
probability theory studying
events of small probability (
tail events ).
Large sample theory
also known as asymptotic theory
Lattice theory
the study of
lattices , being important in
order theory and
universal algebra
Lie algebra theory
Lie group theory
Lie sphere geometry
geometrical theory of
planar or
spatial geometry in which the fundamental concept is the
circle or
sphere .
Lie theory
Line geometry
Linear algebra
a branch of algebra studying
linear spaces and
linear maps . It has applications in fields such as abstract algebra and
functional analysis ; it can be represented in analytic geometry and it is generalized in
operator theory and in
module theory . Sometimes
matrix theory is considered a branch, although linear algebra is restricted to only finite dimensions. Extensions of the methods used belong to
multilinear algebra .
Linear functional analysis
Linear programming
a method to achieve the best outcome (such as maximum profit or lowest cost) in a
mathematical model whose requirements are represented by
linear relationships .
List of graphical methods
Included are diagram techniques, chart techniques, plot techniques, and other forms of visualization.
Local algebra
a term sometimes applied to the theory of
local rings .
Local class field theory
the study of
abelian extensions of
local fields .
Low-dimensional topology
the branch of
topology that studies
manifolds , or more generally topological spaces, of four or fewer
dimensions .
M
Malliavin calculus
a set of mathematical techniques and ideas that extend the mathematical field of
calculus of variations from deterministic functions to
stochastic processes .
Mathematical biology
the
mathematical modeling of biological phenomena.
Mathematical chemistry
the
mathematical modeling of chemical phenomena.
Mathematical economics
the application of mathematical methods to represent theories and analyze problems in
economics .
Mathematical finance
a field of
applied mathematics , concerned with mathematical modeling of
financial markets .
Mathematical logic
a subfield of
mathematics exploring the applications of formal
logic to mathematics.
Mathematical optimization
Mathematical physics
The development of mathematical methods suitable for application to problems in
physics .
[13]
Mathematical psychology
an approach to
psychological research that is based on
mathematical modeling of perceptual, thought, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characteristics with quantifiable behavior.
Mathematical sciences
refers to
academic disciplines that are mathematical in nature, but are not considered proper subfields of mathematics. Examples include
statistics ,
cryptography ,
game theory and
actuarial science .
Mathematical sociology
the area of sociology that uses mathematics to construct social theories.
Mathematical statistics
the application of
probability theory , a branch of
mathematics , to
statistics , as opposed to techniques for collecting statistical data.
Mathematical system theory
Matrix algebra
Matrix calculus
Matrix theory
Matroid theory
Measure theory
Metric geometry
Microlocal analysis
Model theory
the study of classes of mathematical
structures (e.g.
groups ,
fields ,
graphs , universes of
set theory ) from the perspective of
mathematical logic .
Modern algebra
Occasionally used for
abstract algebra . The term was coined by
van der Waerden as the title of his book
Moderne Algebra , which was renamed Algebra in the latest editions.
Modern algebraic geometry
the form of algebraic geometry given by
Alexander Grothendieck and
Jean-Pierre Serre drawing on
sheaf theory .
Modern invariant theory
the form of
invariant theory that analyses the decomposition of
representations into irreducibles.
Modular representation theory
a part of
representation theory that studies
linear representations of
finite groups over a
field K of positive
characteristic p , necessarily a prime number.
Module theory
Molecular geometry
Morse theory
a part of differential topology, it analyzes the
topological space of a manifold by studying
differentiable functions on that manifold.
Motivic cohomology
Multilinear algebra
an extension of linear algebra building upon concepts of
p-vectors and
multivectors with
Grassmann algebra .
Multiplicative number theory
a subfield of analytic number theory that deals with
prime numbers ,
factorization and
divisors .
Multivariable calculus
the extension of
calculus in one
variable to calculus with
functions of several variables : the
differentiation and
integration of functions involving several variables, rather than just one.
Multiple-scale analysis
N
Neutral geometry
See
absolute geometry .
Nevanlinna theory
part of complex analysis studying the value distribution of
meromorphic functions . It is named after
Rolf Nevanlinna
Nielsen theory
an area of mathematical research with its origins in
fixed point topology , developed by
Jakob Nielsen
Non-abelian class field theory
Non-classical analysis
Non-Euclidean geometry
Non-standard analysis
Non-standard calculus
Nonarchimedean dynamics
also known as p-adic analysis or local arithmetic dynamics
Noncommutative algebra
Noncommutative algebraic geometry
a direction in
noncommutative geometry studying the geometric properties of formal duals of non-commutative algebraic objects.
