SafekipediaGlossary of areas of mathematics
Adapted from Wikipedia · Adventurer experience
Mathematics is a big and interesting subject. It has many different areas or branches. These areas are grouped by what they study, the ways they work, or both. For example, analytic number theory is a part of number theory. It uses tools from analysis to learn about natural numbers.
This glossary lists many of these areas in alphabetical order. This means it does not always show how they are connected. If you want to learn about the main parts of mathematics, visit Mathematics § Areas of mathematics. There is also a detailed list called the Mathematics Subject Classification. Mathematicians made this list to help organize books and articles about math. This system helps experts and readers find the exact part of math they want to learn about.
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. Sometimes called modern algebra.
Abstract analytic number theory
The study of arithmetic semigroups to extend ideas from classical analytic number theory.
Abstract differential geometry
A form of differential geometry without the idea of smoothness. It uses sheaf theory and sheaf cohomology.
A modern part of harmonic analysis that builds on Fourier transforms on locally compact groups.
A part of topology that looks at functions between spaces that can be changed into each other.
A field that uses math and statistics to assess risk in insurance, finance, and other areas.
A part of arithmetic combinatorics focused on addition and subtraction.
A part of number theory that looks at groups of whole numbers and how they behave when added.
A branch of geometry that looks at properties that do not depend on distances or angles, like alignment and parallelism.
The study of curve properties that stay the same under affine transformations.
A type of differential geometry focused on features that stay the same under volume-preserving affine transformations.
A part of complex analysis that is the geometric side of Nevanlinna theory. It was created by Lars Ahlfors.
One of the big areas of mathematics. It is the skill of working with operations on symbols called variables that stand for unknown numbers or other mathematical objects, like vectors, matrices, or parts of algebraic structures.
Based on systems of linear partial differential equations, it is a part of algebraic geometry and algebraic topology that uses methods from sheaf theory and complex analysis. It began with Mikio Sato.
A field that uses tools from abstract algebra to solve problems in combinatorics.
An older name for computer algebra.
A branch that mixes tools from abstract algebra with geometry. It mainly studies algebraic varieties.
A part of graph theory that uses algebra to study graphs. It often uses group theory and linear algebra.
An important part of homological algebra that defines and uses a sequence of functors from rings to abelian groups.
The part of number theory that uses algebraic methods, mainly from commutative algebra, to study number fields and their rings of integers.
The use of algebra to help with statistics, sometimes focusing on algebraic geometry and commutative algebra.
A branch that uses tools from abstract algebra to study topological spaces.
Also called computational number theory, it is the study of algorithms for doing number theoretic computations.
A field based on ideas from Alexander Grothendieck in the 1980s that describes how a geometric object can be linked to another, without using abelian groups.
A big area of mathematics that studies continuous functions and includes differentiation, integration, limits, and series.
A part of enumerative combinatorics that uses complex analysis on generating functions.
1. Also called Cartesian geometry, the study of Euclidean geometry using Cartesian coordinates. 2. Like differential geometry, but where differentiable functions are swapped for analytic functions. It is part of complex analysis and algebraic geometry.
A part of number theory that uses methods from mathematical analysis to solve problems about whole numbers.
Analytic theory of L-functions
A mix of many parts of mathematics used to solve real-world and theory problems. It often helps in science, engineering, finance, economics, and logistics.
A part of analysis that looks at how well functions can be matched by simpler ones like polynomials or trigonometric polynomials.
Also known as Arakelov theory
A way to study Diophantine equations in higher dimensions using algebraic geometry. It is named after Suren Arakelov.
1. Also known as elementary arithmetic, the rules for working with addition, subtraction, multiplication and division of numbers. 2. Also known as higher arithmetic, another name for number theory.
See arithmetic geometry.
The study of ideas from combinatorics linked to arithmetic operations like addition, subtraction, multiplication and division.
Arithmetic dynamics studies the properties of whole numbers, fractions, and other number types when repeatedly used in polynomials or rational functions. It aims to explain these properties using geometric structures.
The use of algebraic geometry, especially scheme theory, to solve problems in number theory.
A mix of algebraic number theory and topology that studies links between prime ideals and knots
Arithmetical algebraic geometry
Another name for arithmetic algebraic geometry
It uses the shape of objects to find formulas for their generating functions and then uses complex analysis to find approximations.
The study of asymptotic expansions
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 reach conclusions instead of analytic or algebraic methods.
