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# A Gentle Introduction to the Mathematics Subject Classification Scheme

Here is an introductory guide to the Mathematics Subject Classification (MSC) scheme generally used to classify newly-released mathematics resources. This page is intended for a person with approximately the training of an undergraduate mathematics student; links from here lead to pages at the Mathematical Atlas website which assume somewhat greater familiarity with the sub-disciplines.

You might prefer to take a tour of the mathematical landscape, which parallels the information below at a more leisurely pace.

The major divisions of the MSC hardly provide an equal division of the current mathematics spectrum. Of course what really would be an "equal division" is open to interpretation. The welcome page for this site shows an image of the areas of mathematics which shows the relative numbers of recent papers in each area (arranged so as to illustrate the affinities among related areas).

The MSC does not include classifications for elementary material. Since there are some materials at this site which border on the elementary (e.g. plane geometry and elementary calculus), we have made the best fit possible, but this implies a slight extension of the MSC system. There are also topics within research mathematics which do not fit so neatly into the MSC. We have collected a few pages of information on such non-MSC topics on a separate page.

## What is mathematics, anyway?

Any attempt to distinguish the parts of mathematics must begin with a decision about what constitutes mathematics in the first place! We try to keep the broad definition here, that mathematics includes all the related areas which touch on quantitative, geometric, and logical themes. This includes Statistics, Computer Science, Logic, Applied Mathematics, and other fields which are frequently considered distinct from mathematics. We draw the line only at experimental sciences, philosophy, and computer applications. Personal perspectives vary widely, of course! Probably the only absolute definition of mathematics: that which mathematicians do.

Contrary to common perception, mathematics does not consist of "crunching numbers" or "solving equations". As we shall see there are branches of mathematics concerned with setting up equations, or analyzing their solutions, and there are parts of mathematics devoted to creating methods for doing computations. But there are also parts of mathematics which have nothing at all to do with numbers and equations.

## How many parts of mathematics -- Two? Eight? Sixty-three?

One way to divide the mathematics literature is to decide which books and articles are designed to reveal the structure of mathematics itself, and which are intended to apply mathematics to closely allied areas.

The first group divides roughly into just a few broad overlapping areas:

• Foundations considers questions in logic or set theory -- the very language of mathematics.
• Algebra is principally concerned with symmetry, patterns, discrete sets, and the rules for manipulating arithmetic operations; one might think of this as the outgrowth of arithmetic and algebra classes in primary and secondary school.
• Geometry is concerned with shapes and sets, and the properties of them which are preserved under various kinds of motions. Naturally this is related to elementary geometry and analytic geometry.
• Analysis studies functions, the real number line, and the ideas of continuity and limit; this is perhaps the natural successor to courses in graphing, trigonometry, and calculus. (This is a very large area; we subdivide it below.)
Of course, the division of the subject areas into these broad headings is a little fuzzy: combinatorics is only weakly associated to the rest of "algebra"; algebraic groups are arguably a part of analysis or topology instead of algebra, differential geometry is in practice closer to analysis than geometry, and so on.

The second broad part of the mathematics literature includes those areas which could be considered either independent disciplines or central parts of mathematics, as well as those areas which clearly use mathematics but are interested in non-mathematical ideas too. It is important to note that the MSC, as well as the collection of files at this site, covers only the mathematical aspects of these subjects; we provide only cursory links to observational and experimental data, mathematically routine applications, computer paradigms, and so on.

• Probability and Statistics, for example, has a dual nature -- mathematical and experimental. This classification scheme focuses on the former -- the study of the validity of the measurements one might make.
• Computer sciences have obviously flourished in the last half-century, and consider algorithms and information handling. Here we are concerned with what might be computed, not with compilers, architectures, and so on.
• Significant mathematics must be developed to formulate ideas in the physical sciences, engineering, and other branches of science. Again it is the theoretical underpinnings which concern us here rather than the experiment or tangible construction.
The division between mathematics and its applications is of course vague. In the Math Reviews database, for example, papers in these areas are perhaps over-represented in terms of the number included, yet under-represented in the number carefully reviewed.

Finally note that every branch of mathematics has its own history, collections of important works -- reference, research, biographical, or expository -- and in many cases a suite of important algorithms. The classification allows these topics to be included within each major heading at a secondary level, although there is always some material which cannot otherwise be classified.

The MSC scheme now breaks down these general areas into 61 numbered subject classifications (with widely varying characteristics). We adhere to the polite fiction that these areas are more distinct than the subfields of some of the larger areas; more detail is available in the pages for the various areas.

## Logic and set theory

These areas consider the framework in which mathematics itself is carried out. To the extent that this considers the nature of proof and of mathematical reality, it borders on philosophy. But standard mathematical perspectives are used in most topics covered in the MSC.

• 03: Mathematical logic lies at the heart of the discipline, but a good understanding of the rules of logic came only after their first use. Besides basic propositional logic used formally in computer science and philosophy as well as mathematics, this field covers general logic and proof theory, leading to Model theory. Here we find celebrated results such as the Gödel incompleteness theorem and Church's thesis in recursion theory. Applications to set theory include the use of forcing to determine the independence of the Continuum hypothesis. Applications to analysis include Nonstandard analysis, an alternate perspective for calculus. Undecidability issues permeate algebra and geometry as well. This heading includes Set Theory as well: axiomatizations of sets, cardinal and ordinal arithmetic, and even Fuzzy Set theory.

## Algebraic areas

The algebraic areas of mathematics developed from abstracting key observations about our counting, arithmetic, algebraic manipulations, and symmetry. Typically these fields define their objects of study by just a few axioms, then consider examples, structure, and application of these objects. Other fairly algebraic areas include Algebraic topology (55), Information and communication (94), and perhaps Numerical analysis (65).

• 11: Number theory is one of the oldest branches of pure mathematics, and one of the largest. Of course, it asks questions about numbers, usually meaning whole numbers or rational numbers (fractions). Besides elementary topics involving congruences, divisibility, primes, and so on, number theory now includes highly algebraic studies of rings and fields of numbers; analytical methods applied to asymptotic estimates and special functions; and geometric topics (e.g. the geometry of numbers) Important connections exist with cryptography, mathematical logic, and even the experimental sciences.
• 20: Group theory studies those sets in which an invertible associative "product" operation is defined. This includes the sets of symmetries of other mathematical objects, giving group theory a place in all the rest of mathematics. Finite groups are perhaps the best understood, but groups of matrices and symmetries of geometric patterns also give central examples of groups.
• 22: Lie groups are an important special branch of group theory. They have algebraic structure, of course, and yet are also subsets of space, and so have a geometry; moreover, portions of them look just like Euclidean space, making it possible to do analysis on them (e.g. solve differential equations). Thus Lie groups and other topological groups lie at the convergence of the different areas of pure mathematics. (They are quite useful in application of mathematics to the sciences as well!)
• 13: Commutative rings are sets like the set of integers, allowing addition and multiplication. Of particular interest are several classes of rings of interest in number theory, field theory, and related areas; however, other classes of rings arise, and a rich structure theory arises to analyze commutative rings in general, using the concepts of ideals, localizations, and homological algebra.
• 16: Associative ring theory may be considered the non-commutative analogue of the previous paragraph. This includes the study of matrix rings, division rings such as the quaternions, and rings of importance in group theory. As in the previous paragraph, various tools are studied to enable consideration of general rings.
• 17: Nonassociative ring theory widens the scope further. Here the general theory is much weaker, but special cases of such rings are of key importance: Lie algebras in particular, as well as Jordan algebras and other types.
• 12: Field theory looks at sets, such as the real number line, on which all the usual arithmetic properties hold, including, now, those of division. The study of multiple fields is important for the study of polynomial equations, and thus has applications to number theory and group theory.
• 08: General algebraic systems include those structures with a very simple axiom structure, as well as those structures not easily included with groups, rings, fields, or the other algebraic systems.
• 14: Algebraic geometry combines the algebraic with the geometric for the benefit of both. Thus the recent proof of "Fermat's Last Theorem" -- ostensibly a statement in number theory -- was proved with geometric tools. Conversely, the geometry of sets defined by equations is studied using quite sophisticated algebraic machinery. This is an enticing area but the important topics are quite deep. This area includes elliptic curves.
• 15: Linear algebra, sometimes disguised as matrix theory, considers sets and functions which preserve linear structure. In practice this includes a very wide portion of mathematics! Thus linear algebra includes axiomatic treatments, computational matters, algebraic structures, and even parts of geometry; moreover, it provides tools used for analyzing differential equations, statistical processes, and even physical phenomena.
• 18: Category theory, a comparatively new field of mathematics, provides a universal framework for discussing fields of algebra and geometry. While the general theory and certain types of categories have attracted considerable interest, the area of homological algebra has proved most fruitful in areas of ring theory, group theory, and algebraic topology.
• 19: K-theory is an interesting blend of algebra and geometry. Originally defined for (vector bundles over) topological spaces it is now also defined for (modules over) rings, giving extra algebraic information about those objects.
• 05: Combinatorics, or Discrete Mathematics, looks at the structure of sets in which certain subsets are distinguished. For example, a graph is a set of points in which some edges -- sets of two points -- are given. Other combinatorial questions ask for a count of the subsets of a set having a given property. This is a large field, of great interest to computer scientists and others outside mathematics.
• 06: Ordered sets, or lattices, give a uniform structure to, for example, the set of subfields of a field. Various special types of lattices have particularly nice structure and have applications in group theory and algebraic topology, for example.

