SafekipediaGlossary of areas of mathematics
Adapted from Wikipedia · Discoverer experience
Mathematics is a broad and fascinating subject that can be divided into many different areas or branches. These areas are defined by what they study, the methods they use, or both. For example, analytic number theory is a part of number theory that uses techniques from analysis to explore natural numbers.
This glossary lists many of these areas in alphabetical order. While this makes it easy to find terms, it doesn’t always show how the different areas connect to each other. For a look at the widest categories of mathematics, you can visit Mathematics § Areas of mathematics. There is also a detailed list called the Mathematics Subject Classification, created by mathematicians to organize subjects and topics in their field. This system helps publishers sort and share mathematical research around the world.
A
Absolute differential calculus
An older name of Ricci calculus.
Also called neutral geometry, a synthetic geometry similar to Euclidean geometry but without the parallel postulate.
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.
The discipline that applies mathematical and statistical methods to assess risk in insurance, finance and other industries and professions.
The part of arithmetic combinatorics devoted to the operations of addition and subtraction.
A part of number theory that studies subsets of integers and their behaviour under addition.
A branch of geometry that deals with properties that are independent from distances and angles, such as alignment and parallelism.
The study of curve properties that are invariant under affine transformations.
A part of complex analysis being the geometric counterpart of Nevanlinna theory. It was invented by Lars Ahlfors.
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.
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.
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.
An older name of computer algebra.
a branch that combines techniques from abstract algebra with the language and problems of geometry. Fundamentally, it studies algebraic varieties.
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.
an important part of homological algebra concerned with defining and applying a certain sequence of functors from rings to abelian groups.
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.
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.
a branch that uses tools from abstract algebra for topology to study topological spaces.
also known as computational number theory, it is the study of algorithms for performing number theoretic computations.
A wide area of mathematics centered on the study of continuous functions and including such topics as differentiation, integration, limits, and series.
part of enumerative combinatorics where methods of complex analysis are applied to generating functions.
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Also known as Cartesian geometry, the study of Euclidean geometry using Cartesian coordinates.
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Analogue to differential geometry, where differentiable functions are replaced with analytic functions. It is a subarea of both complex analysis and algebraic geometry.
An area of number theory that applies methods from mathematical analysis to solve problems about integers.
Analytic theory of L-functions
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.
part of analysis that studies how well functions can be approximated by simpler ones (such as polynomials or trigonometric polynomials)
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Also known as elementary arithmetic, the methods and rules for computing with addition, subtraction, multiplication and division of numbers.
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Also known as higher arithmetic, another name for number theory.
the study of the estimates from combinatorics that are associated with arithmetic operations such as addition, subtraction, multiplication and division.
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.
The use of algebraic geometry and more specially scheme theory for solving problems of number theory.
a combination of algebraic number theory and topology studying analogies between prime ideals and knots
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.
the study of systems of axioms in a context relevant to set theory and mathematical logic.
B
Bifurcation theory studies how the way things are arranged can change, and it is part of dynamical systems theory.
Biostatistics uses statistical methods to help understand topics in biology. Birational geometry is a part of algebraic geometry that looks at shapes based on their function field. Bolyai–Lobachevskian geometry is another name for hyperbolic geometry.
C
This is a type of algebra that studies certain kinds of mathematical structures called operators on special spaces.
This is another name for a way of describing shapes and points using numbers, also known as analytic geometry.
Calculus is a branch of mathematics that studies how things change. It is connected by the fundamental theorem of calculus.
This is a part of calculus that uses very tiny numbers, called infinitesimals, to study change.
This extends a part of mathematics called tensor calculus to study surfaces that change shape.
This area of math focuses on finding the best way to maximize or minimize certain quantities, called functionals.
This studies sudden changes in systems, using ideas from geometry.
This connects two areas of math: category theory and mathematical logic, using ideas from type theory.
This studies the basic ideas in mathematics by looking at objects and the relationships between them.
This looks at how systems can behave in unpredictable ways when they are very sensitive to small changes.
This is a part of group theory that studies certain features of group representations.
This is a part of algebraic number theory that studies certain kinds of extensions of number fields.
Classical differential geometry
Also known as Euclidean differential geometry.
This usually means the traditional parts of analysis, like real analysis and complex analysis, without using more modern techniques.
Classical analytic number theory
Classical differential calculus
Classical Diophantine geometry
See Euclidean geometry.