The study of systems of axioms related to set theory and mathematical logic.
B
Bifurcation theory studies how patterns change shape and look. It is part of dynamical systems theory, which looks at how systems change over time.
Biostatistics uses statistical ways to study biology, like plants and animals.
Birational geometry is part of algebraic geometry that looks at shapes and how they relate to each other.
Bolyai–Lobachevskian geometry is another name for hyperbolic geometry.
C
This is a type of math that studies special kinds of algebras. Algebras are sets with rules for combining elements.
See analytic geometry
Calculus is a part of math that studies how things change. It is connected by the fundamental theorem of calculus.
Also called infinitesimal calculus
This is a foundation of calculus that was developed in the 1600s. It uses very small numbers to study change.
This extends the theory of tensor calculus to include surfaces that change shape over time.
This field is about finding the best way to maximize or minimize certain mathematical expressions. It used to be called functional calculus.
This is a part of bifurcation theory from dynamical systems theory. It is also a special case of singularity theory. It looks at the shapes that can appear in certain situations.
This is a part of category theory that connects to mathematical logic. It is based on type theory for intuitionistic logics.
This studies the properties of mathematical ideas by organizing them into collections of objects and arrows that show relationships.
This looks at how systems behave when they are very sensitive to small changes.
This is a part of group theory that studies special features of group representations.
This is a part of algebraic number theory that studies special kinds of number fields.
Classical differential geometry
Also known as Euclidean differential geometry. See Euclidean differential geometry
See algebraic topology
This usually means the traditional parts of analysis, such as real analysis. It includes work that does not use techniques from functional analysis.
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 part of invariant theory describes polynomial functions that stay the same even when you change them using certain rules.
Classical mathematics
This is the usual way of doing math using classical logic.
This studies special kinds of operators in geometry and analysis using clifford algebras.
This is a part of representation theory.
This looks at the properties of special kinds of messages called codes.
Combinatorial commutative algebra
This is a mix of commutative algebra and combinatorics.
This is a part of combinatorics that deals with creating and studying systems of finite sets.
See discrete geometry
This theory studies free groups. It is closely related to geometric group theory.
This area of math is mostly about counting. It looks at both the process of counting and what it tells us about certain properties.
Also known as Infinitary combinatorics. See infinitary combinatorics
This used to be an old name for algebraic topology.
This is a part of discrete mathematics that deals with countable structures. It includes enumerative combinatorics, combinatorial design theory, matroid theory, and algebraic combinatorics.
This is a part of abstract algebra that studies commutative rings.
This is the main part of algebraic geometry that studies the complex points of algebraic varieties.
This is a part of analysis that deals with functions that have complex numbers as inputs.
This is a part of complex dynamics that studies dynamic systems.
This applies complex numbers to plane geometry.
This is a part of differential geometry that studies complex manifolds.
This studies dynamical systems defined by iterated functions.
This studies complex manifolds and functions with complex numbers.
This studies complex systems.
This looks at which parts of real analysis can be done using computers.
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. It now also includes the study of more general kinds of computability.
Computational algebraic geometry
Computational complexity theory
This is a part of math and theoretical computer science that classifies computational problems based on how hard they are.
This is a part of computer science that studies algorithms that can be described using geometry.
This studies groups using computers.
This is math research in areas where computing is important.
Also known as algorithmic number theory, this studies algorithms for doing number theoretic computations.
Computational synthetic geometry
See symbolic computation
This studies conformal transformations on a space.
This is analysis done using the principles of constructive mathematics.
This is a part of analysis that is closely related to approximation theory.
This kind of math tends to use intuitionistic logic.
Constructive quantum field theory
This is a part of mathematical physics that shows that quantum theory works mathematically.
This is an approach to mathematical constructivism.
This is a part of differential geometry and topology. It is closely related to symplectic geometry.
This studies the properties of convex functions.
This part of geometry studies convex sets.
See analytic geometry
This is a part of differential geometry that studies CR manifolds.
D
Derived noncommutative algebraic geometry
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.
The branch of calculus contrasted to integral calculus, and concerned with derivatives.
the study of the Galois groups of differential fields.
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.
a branch of topology that deals with differentiable functions on differentiable manifolds.
in general the study of algebraic varieties over fields that are finitely generated over their prime fields.
