Areas of mathematics

https://en.wikipedia.org/wiki/Glossary_of_areas_of_mathematics https://en.wikipedia.org/wiki/Areas_of_mathematics

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Absolute differential calculus : the original name for tensor calculus developed around 1890.

Absolute geometry : an extension of ordered geometry that is sometimes referred to as neutral geometry because its axiom system is neutral to the parallel postulate.

Abstract algebra : the study of algebraic structures and their properties. Originally it was known as modern algebra.

Abstract analytic number theory : a branch of mathematics that takes ideas from classical analytic number theory and applies them to various other areas of mathematics.

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 is centered on the study of geometric properties that remain unchanged by affine transformations. It can be described as a generalization of Euclidean geometry.

Affine geometry of curves : the study of curves in affine space.

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 : a major part of pure mathematics centered on operations and relations. Beginning with elementary algebra, it introduces the concept of variables and how these can be manipulated towards problem solving; known as equation solving. Generalizations of operations and relations defined on sets have led to the idea of an algebraic structure which are studied in abstract algebra. Other branches of algebra include universal algebra, linear algebra and multilinear algebra.

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 : see symbolic computation.

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: a part of algebraic geometry devoted to the study of the points of the algebraic varieties whose coordinates belong to an algebraic number field. It is a major branch of number theory and is also said to study algebraic structures related to algebraic 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 rigorous branch of pure mathematics that had its beginnings in the formulation of infinitesimal calculus. (Then it was known as infinitesimal analysis.) The classical forms of analysis are real analysis and its extension complex analysis, whilst more modern forms are those such as functional analysis.

Analytic combinatorics: part of enumerative combinatorics where methods of complex analysis are applied to generating functions.

Analytic geometry: usually this refer to the study of geometry using a coordinate system (also known as Cartesian geometry). Alternatively it can refer to the geometry of analytic varieties. In this respect it is essentially equivalent to real and complex algebraic geometry.

Analytic number theory: part of number theory using methods of analysis (as opposed to algebraic number theory)

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: to most people this refers to the branch known as elementary arithmetic dedicated to the usage of addition, subtraction, multiplication and division. However arithmetic also includes higher arithmetic referring to advanced results from 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 study of schemes of finite type over the spectrum of the ring of integers

Arithmetic topology: a combination of algebraic number theory and topology studying analogies between prime ideals and knots

Arithmetical algebraic geometry: an alternative name for arithmetic algebraic geometry

Assignment problem

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 geometric analysis

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 homology theory

Axiomatic set theory: the study of systems of axioms in a context relevant to set theory and mathematical logic.