1. Classification Systems

• Wikipedia uses a Category: Mathematics system on its articles, and also has a list of mathematics lists.
• The Mathematics Subject Classification (MSC) is produced by the staff of the review databases Mathematical Reviews and Zentralblatt MATH. Many mathematics journals ask authors to label their papers with MSC subject codes. The MSC divides mathematics into over 60 areas, with further subdivisions within each area.
• In the Library of Congress Classification, mathematics is assigned the many subclass QA within the class Q (Science). The LCC defines broad divisions, and individual subjects are assigned specific numerical values.
• The Dewey Decimal Classification assigns mathematics to division 510, with subdivisions for Algebra & Number theory, Arithmetic, Topology, Analysis, Geometry, Numerical analysis, and Probabilities & Applied mathematics.
• The Categories within Mathematics list is used by the arXiv for categorizing preprints. It differs from MSC; for example, it includes things like Quantum algebra.
• The IMU uses its programme structure for organizing the lectures at its ICM every four years. One top-level section that MSC doesn't have is Lie theory.
• The ACM Computing Classification System includes a couple of mathematical categories F. Theory of Computation and G. Mathematics of Computing.
• MathOverflow has a tag system.
• Mathematics book publishers such as Springer (subdisciplines), Cambridge University Press (Browse Mathematics and statistics) and the AMS (subject area) use their own subject lists on their websites to enable customers to browse books or filter searches by subdiscipline, including topics such as mathematical biology and mathematical finance as top-level headings.
• Schools and other educational bodies have syllabuses.
• SIAM divides the areas of applied mathematics in activity groups.

2. Major Divisions of Mathematics

2.1. Pure Mathematics

Foundations of mathematics

Mathematicians have always worked with logic and symbols, but for centuries the underlying laws of logic were taken for granted, and never expressed symbolically. Mathematical logic, also known as symbolic logic, was developed when people finally realized that the tools of mathematics can be used to study the structure of logic itself. Areas of research in this field have expanded rapidly, and are usually subdivided into several distinct subfields.

• Proof theory and constructive mathematics: Proof theory grew out of David Hilbert's ambitious program to formalize all the proofs in mathematics. The most famous result in the field is encapsulated in Gödel's incompleteness theorems. A closely related and now quite popular concept is the idea of Turing machines. Constructivism is the outgrowth of Brouwer's unorthodox view of the nature of logic itself; constructively speaking, mathematicians cannot assert "Either a circle is round, or it is not" until they have actually exhibited a circle and measured its roundness.

• Model theory: Model theory studies formal mathematical structures in a general framework by combining mathematical logic and tools from various areas in mathematics, particularly algebra. Its main tool is first-order logic.

• Computability theory: Also known as recursion theory, computability theory formally studies computation, algorithms and computable functions. Its origin lies in the Entscheidungsproblem of David Hilbert and its solution via the Church–Turing thesis that basically describes how any function that is computable by an algorithm is a computable function. It has close ties with many topics in theoretical computer science.

• Set theory: A set can be thought of as a collection of distinct things united by some common feature. Set theory is subdivided into three main areas. Naive set theory is the original set theory developed by mathematicians at the end of the 19th century. Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naive set theory. It treats sets as "whatever satisfies the axioms", and the notion of collections of things serves only as motivation for the axioms. Internal set theory is an axiomatic extension of set theory that supports a logically consistent identification of illimited (enormously large) and infinitesimal (unimaginably small) elements within the real numbers. See also List of set theory topics.

• Category theory: A category expands on the notion of sets to consider both collections of distinct objects and relationships between them, referred to as "maps," "morphisms," or "arrows," organizing mathematical information into the structure of a labeled directed graph. As an organizational structure, category theory finds common usage throughout many areas of pure mathematics, particularly in subjects with algebraic connections such as representation theory, algebraic topology, algebraic geometry, homology theory, homotopy theory, and homological algebra. However, due to its ubiquitous structure, category theory has seen applications in many other areas of mathematics, as well as in computer science (particularly functional programming), physics, and other subjects.

Analysis

Within mathematics, analysis is the branch that focuses on functions, limits, derivatives, rates of change, integrals, and multiple things changing relative to (or independently of) one another.