Discrete differential geometry
a branch of geometry that studies combinatorial properties and constructive methods of discrete geometric objects.
the study of mathematical structures that are fundamentally discrete rather than continuous.
a combinatorial adaption of Morse theory.
a branch that studies special kinds of partially ordered sets (posets) commonly called domains.
the study of smooth 4-manifolds using gauge theory.
an area used to describe the behavior of the complex dynamical systems, usually by employing differential equations or difference equations.
E
Econometrics uses math and statistical methods to study economic data.
Elementary algebra builds on elementary arithmetic by introducing variables. Elementary arithmetic is the basic math you learn in early school, like adding, subtracting, multiplying, and dividing with whole numbers, fractions, and negative numbers.
Elementary mathematics covers math topics from primary and secondary school, such as elementary arithmetic, geometry, probability, and statistics. It also includes elementary algebra and trigonometry, but not usually calculus.
Elliptic geometry is a type of non-Euclidean geometry based on spherical geometry and does not follow Euclid's parallel postulate.
Enumerative combinatorics counts how many ways certain patterns can be formed. Extremal combinatorics looks at the biggest or smallest possible sizes of groups of objects under certain rules.
F
This area of algebra studies fields, a special type of algebraic structure.
This is a part of model theory that looks at interpretations on finite structures.
This is a part of differential geometry that studies Finsler manifolds, which are a more general idea than Riemannian manifolds.
This studies how general functions can be shown using sums of trigonometric functions.
This area of analysis looks at using real or complex powers of the differentiation operator.
This examines how objects and systems behave when described using differentiation and integration of fractional orders, using methods from fractional calculus.
This is part of spectral theory that studies integral equations.
This term usually means mathematical analysis.
This is a part of mathematical analysis that mainly studies function spaces, which are types of topological vector spaces.
Originally, this term meant the same as calculus of variations, but today it refers to a part of functional analysis linked to spectral theory.
This branch of mathematics is based on fuzzy set theory and fuzzy logic.
This form of set theory studies fuzzy sets, which are sets where items can have different levels of membership.
G
Galois cohomology is a way to study groups using algebra. It looks at how groups work with math structures.
Galois theory is named after Évariste Galois. It connects two parts of algebra: fields and groups. It helps us understand how these two are related.
Game theory studies how people or things make choices when they depend on each other. It uses math to show these strategies.
General topology is a part of topology. It studies spaces and their properties, without needing them to look like normal shapes.
Geometric algebra is a way to use algebra to describe geometry. It shows how shapes and math objects relate.
Geometric combinatorics is a part of math that looks at geometric shapes and their properties, like the faces of solid shapes or where shapes meet.
Graph theory is the study of graphs. Graphs are made of points and lines that connect them. It has many real-life uses, like showing networks and relationships.
Group theory is the study of groups. Groups are special math structures that follow certain rules.
Geometry is a part of math that studies shapes and spaces. It started with looking at length, area, and volume. It now includes many kinds of geometry like Euclidean, projective, and non-Euclidean geometry.
H
see classical analysis
This area of mathematics shows how functions can be drawn using waves. It uses ideas from Fourier series and Fourier transforms, which are parts of Fourier analysis.
This part of category theory looks closer at how things connect, using special arrows to study their shapes.
This studies structures that become more detailed.
This method helps us understand smooth shapes by using partial differential equations.
Holomorphic functional calculus
This branch of mathematics starts with holomorphic functions.
This studies patterns in algebra using homology.
Also called Lobachevskian geometry or Bolyai-Lobachevskian geometry, this is a type of non-Euclidean geometry that looks at hyperbolic space.
hyperbolic trigonometry
This studies hyperbolic triangles in hyperbolic geometry, or special hyperbolic functions in regular geometry. Other types include gyrotrigonometry and universal hyperbolic trigonometry.
This extends the study of functions to include more complex numbers.
I
Ideal theory was an old name for what we now call commutative algebra. It studies ideals inside commutative rings.
Idempotent analysis looks at special number systems called idempotent semirings, like the tropical semiring.
Incidence geometry studies how shapes like curves and lines relate to each other.
Infinitary combinatorics expands combinatorics — the study of arrangements — to include infinite sets.
Infinitesimal analysis and infinitesimal calculus are older names for the calculus of infinitesimals.
Information geometry mixes ideas from differential geometry to study probability theory and statistics. It looks at special spaces called statistical manifolds linked to probability distributions.
