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[Algebraic areas of MathMap]

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 of which is arguably a distinctive area of mathematics; the MathMap suggests that these parts of mathematics (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.

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 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org