The Mathematics Subject Classification (MSC) is an alphanumerical

classification scheme In information science and ontology, a classification scheme is the product of arranging things into kinds of things (classes) or into ''groups'' of classes; this bears similarity to categorization, but with perhaps a more theoretical bent, as cla ...

collaboratively produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and

Zentralblatt MATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastruct ...

. The MSC is used by many mathematics journals, which ask authors of

research papers Academic publishing is the subfield of publishing which distributes academic research and scholarship. Most academic work is published in academic journal articles, books or theses. The part of academic written output that is not formally publ ...

and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The current version is MSC2020.

Structure

The MSC is a hierarchical scheme, with three levels of structure. A classification can be two, three or five digits long, depending on how many levels of the classification scheme are used. The first level is represented by a two-digit number, the second by a letter, and the third by another two-digit number. For example: * 53 is the classification for differential geometry * 53A is the classification for classical differential geometry * 53A45 is the classification for

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

and

tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...

analysis

First level

At the top level, 64 mathematical disciplines are labeled with a unique two-digit number. In addition to the typical areas of mathematical research, there are top-level categories for "

History History (derived ) is the systematic study and the documentation of the human activity. The time period of event before the invention of writing systems is considered prehistory. "History" is an umbrella term comprising past events as well ...

and

Biography A biography, or simply bio, is a detailed description of a person's life. It involves more than just the basic facts like education, work, relationships, and death; it portrays a person's experience of these life events. Unlike a profile or ...

", "

Mathematics Education In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge. Although re ...

", and for the overlap with different sciences.

Physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

(i.e. mathematical physics) is particularly well represented in the classification scheme with a number of different categories including: *

Fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...

Quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...

Geophysics Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' so ...

Optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...

and

electromagnetic theory In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...

All valid MSC classification codes must have at least the first-level identifier.

Second level

The second-level codes are a single letter from the Latin alphabet. These represent specific areas covered by the first-level discipline. The second-level codes vary from discipline to discipline. For example, for differential geometry, the top-level code is 53, and the second-level codes are: * A for classical differential geometry * B for local differential geometry * C for global differential geometry * D for symplectic geometry and contact geometry In addition, the special second-level code "-" is used for specific kinds of materials. These codes are of the form: * 53-00 General reference works (handbooks, dictionaries, bibliographies, etc.) * 53-01 Instructional exposition (textbooks, tutorial papers, etc.) * 53-02 Research exposition (monographs, survey articles) * 53-03 Historical (must also be assigned at least one classification number from Section 01) * 53-04 Explicit machine computation and programs (not the theory of computation or programming) * 53-06 Proceedings, conferences, collections, etc. The second and third level of these codes are always the same - only the first level changes. For example, it is not valid to use 53- as a classification. Either 53 on its own or, better yet, a more specific code should be used.

Third level

Third-level codes are the most specific, usually corresponding to a specific kind of mathematical object or a well-known problem or research area. The third-level code 99 exists in every category and means ''none of the above, but in this section''.

Using the scheme

The AMS recommends that papers submitted to its journals for publication have one primary classification and one or more optional secondary classifications. A typical MSC subject class line on a research paper looks like MSC Primary 03C90; Secondary 03-02;

History

According to the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

(AMS) help page about MSC, the MSC has been revised a number of times since 1940. Based on a scheme to organize AMS's ''Mathematical Offprint Service'' (MOS scheme), the ''AMS Classification'' was established for the classification of reviews in ''Mathematical Reviews'' in the 1960s. It saw various ad-hoc changes. Despite its shortcomings, Zentralblatt für Mathematik started to use it as well in the 1970s. In the late 1980s, a jointly revised scheme with more formal rules was agreed upon by Mathematical Reviews and Zentralblatt für Mathematik under the new name Mathematics Subject Classification. It saw various revisions as ''MSC1990'', ''MSC2000'' and ''MSC2010''. In July 2016, Mathematical Reviews and zbMATH started collecting input from the mathematical community on the next revision of MSC, which was released as MSC2020 in January 2020.MSC2020 available now
/ref> The original classification of older items has not been changed. This can sometimes make it difficult to search for older works dealing with particular topics. Changes at the first level involved the subjects with (present) codes 03, 08, 12-20, 28, 37, 51, 58, 74, 90, 91, 92.

Relation to other classification schemes

For physics papers the Physics and Astronomy Classification Scheme (PACS) is often used. Due to the large overlap between mathematics and physics research it is quite common to see both PACS and MSC codes on research papers, particularly for multidisciplinary journals and repositories such as the

arXiv arXiv (pronounced "archive"—the X represents the Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not peer review. It consists of ...

. The

ACM Computing Classification System The ACM Computing Classification System (CCS) is a subject classification system for computing devised by the Association for Computing Machinery (ACM). The system is comparable to the Mathematics Subject Classification (MSC) in scope, aims, and s ...

(CCS) is a similar

hierarchical classification Hierarchical classification is a system of grouping things according to a hierarchy. In the field of machine learning, hierarchical classification is sometimes referred to as instance space decomposition, which splits a complete multi-class pro ...

scheme for

computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...

. There is some overlap between the AMS and ACM classification schemes, in subjects related to both mathematics and computer science, however the two schemes differ in the details of their organization of those topics. The classification scheme used on the arXiv is chosen to reflect the papers submitted. As arXiv is multidisciplinary its classification scheme does not fit entirely with the MSC, ACM or PACS classification schemes. It is common to see codes from one or more of these schemes on individual papers.

First-level areas

*00: General (Includes topics such as

recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...