Integral calculus is a part of calculus that works with integralss, unlike differential calculus.
Integral geometry studies measures in space that stay the same even when the space is moved or turned.
Intersection theory is a part of both algebraic geometry and algebraic topology.
Intuitionistic type theory is a way to build the foundations of mathematics.
Invariant theory looks at how functions change — or don’t change — when group actions are applied to shapes.
Inversive geometry studies properties that stay the same after a special change called inversion.
Inversive plane geometry is inversive geometry but only in flat, two-dimensional space.
Itô calculus expands calculus to work with unpredictable processes like Brownian motion, useful in mathematical finance and stochastic differential equations.
Iwasawa theory studies number theory problems over endless chains of number fields.
J
Job shop scheduling is a way to organize tasks and machines so that work gets done faster and better. It helps decide when and where each job should be done. This is often used in factories to keep things running well.
Main article: Job shop scheduling
K
K-theory is a way to study special shapes called vector bundles. In topology, it is called topological K-theory. In algebra and geometry, it is called algebraic K-theory. K-theory is also used in some parts of physics.
Other related ideas include K-homology, which looks at properties of spaces, and Kähler geometry, which studies special shapes. There is also knot theory, which is part of topology and looks at knots.
L
L-theory is connected to the K-theory of quadratic forms.
Large deviations theory is a part of probability theory that studies very rare events, called tail events. Large sample theory, also known as asymptotic theory, looks at lattices. These are important in order theory and universal algebra. This area includes Lie algebra theory, Lie group theory, and Lie sphere geometry. Lie sphere geometry is a geometrical theory of planar or spatial geometry that focuses on the circle or sphere.
Line geometry and Linear algebra study linear spaces and linear maps. They are useful in many areas, including abstract algebra. Linear programming is a way to find the best result, like the most profit or least cost, in a mathematical model with linear relationships. There is also a List of graphical methods that shows different ways to display information using diagrams, charts, and plots.
M
Malliavin calculus is a set of tools that extends calculus of variations to work with stochastic processes.
Mathematical biology uses mathematical modeling to study living things. Mathematical chemistry uses modeling for chemical reactions. Mathematical economics applies math to understand economic ideas, while mathematical finance uses applied mathematics to model financial markets.
Mathematical logic studies how formal logic works in math. Mathematical optimization and mathematical physics create math methods to solve physics problems. Mathematical psychology uses mathematical modeling to study how people think and act. Mathematical sciences includes areas like statistics, cryptography, game theory, and actuarial science.
Mathematical sociology uses math to build ideas about society. Mathematical statistics uses probability theory to work with data. Matrix algebra, matrix calculus, and matrix theory all focus on matrices, which are tables of numbers used in many parts of math.
N
Neutral geometry is related to absolute geometry.
Nevanlinna theory is a part of complex analysis. It looks at how meromorphic functions behave. It is named after Rolf Nevanlinna.
Nielsen theory is an area of math. It started from fixed point topology and was developed by Jakob Nielsen.
Non-abelian class field theory, Non-classical analysis, Non-Euclidean geometry, Non-standard analysis, Non-standard calculus, and Nonarchimedean dynamics are special areas of math study.
Noncommutative algebra includes Noncommutative algebraic geometry. This is a part of noncommutative geometry. It looks at the geometric properties of certain math objects.
Noncommutative geometry and Noncommutative harmonic analysis and Noncommutative topology are areas of math research.
Nonlinear analysis and Nonlinear functional analysis study complex mathematical functions.
Number theory is a branch of pure mathematics. It mainly studies the integers. It was originally called arithmetic or higher arithmetic.
Numerical analysis and Numerical linear algebra are important areas of math. They help solve problems with numbers.
O
Operad theory is a part of algebra that studies basic structures. Operation research looks at ways to make processes better. Operator K-theory and Operator theory are parts of functional analysis that focus on special math objects called operators. Optimal control theory extends ideas from the calculus of variations. Optimal maintenance deals with keeping things working best. Orbifold theory is another area of abstract math. Order theory studies how things can be arranged. Ordered geometry is a type of geometry that looks at how points sit between each other. It helps connect affine geometry, absolute geometry, and hyperbolic geometry. Oscillation theory is also a part of mathematics.
P
p-adic analysis is a part of number theory that looks at functions using special numbers called p-adic numbers. p-adic dynamics uses these ideas to study certain types of equations. Paraconsistent mathematics is an area that tries to build math using different rules.
