kb/data/en.wikipedia.org/wiki/Glossary_of_areas_of_mathematics-0.md

title	chunk	source	category	tags	date_saved	instance
Glossary of areas of mathematics	1/7	https://en.wikipedia.org/wiki/Glossary_of_areas_of_mathematics	reference	science, encyclopedia	2026-05-05T07:50:22.568786+00:00	kb-cron

Mathematics is a broad subject that is commonly divided in many areas or branches 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, a synthetic geometry similar to Euclidean geometry but without the parallel postulate. Abstract algebra The part of algebra devoted to the study of algebraic structures in themselves. 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. 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. 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. 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
3. Also known as elementary arithmetic, the methods and rules for computing with addition, subtraction, multiplication and division of numbers.
4. 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.