Mathematics Subject Classification

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The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme collaboratively produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH. It 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 MSC2010.


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 2 digit number. As well as 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:

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 sympletic 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. It is not valid to put 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;


According to the American Mathematical Society help page about MSC,[1] the MSC has been revised a number of times since 1940, but the original classification of older items has not been reclassified. 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 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

The top level subjects under the MSC are, grouped here by common area names that are not part of the MSC:

General/foundations [Study of foundations of mathematics and logic]

Discrete mathematics/algebra [Study of structure of mathematical abstractions]

Analysis [Study of change and quantity]

Geometry and topology [Study of space]

Applied mathematics / other [Study of applications of mathematical abstractions]

See also


External links