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{{Short description|Classification scheme for mathematics}}
The '''Mathematics Subject Classification''' ('''MSC''') is an alphanumerical [[classification scheme]] that has collaboratively been 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 [[Academic journal|journals]], which ask authors of [[Academic paper|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 [[Euclidean vector|vector]] and [[tensor]] analysis

=== First level ===
At the top level, 63 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]]", "[[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]]
* [[Quantum mechanics]]
* [[Geophysics]]
* [[Optics]] and [[electromagnetic theory]]

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;

=={{anchor|MOS|AMS Classification|MSC1990|MSC2000|MSC2010}}History==
{{expand section|date=January 2014}}
According to the [[American Mathematical Society]] (AMS) help page about MSC,<ref>[https://www.ams.org/mathscinet/help/field_help.html#mscp MR: Help: MSC Primary]</ref> 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''.<ref>[[Bernd Wegner (mathematician)|Bernd Wegner]]. ''Indexierung mathematischer Literatur Die Revision der Mathematics Subject Classification MSC''. Institute of Mathematics, TU Berlin. http://fidmath.de/fileadmin/download/graz_wegner.ppt</ref> In July 2016, Mathematical Reviews and zbMATH started collecting input from the mathematical community on the next revision of MSC,<ref>[https://msc2020.org Announcement of the plan to revise the Mathematics Subject Classification]</ref> which was released as MSC2020
<ref>{{cite journal |journal=EMS Newsletter
|title=Mathematics Subject Classification 2020
|volume=115 |publication-date=4 March 2020 |pp=5-6
|first1=Edward |last1=Dunne |first2=Klaus |last2=Hulek
|url=https://zbmath.org/static/msc2020.pdf
|publisher=Mathematical Reviews and zbMATH Open
|doi=10.4171/NEWS/115/2
|access-date=25 Mar 2025}}</ref> in January 2020.

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]].

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]] and [[mathematical model]]ing.)
*01: [[History of mathematics|History]] and [[List of mathematicians|biography]]
*03: [[Mathematical logic]] and [[Foundations of mathematics|foundations]] (including [[model theory]], [[computability theory]], [[set theory]], [[proof theory]], and [[algebraic logic]])
*05: [[Combinatorics]]
*06: [[Order theory|Order]], lattices, ordered algebraic structures
*08: General [[algebraic system]]s
*11: [[Number theory]]
*12: [[Field theory (mathematics)|Field theory]] and [[polynomial]]s
*13: [[Commutative algebra]] ([[Commutative ring]]s and [[commutative algebra (structure)|algebras]])
*14: [[Algebraic geometry]]
*15: [[Linear algebra|Linear]] and [[multilinear algebra]]; [[Matrix (mathematics)|matrix theory]]
*16: [[Associative ring]]s and [[associative algebra|(associative) algebra]]s
*17: [[Non-associative ring]]s and [[non-associative algebra|(non-associative) algebra]]s
*18: [[Category theory]]; [[homological algebra]]
*19: [[K-theory|{{math|''K''}}-theory]]
*20: [[Group theory]] and generalizations
*22: [[Topological group]]s, [[Lie group]]s (and analysis upon them)
*26: [[Real function]]s (including [[derivative]]s and [[integral]]s)
*28: [[Measure (mathematics)|Measure]] and [[Integral|integration]]
*30: [[Complex function|Functions of a complex variable]] (including [[approximation theory]] in the [[complex number|complex domain]])
*31: [[Potential theory]]
*32: [[Several complex variables]] and [[analytic space]]s
*33: [[Special functions]]
*34: [[Ordinary differential equation]]s
*35: [[Partial differential equation]]s
*37: [[Dynamical system]]s and [[ergodic theory]]
*39: [[Difference equation]]s and [[functional equation]]s
*40: [[Sequence]]s, [[series (mathematics)|series]], [[summability]]
*41: [[Approximation theory|Approximations]] and [[series expansion|expansions]]
*42: [[Harmonic analysis]] on [[Euclidean space]]s (including [[Fourier analysis]], [[Fourier transform]]s, [[trigonometric approximation]], [[trigonometric interpolation]], and [[orthogonal function]]s)
*43: Abstract [[harmonic analysis]]
*44: [[Integral transform]]s, [[operational calculus]]
*45: [[Integral equation]]s
*46: [[Functional analysis]] (including [[infinite-dimensional holomorphy]], [[integral transform]]s in [[distribution space]]s)
*47: [[Operator theory]]
*49: [[Calculus of variations]] and [[optimal control]]; [[Optimization (mathematics)|optimization]] (including [[geometric integration theory]])
*51: [[Geometry]]
*52: [[Convex geometry|Convex]] and [[discrete geometry]]
*53: [[Differential geometry]]
*54: [[General topology]]
*55: [[Algebraic topology]]
*57: [[Manifold]]s and cell complexes
*58: [[Global analysis]], [[analysis on manifolds]] (including [[infinite-dimensional holomorphy]])
*60: [[Probability theory]] and [[stochastic processes]]
*62: [[Statistics]]
*65: [[Numerical analysis]]
*68: [[Computer science]]
*70: [[Mechanics]] of particles and systems (including [[particle mechanics]])
*74: [[Elasticity theory|Mechanics of deformable solids]]
*76: [[Fluid mechanics]]
*78: [[Optics]], [[electromagnetic theory]]
*80: Classical [[thermodynamics]], [[heat transfer]]
*81: [[Quantum mechanics|Quantum theory]]
*82: [[Statistical mechanics]], structure of matter
*83: [[Theory of relativity|Relativity]] and [[gravitational theory]] (including [[relativistic mechanics]])
*85: [[Astronomy]] and [[astrophysics]]
*86: [[Geophysics]]
*90: [[Operations research]], [[mathematical programming]]
*91: [[Game theory]], [[mathematical economics|economics]], [[mathematical sociology|social]] and [[mathematical psychology|behavioral sciences]]
*92: [[Biology]] and other [[natural science]]s
*93: [[Systems theory]]; [[control theory|control]] (including [[optimal control]])
*94: [[Information]] and [[communication]], [[electrical network|circuits]]
*97: [[Mathematics education]]

==See also==
{{Wikidata property|P3285}}
* [[Areas of mathematics]]
* [[Mathematical knowledge management]]
* [[MathSciNet]]

==References==
{{reflist}}

==External links==
*[https://zbmath.org/static/msc2020.pdf MSC2020-Mathematical Sciences Classification System] (PDF of MSC2020)
*The [[Zentralblatt MATH]] page on the [https://zbmath.org/classification/ Mathematics Subject Classification]. MSC2020 can be seen here.
*[http://msc2010.org/mscwiki/index.php?title=MSC2010 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]] form can be had also.
*The [[American Mathematical Society]] page on [https://www.ams.org/msc/ 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 |archive-url=https://web.archive.org/web/20150516045812/http://www.math.niu.edu/~rusin/known-math/index/beginners.html |archive-date=2015-05-16}}

[[Category:Fields of mathematics]]
[[Category:Mathematical classification systems]]

Mathematics Subject Classification - Revision history

imported>Jemiamapus: /* First-level areas */Redundant