## Geometric Areas

One of the oldest areas of mathematical discovery, geometry has undergone several rebirths over the centuries. At one extreme, geometry includes the very precise study of rigid structures first seen in Euclid's Elements; at the other extreme, general topology focuses on the very fundamental kinships among shapes. (There is also a more subtle notion of "geometry" implied in Algebraic Geometry (14), which is frankly quite algebraic; see above.) Other fairly geometric areas are K-theory (19), Lie groups (22), Several complex variables (32), Calculus of variations (49), Global analysis (58).

• 51: Geometry is studied from many perspectives! This large area includes classical Euclidean geometry and synthetic (non-Euclidean) geometries; analytic geometry; incidence geometries (including projective planes); metric properties (lengths and angles); and combinatorial geometries such as those arising in finite group theory. Many results in this area are basic in either the sense of simple, or useful, or both!
• 52: Convex and discrete geometry includes the study of convex subsets of Euclidean space. A wealth of famous results distinguishes this family of sets (e.g. Brouwer's fixed-point theorem, the isoperimetric problems). This classification also includes the study of polygons and polyhedra, and frequently overlaps discrete mathematics and group theory; through piece-wise linear manifolds, it intersects topology. This area also includes tilings and packings in Euclidean space.
• 53: Differential geometry is the language of modern physics as well as an area of mathematical delight. Typically, one considers sets which are manifolds (that is, locally resemble Euclidean space) and which come equipped with a measure of distances. In particular, this includes classical studies of the curvature of curves and surfaces. Local questions both apply and help study differential equations; global questions often invoke algebraic topology.
• 54: General topology studies spaces on which one has only a loose notion of "closeness" -- enough to decide which functions are continuous. Typically one studies spaces with some additional structure -- metric spaces, say, or compact Hausdorff spaces -- and looks to see how properties such as compactness are shared with subspaces, product spaces, and so on. Widely applicable in geometry and analysis, topology also allows for some bizarre examples and set-theoretic conundra.
• 55: Algebraic topology is the study of algebraic objects attached to topological spaces; the algebraic invariants illustrate some of the rigidity of the spaces. This includes various (co)homology theories, homotopy groups, and groups of maps, as well as some rather more geometric tools such as fiber bundles. The algebraic machinery (mostly derived from homological algebra) is powerful if rather daunting.
• 57: Manifolds are spaces like the sphere which look locally like Euclidean space. In particular, these are the spaces in which we can discuss (locally-)linear maps, and the spaces in which to discuss smoothness. They include familiar surfaces. Cell complexes are spaces made of pieces which are part of Euclidean space, generalizing polyhedra. These types of spaces admit very precise answers to questions about existence of maps and embeddings; they are particularly amenable to calculations in algebraic topology; they allow a careful distinction of various notions of equivalence. These are the most classic spaces on which groups of transformations act. This is also the setting for knot theory.

## Analytic areas

Analysis looks carefully at the results obtained in calculus and related areas. One might characterize algebra and geometry as the search for elegant conclusions from small sets of axioms; in analysis on the other hand the measure of success is more frequently the ability to hone a tool which could be applied throughout science. Thus in particular, most of the calculations are done with the real numbers or complex numbers being implicitly understood.

Analysis includes many of the MSC primary headings, a large portion of the mathematics literature, and much of the most easily applied mathematics. Perhaps, then, it is appropriate to subdivide this topic (although schemes for this subdivision are not very standard):

• Calculus and real analysis: differentiation, integration, series, and so on.
• Complex variables: considers those aspects of analytic behaviour unique to complex functions. (Complex variables are also often accepted in other parts of analysis when this causes no essential change in the theory).
• Differential and integral equations: seeks to find functions f knowing relationships between values of f and its derivatives or integrals; study of differential operators and their applications in mathematics
• Functional analysis: study of vector spaces of functions, bases (e.g. Fourier analysis), and linear maps (e.g. integral transforms)
• Numerical analysis and optimization

### Calculus and Real Analysis

• 26: Real functions are those studied in calculus classes; the focus here is on their derivatives and integrals, and general inequalities. This category includes familiar functions such as rational functions. This seems the most appropriate area to receive questions concerning elementary calculus.
• 28: Measure theory and integration is the study of lengths, surface area, and volumes in general spaces. This is a critical feature of a full development of integration theory; moreover, it provides the basic framework for probability theory. Measure theory is a meeting place between the tame applicability of real functions and the wild possibilities of set theory. This is the setting for fractals.
• 33: Special functions are just that: specialized functions beyond the familiar trigonometric or exponential functions. The ones studied (hypergeometric functions, orthogonal polynomials, and so on) arise very naturally in areas of analysis, number theory, Lie groups, and combinatorics. Very detailed information is often available.
• 39: Finite differences and functional equations both involve deducing functions, as in differential equations, but the premises are different: with difference equations, the defining relation is not a differential equation but a difference of values of the function. Functional equations have as premises (usually) algebraic relationships among the values of a function at several points.
• 40: Sequences and series are really just the most common examples of limiting processes; convergence criteria and rates of convergence are as important as finding "the answer". (In the case of sequences of functions, it's also important to find "the question"!) Particular series of interest (e.g. Taylor series of known functions) are of interest, as well as general methods for computing sums rapidly, or formally. Series can be estimated with integrals, their stability can be investigated with analysis. Manipulations of series (e.g. multiplying or inverting) are also of importance.

### Complex variables

• 30: Complex variables studies the effect of assuming differentiability of functions defined on complex numbers. Fascinatingly, the effect is markedly different than for real functions; these functions are much more rigidly constrained, and in particular it is possible to make very definite comments about their global behaviour, convergence, and so on. This area includes Riemann surfaces, which look locally like the complex plane but aren't the same space. Complex-variable techniques have great use in applied areas (including electromagnetics, for example).
• 31: Potential theory studies harmonic functions (and their allies). Mathematically, these are solutions to the Laplace equation Del(u)=0; physically, they are the functions giving the potential energy throughout space resulting from some masses or electric charges.
• 32: Several complex variables is, naturally, the study of (differentiable) functions of more than one complex variable. The rigid constraints imposed by complex differentiability imply that, at least locally, these functions behave almost like polynomials. In particular, study of the related spaces tends to resemble algebraic geometry, except that tools of analysis are used in addition to algebraic constructs. Differential equations on these spaces and automorphisms of them provide useful connections with these other areas.

### Differential and integral equations

• 34: Ordinary differential equations are equations to be solved in which the unknown element is a function, rather than a number, and in which the known information relates that function to its derivatives. Few such equations admit an explicit answer, but there is a wealth of qualitative information describing the solutions and their dependence on the defining equation. There are many important classes of differential equations for which detailed information is available. Applications to engineering and the sciences abound. Numerical solutions are actively studied.
• 35: Partial differential equations begin with much the same formulation as ordinary differential equations, except that the functions to be found are functions of several variables. Again, one generally looks for qualitative statements about the solution. For example, in many cases, solutions exist only if some of the parameters lie in a specific set (say, the set of integers). Various broad families of PDE's admit general statements about the behaviour of their solutions. This area has a long-standing close relationship with the physical sciences, especially physics, thermodynamics, and quantum mechanics.
• 37: Dynamical systems is the study of iteration of functions from a space to itself -- in discrete repetitions or in a continuous flow of time. Thus in principle this field is closely allied to differential equations on manifolds, but in practice the focus is on the underlying sets (invariant sets or limit sets) and on the chaotic behaviour of limiting systems. [new in 2000]
• 45: Integral equations, naturally, seek functions which satisfy relationships with their integrals. For example, the value of a function at each time may be related to its average value over all preceding time. Included in this area are equations mixing integration and differentiation. Many of the themes from differential equations recur: qualitative questions, methods of approximation, specific types of equations of interest, transforms and operators useful for simplifying the problems.
• 49: Calculus of variations and optimization seek functions or geometric objects which are optimize some objective function. Certainly this includes a discussion of techniques to find the optima, such as successive approximations or linear programming. In addition, there is quite a lot of work establishing the existence of optima and characterizing them. In many cases, optimal functions or curves can be expressed as solutions to differential equations. Common applications include seeking curves and surfaces which are minimal in some sense. However, the spaces on which the analysis are done may represent configurations of some physical system, say, so that this field also applies to optimization problems in economics or control theory for example.
• 58: Global analysis, or analysis on manifolds, studies the global nature of differential equations on manifolds. In addition to local tools from ordinary differential equation theory, global techniques include the use of topological spaces of mappings. In this heading also we find general papers on manifold theory, including infinite-dimensional manifolds and manifolds with singularities (hence catastrophe theory), as well as optimization problems (thus overlapping the Calculus of Variations, above).

### Functional analysis

• 46: Functional analysis views the big picture in differential equations, for example, thinking of a differential operator as a linear map on a large set of functions. Thus this area becomes the study of (infinite-dimensional) vector spaces with some kind of metric or other structure, including ring structures (Banach algebras and C-* algebras for example). Appropriate generalizations of measure, derivatives, and duality also belong to this area.
• 42: Fourier analysis studies approximations and decompositions of functions using trigonometric polynomials. Of incalculable value in many applications of analysis, this field has grown to include many specific and powerful results, including convergence criteria, estimates and inequalities, and existence and uniqueness results. Extensions include the theory of singular integrals, Fourier transforms, and the study of the appropriate function spaces. This heading also includes approximations by other orthogonal families of functions, including orthogonal polynomials and wavelets.
• 43: Abstract harmonic analysis: if Fourier series is the study of periodic real functions, that is, real functions which are invariant under the group of integer translations, then abstract harmonic analysis is the study of functions on general groups which are invariant under a subgroup. This includes topics of varying level of specificity, including analysis on Lie groups or locally compact Abelian groups. This area also overlaps with representation theory of topological groups.
• 44: Integral transforms include the Fourier transform (see above) as well as the transforms of Laplace, Radon, and others. (The general theory of transformations between function spaces is part of Functional Analysis, above.) Also includes convolution operators and operational calculi.
• 47: Operator theory studies transformations between the vector spaces studied in Functional Analysis, such as differential operators or self-adjoint operators. The analysis might study the spectrum of an individual operator or the semigroup structure of a collection of them.

### Numerical analysis and optimization

• 65: Numerical analysis involves the study of methods of computing numerical data. In many problems this implies producing a sequence of approximations; thus the questions involve the rate of convergence, the accuracy (or even validity) of the answer, and the completeness of the response. (With many problems it is difficult to decide from a program's termination whether other solutions exist.) Since many problems across mathematics can be reduced to linear algebra, this too is studied numerically; here there are significant problems with the amount of time necessary to process the initial data. Numerical solutions to differential equations require the determination not of a few numbers but of an entire function; in particular, convergence must be judged by some global criterion. Other topics include numerical simulation, optimization, and graphical analysis, and the development of robust working code.
• 41: Approximations and expansions primarily concern the approximation of classes of real functions by functions of special types. This includes approximations by linear functions, polynomials (not just the Taylor polynomials), rational functions, and so on; approximations by trigonometric polynomials is separated into Fourier analysis (below). Topics include criteria for goodness of fit, error bounds, stability upon change of approximating family, and preservation of functional characteristics (e.g. differentiability) under approximation. Effective techniques for specific kinds of approximation are also prized. This is also the area covering interpolation and splines.
• 90: Operations research may be figuratively described as the study of optimal resource allocation. Depending on the options and constraints in the setting, this may involve linear programming, or quadratic-, convex-, integer-, or boolean-programming. This category also includes game theory, which is actually not about games at all but rather about optimization; which combination of strategies leads to an optimal outcome. This area also includes mathematical economics. [Starting in the year 2000, a new section 91: Game theory, economics, social and behavioral sciences

We turn now to the parts of mathematics most concerned with developing mathematical tools applicable outside of mathematics.

## Probability and Statistics

These areas consider the use of numerical information to quantify observations about events. The tools and development are clearly mathematical; these areas overlap with analysis in particular. On the other hand, the use of the ideas developed here is primarily in non-mathematical areas.

• 60: Probability theory is simply enumerative combinatorial analysis when applied to finite sets; thus the techniques and results resemble those of discrete mathematics. The theory comes into its own when considering infinite sets of possible outcomes. This requires much measure theory (and a careful interpretation of results!) More analysis enters with the study of distribution functions, and limit theorems implying central tendencies. Applications to repeated transitions or transitions over time lead to Markov processes and stochastic processes. Probability concepts are applied across mathematics when considering random structures, and in particular lead to good algorithms in some settings even in pure mathematics.
• 62: Statistics is the science of obtaining, synthesizing, predicting, and drawing inferences from data. Elementary calculations of mean and standard variation suffice to summarize a large, finite, normally-distributed dataset; the field of Statistics exists since data are not usually so nicely given. If we do not know all the elements of the dataset, we must discuss sampling and experimental design; if the data are not normal we must use other parameters to summarize them, or resort to nonparametric methods; if multiple data are involved, we study the measures of interaction among the variables. Other topics include the study of time-dependent data, and the foundations necessary to avoid ambiguity or paradox. Computational methods (e.g. for curve-fitting) are of particular importance in applications to the sciences and engineering as well as financial and actuarial work.

## Computer science and Information theory

By design of the MSC, literature concerning specific computations and algorithms is classified with the area of mathematics to which the computations are applied. But mathematics can return the favor and study the process by which computers carry out their information handling.

• 68: Computer science, today more accurately a separate discipline, considers a number of rather mathematical topics. In addition to computability questions arising from many problems in discrete mathematics, and logic questions related to recursion theory, one must consider scheduling questions, stochastic models, and so on.
• 94: Information and communication includes questions of particular interest to algebraists, especially coding theory (related to linear algebra and finite groups) and encryption (related to number theory and combinatorics). Many topics appropriate to this area can be expressed in graph-theoretic terms, such as network flows and circuit design. Data compression and visualization overlap with statistics.

For mathematical analysis of computation see Numerical Analysis.

## Applications to the sciences

Historically, it has been the needs of the physical sciences which have driven the development of many parts of mathematics, particularly analysis. The applications are sometimes difficult to classify mathematically, since tools from several areas of mathematics may be applied. We comment on these applications not by discussing the nature of their discipline but rather their interaction with mathematics.

I must confess that I have only a cursory acquaintance with most of these fields. -- djr.