Classical geometry
This can mean solid geometry or classical Euclidean geometry. See geometry.
This studies certain kinds of polynomial functions that stay the same under specific transformations.
This is the standard way of doing math, based on classical logic and ZFC set theory.
This studies certain operators from geometry and analysis using clifford algebras.
This is a part of representation theory that comes from Cliffords theorem.
This studies the properties of codes and how well they work for different uses.
Combinatorial commutative algebra
This combines ideas from algebra and combinatorics to solve problems in both areas. Polyhedral geometry is also important here.
This is a part of combinatorics that deals with creating sets with specific intersection properties.
See discrete geometry.
This studies free groups and how groups can be described. It is related to geometric group theory and used in geometric topology.
This area of math is mostly about counting and studying properties of finite structures.
Also known as Infinitary combinatorics.
This was an old name for algebraic topology, when topological invariants were seen as coming from combinatorial breakdowns.
This is a part of discrete mathematics that deals with countable structures. It includes enumerative combinatorics, combinatorial design theory, matroid theory, extremal combinatorics, and algebraic combinatorics.
This studies commutative rings, a type of algebraic structure.
This is the main part of algebraic geometry that studies complex points of algebraic varieties.
This is a part of analysis that deals with functions of complex numbers.
This is a part of complex dynamics that studies dynamic systems defined by analytic functions.
This applies complex numbers to plane geometry.
This studies complex manifolds, which are special kinds of geometric spaces.
This studies dynamical systems defined by repeated functions on complex number spaces.
This studies complex manifolds and functions of complex variables. It includes complex algebraic geometry and complex analytic geometry.
This studies complex systems, including the theory of complex systems.
This looks at which parts of real analysis can be done in a computable way. It is closely related to constructive analysis.
This is a part of model theory that deals with questions about computability.
This is a part of mathematical logic that started in the 1930s with the study of computable functions and Turing degrees. It now also includes generalized computability and definability. It overlaps with proof theory and effective descriptive set theory.
Computational algebraic geometry
Computational complexity theory
This is a part of mathematics and theoretical computer science that classifies computational problems by how hard they are and how they relate to each other.
This is a part of computer science that studies algorithms based on geometry.
This studies groups using computers.
This is mathematical research in areas where computing plays an important role.
Also known as algorithmic number theory, this studies algorithms for number theory computations.
Computational synthetic geometry
See symbolic computation.
This studies conformal transformations on a space.
This is mathematical analysis done using constructive mathematics, which is different from classical analysis.
This is a part of analysis that studies the connection between the smoothness of a function and how well it can be approximated.
This is mathematics that uses intuitionistic logic, which is like classical logic but without assuming the law of the excluded middle.
Constructive quantum field theory
This is a part of mathematical physics that shows quantum theory is mathematically compatible with special relativity.
This is an approach to mathematical constructivism using the usual language of classical set theory.
This is a part of differential geometry and topology that studies a special geometric structure called a contact structure on a differentiable manifold.
This studies the properties of convex functions and convex sets.
This part of geometry studies convex sets.
See analytic geometry.
This is a part of differential geometry that studies CR manifolds.
D
a part of mathematical logic, focusing on special types of spaces called Polish spaces.
The part of calculus that studies how things change, using something called derivatives.
a type of geometry that uses math tools to study shapes and spaces. It looks at smooth surfaces and curves using methods from calculus and algebra.
a part of topology that looks at smooth shapes and how they can change.
the study of math structures that are separate and not continuous, like counting objects.
E
Econometrics is about using math and statistical methods to understand economic data.
Elementary algebra is a basic type of algebra that builds on elementary arithmetic by introducing variables. Elementary arithmetic is the simple math taught in early school, including addition, subtraction, multiplication, and division with natural numbers, as well as fractions and negative numbers.
Elementary mathematics includes the math usually taught in primary and secondary school, such as elementary arithmetic, geometry, probability, and statistics. It also covers elementary algebra and trigonometry, but not usually calculus.
F
Field theory is a part of algebra that studies fields, a special kind of algebraic structure. Finite geometry and finite model theory look at mathematical ideas in small, limited ways. Fourier analysis explores how functions can be shown using sums of trigonometric functions.