Noncommutative geometry
Noncommutative harmonic analysis
see representation theory
Noncommutative topology
Nonlinear analysis
Nonlinear functional analysis
Number theory
a branch of
pure mathematics primarily devoted to the study of the
integers . Originally it was known as arithmetic or higher arithmetic .
Numerical analysis
Numerical linear algebra
O
Operad theory
a type of abstract algebra concerned with prototypical
algebras .
Operation research
Operator K-theory
Operator theory
part of
functional analysis studying
operators .
Optimal control theory
a generalization of the
calculus of variations .
Optimal maintenance
Orbifold theory
Order theory
a branch that investigates the intuitive notion of
order using
binary relations .
Ordered geometry
a form of geometry omitting the notion of
measurement but featuring the concept of
intermediacy . It is a fundamental geometry forming a common framework for
affine geometry , Euclidean geometry,
absolute geometry and
hyperbolic geometry .
Oscillation theory
P
p-adic analysis
a branch of
number theory that deals with the analysis of functions of
p-adic numbers .
p-adic dynamics
an application of
p-adic analysis looking at
p-adic
differential equations .
p-adic Hodge theory
Parabolic geometry
Paraconsistent mathematics
sometimes called inconsistent mathematics , it is an attempt to develop the classical infrastructure of mathematics based on a foundation of
paraconsistent logic instead of
classical logic .
Partition theory
Perturbation theory
Picard–Vessiot theory
Plane geometry
Point-set topology
see general topology
Pointless topology
Poisson geometry
Polyhedral combinatorics
a branch within combinatorics and
discrete geometry that studies the problems of describing
convex polytopes .
Possibility theory
Potential theory
Precalculus
Predicative mathematics
Probability theory
Probabilistic combinatorics
Probabilistic graph theory
Probabilistic number theory
Projective geometry
a form of geometry that studies geometric properties that are
invariant under a
projective transformation .
Projective differential geometry
Proof theory
Pseudo-Riemannian geometry
generalizes
Riemannian geometry to the study of
pseudo-Riemannian manifolds .
Pure mathematics
the part of mathematics that studies entirely abstract concepts.
Q
Quantum calculus
a form of calculus without the notion of
limits .
Quantum geometry
the generalization of concepts of geometry used to describe the
physical phenomena of
quantum physics
Quaternionic analysis
R
Ramsey theory
the study of the conditions in which order must appear. It is named after
Frank P. Ramsey .
Rational geometry
Real algebra
the study of the part of algebra relevant to
real algebraic geometry .
Real algebraic geometry
the part of algebraic geometry that studies
real points of the algebraic varieties.
Real analysis
a branch of mathematical analysis; in particular hard analysis , that is the study of
real numbers and
functions of
Real values. It provides a rigorous formulation of the calculus of
real numbers in terms of
continuity and
smoothness , whilst the theory is extended to the
complex numbers in
complex analysis .
Real Clifford algebra
Real K-theory
Recreational mathematics
the area dedicated to
mathematical puzzles and
mathematical games .
Recursion theory
see computability theory
Representation theory
a subfield of abstract algebra; it studies
algebraic structures by representing their elements as
linear transformations of
vector spaces . It also studies
modules over these algebraic structures, providing a way of reducing problems in abstract algebra to problems in linear algebra.
Representation theory of groups
Representation theory of the Galilean group
Representation theory of the Lorentz group
Representation theory of the Poincaré group
Representation theory of the symmetric group
Ribbon theory
a branch of
topology studying
ribbons .
Ricci calculus
Also called absolute differential calculus .
A foundation of
tensor calculus , developed by
Gregorio Ricci-Curbastro in 1887–1896,
[14] and later developed for its applications to
general relativity and
differential geometry .
[15]
Ring theory
Riemannian geometry
a branch of
differential geometry that is more specifically, the study of
Riemannian manifolds . It is named after
Bernhard Riemann and it features many generalizations of concepts from Euclidean geometry, analysis and calculus.
Rough set theory
the a form of
set theory based on
rough sets .
S
Sampling theory
Scheme theory
the study of
schemes introduced by
Alexander Grothendieck . It allows the use of
sheaf theory to study algebraic varieties and is considered the central part of modern algebraic geometry .
Secondary calculus
Semialgebraic geometry
a part of algebraic geometry; more specifically a branch of
real algebraic geometry that studies
semialgebraic sets .
Set-theoretic topology
Set theory
Sheaf theory
The study of
sheaves , which connect local and global properties of
geometric objects .
[16]
Sheaf cohomology
Sieve theory
Single operator theory
deals with the properties and classifications of single
operators .
Singularity theory
a branch, notably of geometry; that studies the failure of manifold structure.