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 peop ...

and mathematical modeling.) *01:

and

biography A biography, or simply bio, is a detailed description of a person's life. It involves more than just the basic facts like education, work, relationships, and death; it portrays a person's experience of these life events. Unlike a profile or ...

*03:

Mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

and foundations (including model theory, computability theory,

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

, proof theory, and algebraic logic) *05: Combinatorics *06: Order, lattices, ordered algebraic structures *08: General

algebraic system In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...

s *11:

Number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

*12: Field theory and

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...

s *13:

Commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...

( Commutative rings and

algebras In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...

) *14: Algebraic geometry *15:

Linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...

and

multilinear algebra Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p' ...

;

matrix theory In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begi ...

*16:

Associative ring In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying pro ...

s and (associative) algebras *17:

Non-associative ring A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if ...

s and (non-associative) algebras *18: Category theory; homological algebra *19: -theory *20:

Group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

and generalizations *22:

Topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...

s, Lie groups (and analysis upon them) *26:

Real function In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an interv ...

s (including

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

s and

integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...

s) *28: Measure and integration *30: Functions of a complex variable (including approximation theory in the complex domain) *31:

Potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...

*32: Several complex variables and

analytic space An analytic space is a generalization of an analytic manifold that allows singularities. An analytic space is a space that is locally the same as an analytic variety. They are prominent in the study of several complex variables, but they also ...

s *33:

Special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined b ...

*34:

Ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

s *35: Partial differential equations *37:

Dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...

s and ergodic theory *39: Difference (equations) and

functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...

s *40:

Sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...

series Series may refer to: People with the name * Caroline Series (born 1951), English mathematician, daughter of George Series * George Series (1920–1995), English physicist Arts, entertainment, and media Music * Series, the ordered sets used in ...

summability In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must ...

*41:

Approximations An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ...

and expansions *42: Harmonic analysis on

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

s (including Fourier analysis, Fourier transforms, trigonometric approximation, trigonometric interpolation, and orthogonal functions) *43: Abstract harmonic analysis *44:

Integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...

operational calculus Operational calculus, also known as operational analysis, is a technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial equation. History Th ...

*45:

Integral equation In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ...

s *46:

Functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...

(including

infinite-dimensional holomorphy In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values in complex Banach spaces (or Fréchet spaces mor ...

integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ...

s in distribution spaces) *47:

Operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...

*49: Calculus of variations and optimal control;

optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

(including geometric integration theory) *51:

Geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

*52: Convex (geometry) and

discrete geometry Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic ge ...

*53: Differential geometry *54:

General topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometri ...

*55:

Algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...

*57: Manifolds and cell complexes *58:

Global analysis In mathematics, global analysis, also called analysis on manifolds, is the study of the global and topological properties of differential equations on manifolds and vector bundles. Global analysis uses techniques in infinite-dimensional manifold th ...

, analysis on manifolds (including

) *60:

Probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

and stochastic processes *62: Statistics *65:

Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...

*68:

Computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...

*70:

Mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to object ...

of particles and systems (including

particle mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects re ...

) *74: Mechanics of deformable solids *76:

*78:

*80: Classical

thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of th ...

heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, ...

*81:

Quantum theory Quantum theory may refer to: Science *Quantum mechanics, a major field of physics *Old quantum theory, predating modern quantum mechanics * Quantum field theory, an area of quantum mechanics that includes: ** Quantum electrodynamics ** Quantum ...

*82: Statistical mechanics, structure of matter *83: Relativity and gravitational theory (including

relativistic mechanics In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...

) *85:

Astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...

and astrophysics *86:

*90:

Operations research Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve decis ...

mathematical programming Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...

*91: Game theory,

economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...

social Social organisms, including human(s), live collectively in interacting populations. This interaction is considered social whether they are aware of it or not, and whether the exchange is voluntary or not. Etymology The word "social" derives from ...

and behavioral sciences *92:

Biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...

and other natural sciences *93:

Systems theory Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structu ...

;

control Control may refer to: Basic meanings Economics and business * Control (management), an element of management * Control, an element of management accounting * Comptroller (or controller), a senior financial officer in an organization * Controllin ...

(including optimal control) *94:

Information Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random ...

and

communication Communication (from la, communicare, meaning "to share" or "to be in relation with") is usually defined as the transmission of information. The term may also refer to the message communicated through such transmissions or the field of inqui ...

, circuits *97:

Mathematics education In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge. Although re ...

References

External links

MSC2020-Mathematical Sciences Classification System
PDF of MSC2020. *The

page on th
Mathematics Subject Classification
MSC2020 can be seen here.
Mathematics Subject Classification 2010
The site where the MSC2010 revision was carried out publicly in an MSCwiki. A view of the whole scheme and the changes made from MSC2000, as well as PDF files of the MSC and ancillary documents are there. A personal copy of the MSC in

TiddlyWiki TiddlyWiki is a personal wiki and a non-linear notebook for organising and sharing complex information. It is an open-source single page application wiki in the form of a single HTML file that includes CSS, JavaScript, embedded files such as ...

form can be had also. *The

page o
the Mathematics Subject Classification
*{{cite web , last1=Rusin , first1=Dave , title=A Gentle Introduction to the Mathematics Subject Classification Scheme , url=http://www.math.niu.edu/~rusin/known-math/index/beginners.html , website= Mathematical Atlas , archiveurl=https://web.archive.org/web/20150516045812/http://www.math.niu.edu/~rusin/known-math/index/beginners.html , archivedate=2015-05-16 Fields of mathematics Mathematical classification systems