Mathematics Subject Classification


The Mathematics Subject Classification 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 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.
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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 geometry53A is the classification for classical differential geometry53A45 is the classification for 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 is particularly well represented in the classification scheme with a number of different categories including:
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 geometryB for local differential geometryC for global differential geometryD 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 53-01 Instructional exposition 53-02 Research exposition 53-03 Historical 53-04 Explicit machine computation and programs 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 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, 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.
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 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 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 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