Algebraic Areas of Mathematics
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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. We have included here the combinatorial
topics and number theory; each is arguably a
distinctive area of mathematics but (as the MathMap suggests) these
parts of mathematics, shown in shades of red, share definite affinities.
The list on this page includes a rather large number of fields in the MSC scheme.
It is also common to interpret the phrase "abstract algebra" in a more
narrow sense --- to view it as the fields obtained by adding successive
axioms to describe the objects of study. Arguably then, abstract algebra
is limited to sections 20 and 22 (Group Theory), 13, 16, and 17 (Ring Theory),
12 (Field Theory), and 15 (Linear Algebra), taken in this way as a
succession from fewest to most restrictive sets of axioms.
The use of algebra is pervasive in mathematics. This particularly true
of group theory --- symmetry groups arise very naturally in almost every
area of mathematics. For example, Klein's vision of geometry was essentially
to reduce it to a study of the underlying group of invariants; Lie groups
first arose from Lie's investigations of differential equations.
It is also true of linear algebra --- a field which, properly construed,
includes huge portions of Numerical Analysis and Functional Analysis, for
example -- hence that field's central position in the MathMap.
Other fairly algebraic areas include 55: Algebraic Topology and 94: Information and Communication.
Symbolic algebra is a heading under 68: Computer Science.
Numerical linear algebra is treated with 65: Numerical Analysis, and
46: Functional Analysis can very
loosely be described as infinite-dimensional linear algebra.
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, algebraic geometry, 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.
Some parts of algebra are best studied using various constructs from
geometry, hence the significant overlap between
these two broad areas. Algebra also rests heavily on the axiomatic method,
bringing it close to foundations.
You might want to continue the tour with a trip through geometry.
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Last modified 2000/01/25 by Dave Rusin. Mail: