Cohomology and Representation Theory

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Abstract for NSF proposal DMS-0400548 NakanoThe principal investigator will explore problems involving the representation theory and cohomology of algebraic groups, Lie algebras and finite groups. An important object of study will be support varieties which provide a bridge linking the representation theory, cohomology theory and the structure theory (involving conjugacy classes) of Lie algebras. Computations of such varieties will be considered as well as generalizations of the theory to quantum groups and affine Lie algebras. The investigator will also look at computations of cohomology groups/rings of finite Chevalley groupsand Frobenius kernels especially for primes less than the Coxeter number. It is anticipated that the use of extensive computer calculations might be necessary for several of these projects. Algebraic structures such are groups, rings and Lie algebras arise naturally, and the basic understanding of these objects have been used in applications involving biology, physics and chemistry. These algebraic objects have complicated internal symmetries. Information about the representation and cohomology theories allows one to organize and extract vital information that can be used in these various applications. The principal investigator has been actively promoting the working knowledge of these methods. He is currently organizing several conferences in representation/cohomology theory and is a co-organizer of the VIGRE (Vertical Integration of Research and Education) Algebra Group at the University of Georgia which promotes learning through active faculty and student participation.