[Search] |

## Index Pages for topics outside the Mathematics Subject Classification |

The index pages at this site are organized according to the Mathematics Subject Classification (MSC) scheme. (This is the scheme developed by the American Mathematical Society and Zentralblatt für Mathematik. Here's the original.) But some topics don't seem to fit into just one category of the Mathematics Subject Classification scheme, or are outside the scope of the MSC. We provide pages for a few of these which are similar are spirit to the ordinary MSC index pages. In general they refer to the ordinary pages whenever appropriate.

These pages will be identifiable by their | green | backgrounds; |

standard MSC-area pages are marked with the | blue | backgrounds. |

Some topics in mathematics are quite broad, consisting of several (usually a more or less well-defined set of) MSC major headings. There are many books and internet materials which treat these as cohesive units. Many of these seem to coincide with the broad-area classifications the AMS used in the MSC until 1972 (see below for defunct MSC codes):

- abstract algebra (areas 12 through 18, surely, and usually 11: number theory and some other allied topics.)
- analysis (areas 26 through 47, certainly -- even more in the broader sense of "analysis".)
- topology (areas 54, 55, 57, and the more analytical 58 and geometric 53, perhaps.)
- mathematical physics (areas 70 through 86, at least!); there is a newsgroup sci.physics.research (moderated) and unmoderated groups sci.physics and sci.physics.particle.

One way to identify "legitimate" topics in mathematics is to look for subject headings used by other classification schemes. Here are a few other classification systems which have seen significant use:

- Library of Congress
- Dewey Decimal
- Referativnyi Zhurnal [125Kb]
- NSF Mathematics Programs
- "Encyclopedic Dictionary" system
- ArXiv Preprint server
- MAA Basic Library List
- Historical systems (1900, 1930)
- Other views

One might envision other ways to break up the discipline and see how even they are. Sample topics: distribution of SIAM members' fields of interest; distribution of Fields medalists; division of math fields in AMS's annual list of recent PhDs; etc.

Some areas of mathematics cut across boundaries of the MSC. It is actually quite frequently the case that a topic fits equally well in more than one classification; in these cases the two (or more) subfields are usually marked quite clearly by cross-reference (e.g. arithmetic questions in algebraic geometry are just as naturally classified as 11G as well as within section 14). Occasionally however we find topics which are arguably cohesive branches of mathematics, but which have over the last few decades not been accorded an MSC heading of their own. Here are a few which have been suggested:

- See for example the division of material on polynomials, mentioned in the page for 12F: Galois theory.
- Quantum algebras
- Computational geometry
- The sphere FAQ written some time ago does not seem to fit so neatly into these area boundaries
- Likewise the product structures FAQ
- General reference and tables

What about the defunct parts of the MSC itself? A few fields were discarded
after a reorganization: 02 (became 03), 04 (folded into 03), 10 (became 11),
21 (became 22), 50 (became 51), 73 (became 74).
A few fields were for papers unclassifiable
except for broad field: 09(algebra), 27(analysis), 36 (differential
equations), 48(geometry), 56(topology), 69(applied math), 99(misc).
Some numbers were *apparently* never used: 7,23-25,29,38,
59,61,63-64,66-67,75,77,87-89,95-96,98

For more detail on old classifications, see this AMS page

09 (40-72) Classical algebra

09-XX (40-72) Classical algebra 09.0X (40-58) Algebra 09.00 (59-72) Classical algebra 09.1X (40-58) Abstract algebra 09.2X (40-58) Elementary algebra 09.3X (40-58) Rings, fields and algebras

21 (40-58) Topological algebraic structures

21-XX (40-58) Topological algebraic structures 21.0X (40-58) Topological algebraic structures

27 (40-58) Analysis

27-XX (40-58) Analysis 27.0X (40-58) Analysis 27.1X (40-58) Foundations of analysis 27.2X (40-58) Theory of sets, theory of functions of real variables 27.3X (40-58) Theory of functions of real variables, theory of measure and integration 27.4X (40-58) Geometrical analysis 27.5X (40-58) Functions with particular properties

36 (40-58) Differential equations, operational calculus

36-XX (40-58) Differential equations, operational calculus 36.0X (40-58) Differential equations 36.1X (40-58) Differential equations, operational calculus