Smooth infinitesimal analysis
a rigorous reformation of
infinitesimal calculus employing methods of
category theory . As a theory, it is a subset of
synthetic differential geometry .
Solid geometry
Spatial geometry
Spectral geometry
a field that concerns the relationships between geometric structures of manifolds and
spectra of canonically defined
differential operators .
Spectral graph theory
the study of properties of a
graph using methods from
matrix theory .
Spectral theory
part of operator theory extending the concepts of
eigenvalues and
eigenvectors from linear algebra and
matrix theory .
Spectral theory of ordinary differential equations
part of
spectral theory concerned with the
spectrum and
eigenfunction expansion associated with
linear
ordinary differential equations .
Spectrum continuation analysis
generalizes the concept of a
Fourier series to non-periodic
functions .
Spherical geometry
a branch of
non-Euclidean geometry , studying the 2-dimensional surface of a
sphere .
Spherical trigonometry
a branch of
spherical geometry that studies
polygons on the surface of a
sphere . Usually the
polygons are
triangles .
Statistical mechanics
Statistical modelling
Statistical theory
Statistics
although the term may refer to the more general study of
statistics , the term is used in mathematics to refer to the
mathematical study of statistics and related fields . This includes
probability theory .
Steganography
Stochastic calculus
Stochastic calculus of variations
Stochastic geometry
the study of random patterns of points
Stochastic process
Stratified Morse theory
Super linear algebra
Surgery theory
a part of
geometric topology referring to methods used to produce one manifold from another (in a controlled way.)
Survey sampling
Survey methodology
Symbolic computation
also known as algebraic computation and computer algebra . It refers to the techniques used to manipulate
mathematical expressions and
equations in
symbolic form as opposed to manipulating them by the numerical quantities represented by them.
Symbolic dynamics
Symplectic geometry
a branch of
differential geometry and topology whose main object of study is the
symplectic manifold .
Symplectic topology
Synthetic differential geometry
a reformulation of
differential geometry in the language of
topos theory and in the context of an
intuitionistic logic .
Synthetic geometry
also known as axiomatic geometry , it is a branch of geometry that uses
axioms and
logical arguments to draw conclusions as opposed to
analytic and algebraic methods.
Systolic geometry
a branch of
differential geometry studying systolic
invariants of
manifolds and
polyhedra .
Systolic hyperbolic geometry
the study of
systoles in
hyperbolic geometry .
T
Tensor algebra, Tensor analysis, Tensor calculus, Tensor theory
the study and use of
tensors , which are generalizations of
vectors . A
tensor algebra is also an
algebraic structure that is used in the
formal definition of tensors.
Tessellation
when periodic tiling has a repeating pattern.
Theoretical physics
a branch primarily of the
science
physics that uses
mathematical models and
abstraction of
physics to rationalize and predict
phenomena .
Theory of computation
Time-scale calculus
Topology
Topological combinatorics
the application of methods from algebraic topology to solve problems in combinatorics.
Topological degree theory
Topological graph theory
Topological K-theory
Topos theory
Toric geometry
Transcendental number theory
a branch of
number theory that revolves around the
transcendental numbers .
Transformation geometry
Trigonometry
the study of
triangles and the relationships between the
length of their sides, and the
angles between them. It is essential to many parts of
applied mathematics .
Tropical analysis
see idempotent analysis
Tropical geometry
Twisted K-theory
a variation on
K-theory , spanning abstract algebra, algebraic topology and
operator theory .
Type theory
U
Umbral calculus
the study of
Sheffer sequences
Uncertainty theory
a new branch of
mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure
axioms .
Universal algebra
a field studying the formalization of algebraic structures itself.
Universal hyperbolic trigonometry
an approach to
hyperbolic trigonometry based on
rational geometry .
V
Valuation theory
Variational analysis
Vector algebra
a part of linear algebra concerned with the
operations of
vector addition and
scalar
multiplication , although it may also refer to
vector
operations of
vector calculus , including the
dot and
cross product . In this case it can be contrasted with
geometric algebra which generalizes into higher dimensions.
Vector analysis
also known as
vector calculus , see vector calculus .
Vector calculus
a branch of
multivariable calculus concerned with
differentiation and
integration of
vector fields . Primarily it is concerned with 3-dimensional
Euclidean space .
W
Wavelets
See also
References
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Der Ricci-Kalkül – Eine Einführung in die neueren Methoden und Probleme der mehrdimensionalen Differentialgeometrie (Ricci Calculus – An introduction in the latest methods and problems in multi-dimensional differential geometry) . Grundlehren der mathematischen Wissenschaften (in German). Vol. 10. Berlin: Springer Verlag.
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