Modern analysis is a vast and rapidly expanding branch of mathematics that touches almost every other subdivision of the discipline, finding direct and indirect applications in topics as diverse as number theory, cryptography, and abstract algebra. It is also the language of science itself and is used across chemistry, biology, and physics, from astrophysics to X-ray crystallography. Within mathematics itself analysis is commonly used in other mathematical disciplines such as analytic number theory, probability theory, and differential geometry. As one of the largest branches of pure mathematics it has many subfields.

• Real analysis: Real analysis studies real-valued functions and real numbers. Traditionally it only includes topics of a single real variable. Major topics include sequences and series of real numbers and functions, continuity, compactness of the Euclidean line, limits of functions, and the Riemann integral. It is commonly taught at school level in a non-rigorous manner as calculus. It forms the basis for almost all further study in analysis.

• Multivariable real analysis & Vector analysis: Multivariable analysis studies similar topics to real analysis except for studying them in the setting of several variables rather than one. Vector analysis is concerned with differentiation and integration of vector fields and is particularly useful for studying fields in physics and engineering. Major topics include gradient, curl, Green's theorem, Stokes' theorem and the divergence theorem. Like real analysis, these topics are commonly taught at a undergraduate level as multivariable calculus.

• Complex analysis: Complex analysis studies functions of complex numbers. Functions studied here display a particularly "niceness" due the Cauchy–Riemann equations that enable them to be infinitely differentiable and analytic (locally equal to its own Taylor series).Common topics studies include holomorphic functions, Cauchy's integral theorem, the residue theorem and conformal maps. It is widely applicable in physics and engineering, particularly for solving certain kinds of integrals. Like real analysis, an extension of complex analysis to functions of multiple variables exists.

• Fourier analysis: Fourier analysis studies the Fourier series and in general how functions can be decomposed into sums of simpler trigonometric functions on the real line. It has widespread applications in physics and engineering, such as the Fourier transform.

• Harmonic analysis: Harmonic analysis is a generalization of Fourier analysis that studies similar topics but generalized to arbitrary spaces and on various kinds of mathematics structures such as groups. Major topics of study include periodic functions, wavelets, the Peter–Weyl theorem and the Pontryagin duality. It has particularly strong connections to representation theory.

• Measure theory: Measure theory studies measures, a generalization of various kinds of geometric measurements. Technically speaking a measure on a set is a way of giving a number to each subset of that set, thus giving a method of producing the size of a set. Measure theory is foundational in Lebesgue integration through the use of Lebesgue measures and probability theory through the use of probability measures. The field also has important connections with set theory and functional analysis, sometimes being considered a subfield of either real or functional analysis. A major subfield of measure theory is geometric measure theory.

• Functional analysis: Functional analysis deals with infinite-dimensional vector spaces equipped with additional structure such as topology or inner product and study of continuous maps and other topics such as convergence from classical real analysis between them. Alternatively, functional analysis can be described to study function spaces and linear operators between them. However it is important to note that functional analysis also studies nonlinear maps, although this area of study is much smaller compared to linear functional analysis. Major results include the Uniform boundedness theorem, the Hahn–Banach theorem, Open mapping theorem and the various kinds of spectral theorems. The main subdivisions of study include topological vector spaces, the theory of distributions, operator theory and operator algebras. Functional analysis has wide applications to many fields of mathematics and physics ranging from forming a mathematical basis for quantum mechanics and the theory behind partial differential equations to dynamical systems and ergodic theory.

• Integral & Differential equations: Differential equations are equations that contain at least one derivative of a function within it. That is, an equation that contains a term that represents the rate of change of a certain quantity. They are widely used for mathematically modelling various phenomena that continuously change in fields such as physics, economics, engineering and biology. The two main subdivisions of equations of this type are ordinary differential equations and partial differential equations. If the function involved only has one variable that contains derivatives it is called an ordinary differential equation, otherwise, if it contains multiple independent variables that each have partial derivatives, it is known as a partial differential equation (PDE). PDEs can then further be split into Elliptic partial differential equations, Parabolic partial differential equations and Hyperbolic partial differential equations. Partial differential equations have widespread applications in other areas of mathematics, particularly functional analysis and geometric analysis. Integral equations on the other hand are equations that contain a unknown function under an integral sign. They are related to differential equations and have applications to physics.

• Calculus of variations: The calculus of variations is a field of analysis where one tries to find a function where a certain minima or maxima of a certain integral is reached. It is particularly connected to problems of classical physics such as the Euler–Lagrange equation, Dirichlet's principle, Noether's theorem, Plateau's problem, and in general of minimal surfaces. Modern work in this area includes the development of the theory of calculus of variations in a global sense, now known as Morse theory. It can be considered as a "generalization" of traditional differential and integral calculus.

• Numerical analysis: Numerical analysis studies methods of numerically approximating solutions to problems of mathematical analysis rather than symbolically attempting to solve them, particularly through the usage of algorithms. It is commonly applied to problems in science and engineering where problems may be too difficult to solve analytically. Particularly systems of ordinary differential equations and numerical linear algebra are easy to express numerically and problems are reduced to being formed in these fields to then be solved using tools from numerical analysis. Modern numerical analysis heavily utilizes scientific computing however there exists a rich history of numerically approximation before the invention of the computer.

Algebra

The study of structure begins with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of these numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies groups, rings, and fields, structures that generalize the properties possessed by everyday numbers. Long standing questions about compass and straightedge constructions were finally settled by Galois theory. The physically important concept of vectors, generalized to vector spaces, is studied in linear algebra. The study of general algebraic structures by their behavior in vector spaces is known as representation theory. Themes common to all kinds of algebraic structures are studied in universal algebra.

• General algebraic systems: Given a set, different ways of combining or relating members of that set can be defined. If these obey certain rules, then a particular algebraic structure is formed. Universal algebra is the more formal study of these structures and systems.

• Order theory: For any two distinct real numbers, one must be greater than the other. Order theory extends this idea to sets in general. It includes notions like lattices, Boolean algebras and ordered algebraic structures. See also the order theory glossary and the list of order topics.

• Group theory:
  • Finite group:
  • Topological group:
    • Discrete group:
    • Lie group:
  • Algebraic group:

• Ring theory:
  • Commutative rings and algebras: In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, then a×b = b×a. Commutative algebra is the field of study of commutative rings and their ideals, modules and algebras. It is foundational both for algebraic geometry and for algebraic number theory. The most prominent examples of commutative rings are rings of polynomials.
  • Noncommutative rings and algebras:

• Field theory and Galois theory: Field theory studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined. A polynomial is an expression in which constants and variables are combined using only addition, subtraction, and multiplication.

• Modules and vector spaces:

• Representation theory:

• Algebra over a field: commonly known as just an 'algebra', they are vector spaces with a bilinear product. As the axioms of an algebra do not specify associativity, algebras can generally be split into two types, associative algebras and non-associative algebras. Many common rings form associative algebras, for example any commutative ring is an algebra over its subrings, the complex numbers form a two-dimensional commutative algebra over the reals. The standard n-by-n matrices over a field also form an associative algebra. By comparison the non-associative algebras (also known as distributive algebras) are where the multiplication operation is not associative. Examples include the Euclidean space R³ with cross product multiplication, Lie algebras and Jordan algebras. More general classes of non-associative algebras include graded algebras, division algebras and Cayley–Dickson algebras in general.

• Homological algebra:

Number theory

Number theory is the study of numbers and the properties of operations between them. Number theory is traditionally concerned with the properties of integers, but more recently, it has come to be concerned with wider classes of problems that have arisen naturally from the study of integers.

• Arithmetic: An elementary part of number theory that primarily focuses upon the study of natural numbers, integers, fractions, and decimals, as well as the properties of the traditional operations on them: addition, subtraction, multiplication and division. Up until the 19th century, arithmetic and number theory were synonyms, but the evolution and growth of the field has resulted in arithmetic referring only to the elementary branch of number theory.

• Elementary number theory: The study of integers at a higher level than arithmetic, where the term 'elementary' here refers to the fact that no techniques from other mathematical fields are used.

• Analytic number theory: Calculus and complex analysis are used as tools to study the integers.

