Apr 13, 2010

Algebraic Areas of Mathematics


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.

  • 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 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.
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.

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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