48 (40-58) Geometry

48-XX (40-58) Geometry 48.0X (40-58) Geometry

56 (40-58) Topology

56-XX (40-58) Topology 56.0X (40-58) Topology

69 (40-72) General applied mathematics

69-XX (40-72) General applied mathematics 69.00 (59-79) General Applied Mathematics

71 (40-58) Mechanics

71-XX (1940-1958) Mechanics 71.0X (1940-1958) Mechanics

79 (40-58) Mathematical physics, physical applications

79-XX (1940-1958) Mathematical physics, physical applications 79.0X (1940-1958) Mathematical physics, physical applications

84 (40-58) Relativity, astronomy

84-XX (1940-1958) Relativity, astronomy 84.0X (1940-1958) Relativity, astronomy

91 (40-58) Other applications

91-XX (1940-1958) Other applications 91.0X (1940-1958) Other applications

99 (40-58) Miscellaneous

99-XX (40-58) Miscellaneous 99.0X (40-58) Miscellaneous

There are some subjects which, based on reports in the popular press and the presentations in the undergraduate curriculum, one might expect to be major portions of the subject classification system. Did you look for these in vain? Here are a few pointers:

- Trigonometry: Mathematically this topic is largely straightforward and complete, but the trig functions are a major component of Special Functions
- Vectors and Matrices: Nominally the content of Linear Algebra, the topics there are rather beyond questions of determining normal lines to planes and so on.
- Calculus: Elementary calculus is considered in Real functions. Vector calculus (Stokes' theorem and so on) may be in Calculus on manifolds. Specific derivatives and integrals may be discussed in Special functions; the general theory is more likely in Measure and integration. Topics in Approximations and expansions and Sequences and series are usually also included in calculus courses.
- Differential equations: Beside Ordinary Differential Equations, the material treated in an introductory course may include Integral transforms (Laplace, Fourier).
- "Discrete mathematics" is, if not an elementary course, likely to be mostly Combinatorics and Graph Theory.
- Chaos, catastrophe theory, dynamical systems: Try Global analysis
- Fractals: Try Measure theory or General topology
- Inverse problems: try 65J; 35R
- p-adic numbers: some material is in section 11S; areas 43, 54E are also reasonable places to look.

Mathematics is frequently presented to target audiences, with the theory selected, and the examples configured, to match the practical needs of the audience. In some cases (e.g. "woodshop mathematics") the mathematics used is actually quite elementary (see above). In other cases, some more or less sophisticated mathematics is used, gathered together irrespective of the mathematical basis of the tools. We mention in particular

- Financial mathematics (but see some topics in Economics -- 90A09 or Statistics -- 62P05)
- Actuarial science (but see some topics in Statistics -- 62P05)
- Mathematical physics
- Engineering mathematics
- Mathematical modeling (but see some topics in applications of mathematics)
- Vocational and technical mathematics
- Business Mathematics
- Mathematics for lay audiences

We next consider some topics in mathematics which are too elementary for inclusion in the MSC. (Recall that the MSC is designed to classify only new research work. Occasionally expository articles on these topics, or work discussing the teaching of these topics, will be included in the most appropriate subject headings.)

- Pre-calculus (basic arithmetic, elementary algebra and geometry, trigonometry, functions and graphs)
- Calculus (differentiation and integration of real functions, applications, and vector analysis)
- Finite mathematics (counting and elementary probability)
- Recreational mathematics (Games, Puzzles, Contests and Problems)

Next we mention some topics often associated with mathematics, but whose primary content is rather free of mathematical ideas -- in particular, these are topics about which a mathematician is unlikely to offer the best surveys. The nearest match to topics within the purvey of the MSC are shown

- Classroom advice and preparations (Mathematics Education)
- Calculators and their role in society (Numerical Analysis, Computer Science)
- Collected statistics and data (Statistics)
- Computer programming [praxis]
- Experimental sciences (Mechanics through Information theory)
- Philosophy (Mathematical Logic)
- Accounting
- Mathematics and the Arts and Literature
- Professional concerns of mathematicians (Academic life, mathematical typesetting, research and society)

Finally, there are some topics which just aren't mathematics! It's always possible, perhaps, that a person lands at this web site completely by mistake. What non-mathematical query might have brought you here?:

- Group Therapy -- try Psychiatry
- Numerology -- see Fortune-telling
- Arithmonancy -- see Fortune-telling
- Manifolds, automotive
- Lucky numbers and the lottery

Last modified 2000/01/16 by Dave Rusin. Mail: