Mathematical Sciences: Cohomology and Actions of Finite Groups

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This project will incorporate and apply ideas and techniques from representation theory and group actions to the cohomology of groups. The main objectives are as follows: 1) Extend an ongoing project to calculate and interpret the mod-2 cohomology of certain key finite groups. The hope is to find a set of useful examples which will lead to further theoretical developments in the field, as has happened in the past. 2) Analyze the cohomology and K-theory of certain discrete groups such as arithmetic groups and mapping class groups, using methods from modular representation theory as well as a recent splitting result in K-theory. 3) Apply methods from equivariant K-theory to certain G-complexes associated to a finite group G. This is motivated by a conjecture in modular representation theory involving the isotopy groups of the complex. Groups are the abstract mathematical embodiment of symmetry. They are ubiquitous in mathematics and in many mathematical applications to the other sciences, being well nigh indispensable in quantum mechanics. As such, one cannot underestimate the importance of efforts to understand their properties even better than we do now - there is no telling where such understanding will pay off, but mathematical physics is a plausible assumption. This project is in a sense a bootstrap project, for cohomology is already a theory based on groups, the various cohomology groups. The investigator has been very successful in applying the cohomology groups in studying the structure and properties of other groups, particularly certain of the so-called finite sporadic groups which had defeated the efforts of others. His project will advance both by making difficult calculations and by evolving theory to bring order to the results of these calculations.