Other areas include partition theory, perturbation theory, and plane geometry, which focuses on shapes and spaces. Polyhedral combinatorics studies the shapes of simple 3D figures, while probability theory deals with chance. Projective geometry looks at properties of shapes that stay the same even when we change how we view them. Pure mathematics studies ideas that don’t need real-world examples.
Q
Quantum calculus is a special kind of calculus that does not use limits.
Quantum geometry extends geometry to help explain strange behaviors in the tiny world of quantum physics. Quaternionic analysis is a part of mathematics that studies numbers in a unique way.
R
Ramsey theory looks at when order must appear, named after Frank P. Ramsey.
Rational geometry studies parts of algebra related to real algebraic geometry.
Real algebraic geometry focuses on 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. It also extends these ideas to complex numbers in complex analysis.
Recreational mathematics is fun math, including mathematical puzzles and mathematical games.
Representation theory studies algebraic structures by showing their elements as linear transformations of vector spaces.
Ribbon theory is a part of topology that looks at 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 special spaces called Riemannian manifolds, expanding ideas from regular geometry, analysis, and calculus. It is named after Bernhard Riemann.
S
the study of schemes introduced by Alexander Grothendieck. It helps us understand shapes.
a part of geometry that looks at special sets of points.
This looks at how small and large parts of shapes are connected.
studies the properties of single operations.
a part of geometry that looks at where shapes can change suddenly.
a careful way to study very tiny changes.
studies how the shapes of objects relate to their vibrations.
looks at the properties of networks.
part of operator theory that extends ideas of special numbers and directions from simple math to more complex systems.
Spectral theory of ordinary differential equations
part of spectral theory that studies solutions to certain equations.
Spectrum continuation analysis
extends the idea of breaking functions into patterns.
a branch of geometry that studies the surface of a sphere.
a part of spherical geometry that studies shapes on the surface of a sphere, usually triangles.
the math used to study patterns and chances. This includes probability theory.
Stochastic calculus of variations
the study of random patterns of points.
a part of geometry that changes one shape into another.
also known as algebraic computation. It is about working with math expressions using symbols.
a branch of geometry and topology that studies special kinds of shapes.
Synthetic differential geometry
a new way to look at shapes using special logic.
also known as axiomatic geometry. It is a branch of geometry that uses basic rules to learn about shapes.
a branch of geometry that studies special measurements of shapes.
the study of special lines in curved geometry.
T
Tensor algebra, Tensor analysis, Tensor calculus, Tensor theory study and use tensors, which are like general versions of vectors. A tensor algebra is a special math tool used to define tensors.
Tessellation means making patterns where tiles repeat again and again.
Theoretical physics is a part of science that uses math to explain and predict what happens in the world.
Theory of computation, Time-scale calculus, and Topology are areas that explore different math ideas and ways to solve problems.
Topological combinatorics uses ideas from topology to help organize and count things.
Topological degree theory, Topological graph theory, Topological K-theory, and Topos theory are parts of topology that study different math structures.
Toric geometry and Transcendental number theory study special numbers and shapes.
Transformation geometry looks at how shapes change when we move or turn them.
Trigonometry is the study of triangles and how their sides and angles relate. This is important in many areas of math.
Tropical analysis is also called idempotent analysis.
Tropical geometry, Twisted K-theory, and Type theory are advanced parts of math that connect different areas of the subject.
U
The umbral calculus is a special way to study something called Sheffer sequences. Uncertainty theory is a new part of mathematics. It looks at ideas like normality and change.
Universal algebra studies the basic rules of many math shapes. Universal hyperbolic trigonometry helps us understand curved space using ideas from rational geometry.
Main article: Umbral calculus
Main articles: Universal algebra, Universal hyperbolic trigonometry
V
Vector algebra is a part of linear algebra. It focuses on adding vectors and multiplying them by numbers. It also includes special operations like the dot product and cross product. It is different from geometric algebra, which works in more than three dimensions.
Vector analysis, also called vector calculus, is a part of multivariable calculus. It studies how vector fields change and combine in three-dimensional space, using methods like differentiation and integration.
W
Wavelets are special tools in mathematics that help us study data. They let us see patterns and details that are hard to notice. Think of them like zooming in and out on a picture. Wavelets are useful for working with sound, images, and other kinds of information.
Related articles
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