• 70: Mechanics of particles and systems studies dynamics of sets of particles or solid bodies, including rotating and vibrating bodies. Uses variational principles (energy-minimization) as well as differential equations.
• 74: Mechanics of solids considers questions of elasticity and plasticity, wave propagation, engineering, and topics in specific solids such as soils and crystals.
• 76: Fluid mechanics studies air, water, and other fluids in motion: compression, turbulence, diffusion, wave propagation, and so on. Mathematically this includes study of solutions of differential equations, including large-scale numerical methods (e.g the finite-element method).
• 78: Optics, electromagnetic theory is the study of the propagation and evolution of electromagnetic waves, including topics of interference and diffraction. Besides the usual branches of analysis, this area includes geometric topics such as the paths of light rays.
• 80: Classical thermodynamics, heat transfer is the study of the flow of heat through matter, including phase change and combustion. Historically, the source of Fourier series.
• 81: Quantum Theory studies the solutions of the Schrödinger (differential) equation. Also includes a good deal of Lie group theory and quantum group theory, theory of distributions and topics from Functional analysis, Yang-Mills problems, Feynman diagrams, and so on.
• 82: Statistical mechanics, structure of matter is the study of large-scale systems of particles, including stochastic systems and moving or evolving systems. Specific types of matter studied include fluids, crystals, metals, and other solids.
• 83: Relativity and gravitational theory is differential geometry, analysis, and group theory applied to physics on a grand scale or in extreme situations (e.g. black holes and cosmology).
• 85: Astronomy and astrophysics: as celestial mechanics is, mathematically, part of Mechanics of Particles (!), the principal applications in this area appear to be concerning the structure, evolution, and interaction of stars and galaxies.
• 86: Geophysics applications typically involve material in Mechanics and Fluid mechanics, as above, but for large-scale problems (this subject deals with a very big solid and a large pool of fluid!)
• 93: Systems theory; control study the evolution over time of complex systems such as those in engineering. In particular, one may try to identify the system -- to determine the equations or parameters which govern its development -- or to control the system -- to select the parameters (e.g. via feedback loops) to achieve a desired state. Of particular interest are issues in stability (steady-state configurations) and the effects of random changes and noise (stochastic systems). While popularly the domain of "cybernetics" or "robotics", perhaps, this is in practice a field of application of differential (or difference) equations, functional analysis, numerical analysis, and global analysis (or differential geometry).
• 92: Other sciences whose connections merit explicit connection in the MSC scheme include Chemistry, Biology, Genetics, Medicine, Psychology, Sociology, and other social sciences as a group. In chemistry and biochemistry, it is clear that graph theory, differential geometry, and differential equations play a role. Medical technology uses techniques of information transfer and visualization. Biology (including taxonomy and archaeobiology) use statistical inference and other tools. Economics and finance also make use of statistical tools, especially time-series analysis; some topics, such as voting theory, are more combinatorial. (Mathematical economics is classed with operations research for some reason.) The more behavioural sciences (including Linguistics!) use a medley of statistical techniques, including experimental design and other rather combinatorial topics. [Starting in the year 2000, some of these topics will be moved to a new section 91: Game theory, economics, social and behavioral sciences.]

## Other

Some parts of the mathematics literature seem neither to develop nor apply mathematics; rather, they discuss the nature of mathematics or mathematicians, or recount some very basic mathematical concepts. As a rule, this material is poorly tracked in indices and databases of mathematics, but some of it has a genuine place in the MSC scheme.

• 01: History and biography is a legitimate field of study when applied to mathematics, even if it is not usually practiced carefully by mathematicians. (Many favorite tidbits of mathematical folklore are unfortunately untrue!) This section also includes the sociology of mathematics -- how it comes to be the discipline it is. Portions of this material are attached to the relevant subfield of mathematics in the MSC.
• 97: Mathematics Education is an area newly added to the MSC effective with the 2000 revision, but of a long heritage. Topics of discourse cover all levels of mathematics education from pre-school to university level, and can focus on the student (through educational psychology, for example), the teacher (continuing development or assessment), the classroom (and the books and technology used in the classroom), or the larger system (policy analysis, cross-cultural comparisons, and so on). Analysis can range from small case studies to large statistical surveys; perspectives range from the fairly philosophical to the clinical.
• 00: General mathematics has to include the inevitable "none of the above" entries. Besides general textbooks or expository topics, and thematically unrelated collections of papers, this category contains the sub-category "General and miscellaneous specific topics". Some of them (e.g. dimensional analysis, philosophy of mathematics, dictionaries, or recreational mathematics) could legitimately be included in some of the other portions of the MSC table. The prize, however, surely goes to sub-sub-classification 00A99, "Miscellaneous topics", home to an incredible 96 articles in the MathSciNet database! We use classification 00 for some fairly elementary mathematics at this site, too.

More detailed descriptions of these areas of mathematics, including the subdivisions of them, may be obtained through the main index pages. [Only some of them are complete at this time, sorry. -- djr]