Fractal geometry deals with shapes that appear the same no matter how much you zoom in on them. Functional analysis is a part of mathematical analysis that studies spaces of functions, which are like collections of different possible functions. Fuzzy mathematics uses ideas from fuzzy set theory and fuzzy logic to work with sets that have varying degrees of membership.
G
Galois theory is a part of algebra that connects two big ideas in math: fields and groups. It is named after Évariste Galois.
Game theory studies how people or things make decisions when they have to work together or against each other. It uses math to understand these strategies.
Geometry is the study of shapes and space. It started with simple ideas about length, area, and volume and has grown into many different areas like projective geometry and differential geometry.
Graph theory looks at how things can be connected, using simple drawings called graphs to show links between objects, and it has uses in many real-world situations.
Group theory is about a special kind of math structure called groups, which help us understand patterns and symmetry.
H
Harmonic analysis is part of analysis that looks at representing functions using waves. It builds on ideas from Fourier series and Fourier transforms.
Hyperbolic geometry is a type of non-Euclidean geometry that studies spaces with constant negative curvature, different from the flat geometry we usually learn. It includes the study of special triangles called hyperbolic triangles and functions known as hyperbolic functions.
I
Ideal theory was an early name for what we now call commutative algebra. It focuses on ideals within commutative rings.
Idempotent analysis looks at special number systems called idempotent semirings, like the tropical semiring.
Incidence geometry studies how different geometric shapes, such as curves and lines, relate to each other.
Infinitary combinatorics expands regular combinatorics to include infinite sets.
Information geometry combines ideas from differential geometry to study probability theory and statistics. It looks at spaces called statistical manifolds.
Integral calculus is a part of calculus that deals with integrals, unlike differential calculus.
Itô calculus extends regular calculus to work with unpredictable processes like Brownian motion, useful in mathematical finance.
J
Job shop scheduling is a way to organize tasks in factories or other places where different jobs need to be done on different machines. It helps plan which job goes to which machine and when, so that everything gets done efficiently. This is important in many types of manufacturing and production.
K
K-theory is a part of mathematics that studies special kinds of structures called vector bundles over spaces. It has uses in both topology and physics. For example, in physics, it appears in theories that describe the fundamental nature of the universe.
Other related areas include K-homology, which studies certain properties of spaces, and Kähler geometry, which combines different kinds of geometry to study special shapes called Kähler manifolds. Knot theory is another interesting area that looks at how knots can be formed and manipulated in space.
L
L-theory is related to K-theory and focuses on quadratic forms.
Large deviations theory is a part of probability theory that looks at rare events, called tail events. Large sample theory, also known as asymptotic theory, studies lattices which are important in order theory and universal algebra. This includes areas like Lie algebra theory, Lie group theory, and Lie sphere geometry, which is a geometrical theory of planar or spatial geometry focusing on the circle or sphere.
Line geometry and Linear algebra study linear spaces and linear maps, with applications in many areas of math. Linear programming is a way to find the best result, like maximum profit or lowest cost, in a mathematical model with linear relationships. Low-dimensional topology is a branch of topology that studies manifolds or topological spaces with four or fewer dimensions.
M
Mathematical areas beginning with "M" explore many fascinating topics. For example, Malliavin calculus extends regular calculus to handle unpredictable processes. Mathematical biology uses math to understand living things, while mathematical economics applies math to study money and trade.
Other areas include mathematical physics, which creates math tools for physics problems, and mathematical logic, where logic meets math. There are also fields like matrix algebra and multivariable calculus, which study equations with many variables.
N
Neutral geometry is related to absolute geometry.
Other areas of mathematics include Nevanlinna theory, which studies the values of meromorphic functions, Nielsen theory from fixed point topology, Non-abelian class field theory, Non-Euclidean geometry, Non-standard analysis, Non-standard calculus, Nonarchimedean dynamics, also called p-adic analysis or local arithmetic dynamics, and Number theory, a part of pure mathematics focused on the study of the integers. There are also topics like Numerical analysis and Numerical linear algebra.
O
Operad theory is a type of abstract algebra that studies patterns in mathematical structures.
Operator theory is part of functional analysis that looks at special functions called operators. Order theory explores the idea of ordering using relationships between items. Ordered geometry is a type of geometry that uses the idea of betweenness but does not measure distances, helping to connect different kinds of geometries like affine geometry, Euclidean geometry, absolute geometry, and hyperbolic geometry.
P
p-adic analysis is a part of number theory that studies functions using special numbers called p-adic numbers. Partition theory looks at ways to break numbers into smaller parts. Polyhedral combinatorics is a branch of math that studies shapes with flat sides, called convex polytopes, and their properties.
Projective geometry explores geometric shapes and their properties that stay the same even when you change the view in special ways, known as projective transformations. Pure mathematics focuses on studying abstract ideas and concepts without any real-world application in mind.
Q
Quantum calculus is a special kind of calculus that does not use limits, which are usual ways to understand how things change smoothly.
Quantum geometry extends geometry to explain the strange behaviors seen in the tiny world of quantum physics. Quaternionic analysis is another interesting area that uses special numbers to study mathematical problems.
R
Ramsey theory studies when order must appear, named after Frank P. Ramsey.
Rational geometry focuses on parts of algebra related to real algebraic geometry.
Real algebraic geometry looks at real points in algebraic shapes.
Real analysis is a type of math that studies real numbers and their functions, exploring ideas like continuity and smoothness.
Recreational mathematics includes fun mathematical puzzles and mathematical games.
Representation theory is a part of algebra that studies algebraic structures by showing their elements as linear transformations of vector spaces.
Ribbon theory is a part of topology that studies ribbons.
Ricci calculus, also called absolute differential calculus, is a foundation of tensor calculus, created by Gregorio Ricci-Curbastro and used in general relativity and differential geometry.
Riemannian geometry studies Riemannian manifolds and expands ideas from geometry, analysis, and calculus. It is named after Bernhard Riemann.
Rough set theory is a type of set theory based on rough sets.
S
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.
a part of algebraic geometry; more specifically a branch of real algebraic geometry that studies semialgebraic sets.
The study of sheaves, which connect local and global properties of geometric objects.
deals with the properties and classifications of single operators.
a branch, notably of geometry; that studies the failure of manifold structure.
a rigorous reformation of infinitesimal calculus employing methods of category theory. As a theory, it is a subset of synthetic differential geometry.
a field that concerns the relationships between geometric structures of manifolds and spectra of canonically defined differential operators.
the study of properties of a graph using methods from matrix 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.
a branch of non-Euclidean geometry, studying the 2-dimensional surface of a sphere.
a branch of spherical geometry that studies polygons on the surface of a sphere. Usually the polygons are triangles.
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.
Stochastic calculus of variations
the study of random patterns of points
a part of geometric topology referring to methods used to produce one manifold from another (in a controlled way.)
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.
a branch of differential geometry and topology whose main object of study is the symplectic manifold.
Synthetic differential geometry
a reformulation of differential geometry in the language of topos theory and in the context of an intuitionistic logic.
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.
a branch of differential geometry studying systolic invariants of manifolds and polyhedra.
the study of systoles in hyperbolic geometry.
T
Tensor algebra, Tensor analysis, Tensor calculus, Tensor theory
This area looks at tensors, which are like more complex versions of vectors. A tensor algebra is a special kind of mathematical system used to define tensors formally.
This is about patterns made by repeating shapes, like tiles on a floor.
This is a part of science that uses mathematical models and ideas to explain and predict things that happen in the world.
This uses ideas from a part of mathematics called topology to solve problems about organizing and counting things.
This studies special numbers called transcendental numbers.
This is the study of triangles, looking at how the lengths of their sides and their angles are related. It is very useful in many areas of practical math.
see idempotent analysis
This is a variation of K-theory that connects abstract algebra, the study of shapes, and operator theory.
U
Umbral calculus is a fascinating area of math that studies special number patterns called Sheffer sequences.
Other important areas include Uncertainty theory, which is a new branch of mathematics focusing on ideas like normality and self-duality. There is also Universal algebra, a field that examines the basic structures used in algebra, and Universal hyperbolic trigonometry, which applies hyperbolic trigonometry using ideas from rational geometry.
V
Vector algebra is a part of linear algebra that focuses on adding vectors and multiplying them by numbers, known as scalars. It also includes operations like the dot product and cross product. Vector calculus is a branch of multivariable calculus that studies how vector fields change and integrate in three-dimensional space. It deals with differentiation and integration of vectors in Euclidean space.
W
Wavelets are tools used in mathematics to analyze data, especially in images and signals. They help in breaking down complex information into simpler parts that are easier to study and understand.
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