A Gentle Introduction to the Mathematics Subject Classification Scheme
This is intended to be a 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 lead to pages
assuming greater familiarity with the sub-disciplines.
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), I have
made the best fit I could, but this implies a slight extension of the
MSC system. There is a short page of topics
which do not fit neatly into the MSC, including a couple of topics
(rings; spheres) on which I have written detailed summary
articles. (Also included there are links to a few subjects which
do fall within a MSC heading but whose position there may be
hard for a newcomer to find.)
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 collection 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 is updated at the turn of a decade; the changes for the year 2000
are mentioned only briefly below. The total will be 63 major headings, not 61.]
Warning: this is a single-author document and thus reflects the
opinions and deficiencies of that author [Dave Rusin]. (I'm particularly
unsatisfied with the descriptions of applications to the physical
sciences and other allied disciplines.) Helpful suggestions are always
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! I have tried
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!
Some definitions of mathematics heard from others:
- That which mathematicians do.
- The study of statements of the form "P implies Q".
- The branch of science which you could continue to do if you woke up
and the universe was gone.
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
How many parts of mathematics -- Two? Seven? Sixty-one?
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
The first group divides roughly into just a few broad overlapping areas:
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.
- 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
- 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; I subdivide it below.)
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.
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.
- 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.
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.
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.
04: Set theory, in naive form, is taken for
granted in many other parts of mathematics; however, Russell's paradox
illustrates some thought is required for its proper use. Axiomatic set
theory (actually not classed in this section by Math Reviews!)
considers possible hypotheses and models for set theory. Descriptive
set theory considers various classifications of sets. Cardinal
arithmetic considers the "size" of various sets; ordinal arithmetic
refines this to the "ordering" of sets. Fuzzy set theory replaces the
yes/no statement of set membership with a qualitative predicate.
[This small section will be folded into (03) starting in the year 2000].
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
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
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.
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
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
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
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.
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
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.
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
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
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.
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
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, that is,
dynamical systems. 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).
[Starting in the year 2000, a new section 37: "Dynamical systems and ergodic theory" will hold some of this material.]
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
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.
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
[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.
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
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.
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.
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. I can
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.
73: Mechanics of solids considers questions of
elasticity and plasticity, wave propagation, engineering, and topics
in specific solids such as soils and crystals. [Starting in 2000 this
section will be reorganized into a new section 74: Mechanics of deformable solids.]
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
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
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.]
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
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]
You can reach this page through http://www.math-atlas.org/welcome.html
Last modified 1999/05/12 by Dave Rusin. Mail: firstname.lastname@example.org