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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. The MSC is used by many mathematics journals, which ask authors of research papers 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 and 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 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", and for the overlap with different sciences. Physics (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 bio ...
* Quantum mechanics * Geophysics * Optics 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 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 Infrastructu ...
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 The Physics and Astronomy Classification Scheme (PACS) is a scheme developed in 1970 by the American Institute of Physics (AIP) for classifying scientific literature using a hierarchical set of codes. PACS has been used by over 160 international jou ...
(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 s ...
. The ACM Computing Classification System (CCS) is a similar hierarchical classification scheme for computer science. 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,
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 ...
and mathematical modeling.) *01: History 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 and
foundations Foundation may refer to: * Foundation (nonprofit), a type of charitable organization ** Foundation (United States law), a type of charitable organization in the U.S. ** Private foundation, a charitable organization that, while serving a good cause ...
(including model theory,
computability theory Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since ...
, set theory,
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding parts, ...
, and
algebraic logic In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for ...
) *05: Combinatorics *06:
Order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
, 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 ...
s *11: Number theory *12: Field theory and polynomials *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. Prominent ...
(
Commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
s 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 Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
*15: 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 *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 prop ...
s and (associative) algebras *17: Non-associative rings and (non-associative) algebras *18:
Category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
; homological algebra *19: -theory *20:
Group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
and generalizations *22: Topological groups,
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
s (and analysis upon them) *26: Real functions (including derivatives and integrals) *28: Measure and
integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
*30:
Functions of a complex variable Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
(including
approximation theory In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby. Note that what is meant by ''best'' and ''simpler'' ...
in the complex domain) *31: Potential theory *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 a ...
s *33: Special functions *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 equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
s *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 equations *40: Sequences,
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 spaces (including Fourier analysis, Fourier transforms, trigonometric approximation, trigonometric interpolation, and
orthogonal function In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be the integral of the product of functions over the ...
s) *43: Abstract harmonic analysis *44: Integral transforms, operational calculus *45: Integral equations *46: Functional analysis (including infinite-dimensional holomorphy, integral transforms in distribution spaces) *47: Operator theory *49:
Calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
and
optimal control Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering an ...
; optimization (including geometric integration theory) *51: Geometry *52: Convex (geometry) and discrete geometry *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, geometr ...
*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:
Manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
s and cell complexes *58: Global analysis, analysis on manifolds (including infinite-dimensional holomorphy) *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 In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
*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 t ...
*68: Computer science *70: Mechanics of particles and systems (including particle mechanics) *74: Mechanics of deformable solids *76:
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 bio ...
*78: Optics,
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 ...
*80: Classical thermodynamics, heat transfer *81: Quantum theory *82:
Statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, 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, galax ...
and
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the he ...
*86: Geophysics *90: Operations research, mathematical programming *91:
Game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applic ...
, economics, social and
behavioral sciences Behavioral sciences explore the cognitive processes within organisms and the behavioral interactions between organisms in the natural world. It involves the systematic analysis and investigation of human and animal behavior through naturalistic o ...
*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 ...
and other natural sciences *93: Systems theory;
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 * Controlli ...
(including
optimal control Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering an ...
) *94: Information and communication, circuits *97: Mathematics education


See also

*
Areas of mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
* Mathematical knowledge management * MathSciNet


References


External links


MSC2020-Mathematical Sciences Classification System
PDF of MSC2020. *The Zentralblatt MATH 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
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, ...
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