Open Questions: Algebra

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See also: Symmetry -- Algebraic geometry

Introduction

Combinatorial group theory

The word problem

Algebraic K-theory

Commutative algebra

Inverse Galois theory

Category theory

Topos theory


Recommended references: Web sites

Recommended references: Magazine/journal articles

Recommended references: Books

Introduction



Recommended references: Web sites

Site indexes

Math Forum Internet Mathematics Library: Modern Algebra
Alphabetized list of links with extensive annotations.
The Math Forum: Modern Algebra
Selected list of links with extensive annotations.
Mathematics Archives - Abstract Algebra
Extensive annotated list of links.
Open Directory Project: Algebra
Categorized and annotated algebra links. A version of this list is at Google, with entries sorted in "page rank" order.
Galaxy: Algebra
Categorized site directory. Entries usually include descriptive annotations. There are also indexes for group theory, (more here), ring theory, field theory, and linear algebra (more here).

Sites with general resources

Abstract Algebra Online
Contains a list of many links organized by topic, a glossary of algebra terms, and an index of important theorems. Site is maintained by John Beachy and is based on his abstract algebtra textbooks.

Surveys, overviews, tutorials

Abstract algebra
Article from Wikipedia.
Wikibooks: Abstract Algebra
Textbook in the Wikibooks collection. A work in progress, but already contains much useful information. Topics covered include: Groups, Rings, Category theory, Finite fields.
Abstract algebra
Elementary introduction to abstract algebra, by Joseph Mileti.

Group theory

Group (mathematics)
Article from Wikipedia. See also Group theory.
Open Problems in Combinatorial Group Theory
Collected by G.Baumslag, A.Miasnikov and V.Shpilrain.

Ring theory and commutative algebra

Ring (mathematics)
Article from Wikipedia. See also Ring theory, Commutative algebra.

Field theory

Field (mathematics)
Article from Wikipedia. See also Field theory, Galois theory, Finite field.

Linear algebra and matrix theory

Linear algebra
Article from Wikipedia. See also Matrix theory. Algebra over a field, Associative algebra

Category theory

Galaxy: Category Theory
Categorized site directory. Entries usually include descriptive annotations.
Category theory
Article from Wikipedia. See also Universal algebra, Topos.

Miscellaneous algebra topics

Homological algebra
Article from Wikipedia.
K-theory
Article from Wikipedia.


Recommended references: Magazine/journal articles

The Octonions
John C. Baez
Bulletin of the AMS, April 2002, pp. 145-205
There are only four normed divisions algebras: the real numbers, the complex numbers, the quaternions, and the octonions. The latter bridge many areas of mathematics, such as Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and exceptional Lie groups. The also appear in theoretical physics in connectin with quantum logic, special relativity, and supersymmetry.
[Abstract, references, downloadable text]
Recent Developments in the Cohomology of Finite Groups
Alejandro Adem
Notices of the AMS, August 1997, pp. 806-812
Cohomology of finite groups can be described completely algebraically, but it can also be described in topological terms. It forms an important bridge between algebra and topology and touches a number of areas of mathematics. There are now a number of ways to compute group cohomology.
[Article in PDF format]
An Introduction to Computational Group Theory
Ákos Seress
Notices of the AMS, June/July 1997, pp. 671-679
Computational group theory is one of the oldest and most developed branches of computational algebra. Different techniques and problems are associated with specific types, of groups, such as finitely presented groups, polycyclic groups, permutation groups, and matrix groups.
[Article in PDF format]
On Finite Simple Groups and Their Classification
Ron Solomon
Notices of the AMS, February 1995, pp. 231-239
Although the classification theorem for finite simple groups was felt to be complete in 1983, and in spite of about 15,000 pages devoted to the proof, not all details had been published, and loose ends remained. Since then, efforts have continued to clean up the classification and address new problems that it suggested. Interesting questions remain, especially with respect to the sporadic simple groups, and the "Monster" in particular.
[Article in PDF format]


Recommended references: Books


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