• Algebraic number theory: The techniques of abstract algebra are used to study the integers, as well as algebraic numbers, the roots of polynomials with integer coefficients.

• Other number theory subfields : Geometric number theory; combinatorial number theory; transcendental number theory; and computational number theory. See also the list of number theory topics.

Combinatorics

Combinatorics is the study of finite or discrete collections of objects that satisfy specified criteria. In particular, it is concerned with "counting" the objects in those collections (enumerative combinatorics) and with deciding whether certain "optimal" objects exist (extremal combinatorics). It includes graph theory, used to describe interconnected objects (a graph in this sense is a network, or collection of connected points). See also the list of combinatorics topics, list of graph theory topics and glossary of graph theory. A combinatorial flavour is present in many parts of problem solving.

Geometry

Geometry deals with spatial relationships, using fundamental qualities or axioms. Such axioms can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions. See also List of geometry topics.

• Euclidean geometry: Geometry in its classical sense. Deals with basic concepts such as points, lines, planes, angles, triangles, congruence, similarity, circles, solid geometry, and analytic geometry.

• Non-Euclidean geometry: Historically replaces Euclid's parallel postulate with an alternative, resulting in elliptic, spherical and hyperbolic geometries.

• Projective geometry: Introduced in the 16th century by Girard Desargues, it extends Euclidean geometry by adding points at infinity at which parallel lines intersect. This simplifies many aspects of classical geometry by avoiding to have a different treatment for intersecting and parallel lines.

• Affine geometry: The study of properties relative to parallelism and independent from the concept of length.

• Convex geometry: Includes the study of objects such as polytopes and polyhedra, and more generally of convex sets. Has important applications in mathematical optimization. See also List of convexity topics.

• Discrete geometry and combinatorial geometry: The study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation. It includes the study of shapes such as the Platonic solids and the notion of tessellation.

• Differential geometry: The study of geometry using calculus and differentiable functions. It is closely related to differential topology, differing that objects studied in differential geometry have a geometric structure imposed on them. Traditionally covered areas such as curvature, differential geometry of curves and differential geometry of surfaces. More modern topics cover such areas as Riemannian geometry, symplectic manifolds, Finsler manifolds, Poisson manifolds, conformal geometry, contact geometry and geometric analysis. There are significant connections to physics and engineering, particularly in General relativity with Pseudo-Riemannian manifolds, classical mechanics with symplectic geometry and quantum field theory with gauge theories. See also the glossary of differential geometry and topology.

• Algebraic geometry: Given a polynomial of two real variables, the points on a plane where that function is zero will form a curve. An algebraic curve extends this notion to polynomials over a field in a given number of variables. Algebraic geometry may be viewed as the study of these curves. See also the list of algebraic geometry topics and list of algebraic surfaces.
  • Real algebraic geometry: The study of semialgebraic sets, i.e. real number solutions to algebraic inequalities with real number coefficients, and mappings between them.
  • Complex geometry: Studies geometric constructs made with complex numbers and the complex plane. In particular, modern complex geometry focuses on complex manifolds and Riemann surfaces, complex algebraic varieties, complex analytic varieties, holomorphic vector bundles and coherent sheaves. A mix of complex analysis, algebraic geometry and differential geometry, it has connections to theoretical physics and string theory through Calabi–Yau manifolds.
  • Arithmetic geometry: The study of schemes of finite type over the spectrum of the ring of integers. Alternatively defined as the application of the techniques of algebraic geometry to problems in number theory.
  • Diophantine geometry: The study of the points of algebraic varieties with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, and p-adic fields, but not including the real numbers.

• Computational geometry: Computational geometry deals with algorithms and their implementations for manipulating geometrical shapes and objects. Although it is a new area of research, it has many applications in computer vision, digital image processing, computer-aided design, medical imaging and areas of pure mathematics such as computational algebraic geometry.

• Fractal geometry: The study of fractals, geometric objects exhibiting self-similar behavior. Common examples such as the Mandelbrot set and the Cantor set motivate the study of fractal geometry, and the subject enjoys applications toward many other disciplines both within mathematics and the sciences.

Topology

Deals with the properties of a figure that do not change when the figure is continuously deformed. The main areas are point set topology (or general topology), algebraic topology, and the topology of manifolds, defined below.

• General topology: Also called point set topology. Properties of topological spaces. Includes such notions as open and closed sets, compact spaces, continuous functions, convergence, separation axioms, metric spaces, dimension theory. See also the glossary of general topology and the list of general topology topics.

• Algebraic topology: Properties of algebraic objects associated with a topological space and how these algebraic objects capture properties of such spaces. (Some of these algebraic objects are examples of functors.) Contains areas like homology theory, cohomology theory, homotopy theory, and homological algebra. Homotopy deals with homotopy groups (including the fundamental group) as well as simplicial complexes and CW complexes (also called cell complexes). See also the list of algebraic topology topics.

• Differential topology: The field dealing with differentiable functions on differentiable manifolds, which can be thought of as an n:dimensional generalization of a surface in the usual 3-dimensional Euclidean space.

• Geometric topology: Tends to focus on low-dimensional manifolds (manifolds of dimensions 2, 3 and 4) as in higher dimensions algebraic tools make solving problems easier however generally can be defined to include the study of manifolds with some geometric stucture and maps between them. Topics studied include embeddings, knot theory, surgery theory, cobordisms, orientability, handle decompositions and the geometrization and Poincaré conjectures.

2.2. Applied Mathematics

Probability and statistics

• Statistics: The science of making effective use of numerical data from experiments or from populations of individuals. Statistics includes not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments. See also the list of statistical topics.

Computational mathematics

• Numerical analysis: Many problems in mathematics cannot in general be solved exactly. Numerical analysis is the study of iterative methods and algorithms for approximately solving problems to a specified error bound. Includes numerical differentiation, numerical integration and numerical methods; c.f. scientific computing. See also List of numerical analysis topics.

• Computer algebra: This area is also called symbolic computation or algebraic computation. It deals with exact computation, for example with integers of arbitrary size, polynomials or elements of finite fields. It includes also the computation with non numeric mathematical objects like polynomial ideals or series.

Mathematical physics

• Classical Mechanics: Addresses and describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.

• Mechanics of structures: Mechanics of structures is a field of study within applied mechanics that investigates the behavior of structures under mechanical loads, such as bending of a beam, buckling of a column, torsion of a shaft, deflection of a thin shell, and vibration of a bridge.

• Mechanics of deformable solids: Most real-world objects are not point-like nor perfectly rigid. More importantly, objects change shape when subjected to forces. This subject has a very strong overlap with continuum mechanics, which is concerned with continuous matter. It deals with such notions as stress, strain and elasticity.

• Fluid mechanics: Fluids in this sense includes not just liquids, but flowing gases, and even solids under certain situations. (For example, dry sand can behave like a fluid). It includes such notions as viscosity, turbulent flow and laminar flow (its opposite).

• Particle mechanics: In mathematics, a particle is a point-like, perfectly rigid, solid object. Particle mechanics deals with the results of subjecting particles to forces. It includes celestial mechanics—the study of the motion of celestial objects.

Other applied mathematics

• Operations research (OR): Also known as operational research, OR provides optimal or near-optimal solutions to complex problems. OR uses mathematical modeling, statistical analysis, and mathematical optimization.

• Mathematical programming: Mathematical programming (or mathematical optimization) minimizes (or maximizes) a real-valued function over a domain that is often specified by constraints on the variables. Mathematical programming studies these problems and develops iterative methods and algorithms for their solution.

• Computational biology

• Computational linguistics

2.3. Other

Philosophy of mathematics

The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's lives.

History of mathematics and biographies of mathematicians

The history of mathematics is inextricably intertwined with the subject itself. This is perfectly natural: mathematics has an internal organic structure, deriving new theorems from those that have come before. As each new generation of mathematicians builds upon the achievements of their ancestors, the subject itself expands and grows new layers.

Recreational mathematics

From magic squares to the Mandelbrot set, numbers have been a source of amusement and delight for millions of people throughout the ages. Many important branches of "serious" mathematics have their roots in what was once a mere puzzle and/or game.