Cohomology, Geometry and Representation Theory: Algebraic Groups, Quantum Groups and Lie Superalgebras

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The Principal Investigator (PI) will investigate problems involving the connections between representations of algebraic objects and their underlying geometric structures. The basic algebraic structures that the PI proposes to study are Lie superalgebras, algebraic/finite groups, quantum groups, and Frobenius kernels. The algebraic objects have concrete (discrete) realizations, and often times the underlying rich geometric structures arise at the derived level. Cohomological methods are useful for unveiling this geometry. The PI proposes to use new methods involving the Balmer spectrum to describe homological properties of Lie superalgebras. He also plans to make calculations of support varieties for algebraic and quantum groups as a way to connect geometric objects and representation theory. The PI plans to use geometric structures to understand the behavior of the cohomology of finite and algebraic groups. It is well known that algebraic structures such as groups, rings, Lie algebras, and Lie superalgebras manifest themselves naturally in science. The basic understanding of these objects have been used in many different applications in physics and chemistry. These structures are often complicated.Both algebraic and geometric methods are often necessary to extract the important encoded information within these algebraic objects. In terms of broader impacts, the PI has been active nationally in the promotion of integrating research and education. He will continue to direct the NSF funded VIGRE (Vertical Integration of Research and Education) Program at the University of Georgia (UGA). He is also a co-organizer of the VIGRE Algebra Group at UGA which provides practical training in contemporary mathematics to postdoctoral fellows and graduate students. The PI will continue to organize conferences in algebra with an emphasis toward the development of junior mathematicians, and will promote the working knowledge of cohomology and representation theory as an invited speaker at seminars, workshops, and summer schools in the U.S. and abroad.