Cohomology and Actions of Finite Groups

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

The principal investigator will apply techniques from algebra and topology to study a number of basic questions in transformation groups, orbifolds, field theory, group cohomology and related topics. Among other things he proposes the following research projects: (1) construction of finite group actions using methods from group cohomology and homotopy theory; (2) research on representation spaces and K-theory of the braid groups and other discrete groups; (3) algebraic topology of orbifolds: the development and study of topological invariants associated to orbifolds; (4) field theory and cohomology of groups: the analysis of Galois groups using cohomological methods; and (5) computation of the cohomology of low rank simple groups, using a combination of theoretical and computer-assisted methods. This proposal is interdisciplinary in its conception, as it intends to combine methods from algebra and topology to address a broad range of problems. Symmetry is a topic of central interest in mathematics, which can be effectively studied using topological methods. Research on mathematical aspects of orbifold string theory is representative of the emerging fusion between geometry and physics. Many important ideas have originated in physics and they consistently lead to the opening of new frontiers in mathematics. On the other hand the cohomology of groups is a subject involving intricate algebraic invariants which very naturally play a role in representation theory, number theory, field theory, etc. Recent techniques use advanced methods and capabilities from computer algebra, and the proposed calculations will serve as benchmarks for further computational as well as theoretical developments in the field--with many potential applications. This proposal will seek to develop close and fruitful interactions between topology, algebra and physics, helping to consolidate the cohomology of groups as an interactive subject involving both theoretical and computational methods.