Cohomological Methods in the Representation Theory of Algebraic Groups, Quantum Groups and Superalgebras

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The principal investigator (PI) will investigate problems involving cohomological methods with applications to the representation theory of algebraic groups, quantum groups, and Lie superalgebras. More specifically, the PI aims to expand our knowledge of important cohomological calculations for algebraic groups, quantum groups and Lie algebras. Studying line bundle cohomology for the flag variety will play a prominent role in these calculations through a series of steps involving Lie algebra and Frobenius kernel cohomology. The PI proposes to utilize geometric methods to study the representation theory through the use of support varieties. For Lie superalgebras, this approach provides a beautiful homological interpretation of the well-studied combinatorial notions of defect and atypicality. Representation theory emerged about 100 years ago with the pioneering work of Frobenius and Schur, and has become a central area of mathematics because of its connections to combinatorics, algebraic geometry, number theory, and applications to physics. Cohomology theories were developed throughout the 20th century by topologists to construct algebraic invariants for the investigation of manifolds and topological spaces. Cohomology was also defined for algebraic structures like groups and Lie algebras to determine ways in which their representations can be glued together. Even more striking is how the cohomology of algebraic structures can be used to introduce the underlying geometry, which is not seen at the representation theoretic level, into the picture. The PI has been actively promoting the working knowledge of cohomological methods in representation theory and has recently organized several conferences in the area with an emphasis toward the development of junior mathematicians. The PI also co-directs a research group in algebra at the University of Georgia. This group provides practical training in representation theory for postdoctoral fellows and students through the use of computer algebra packages and the publication of research.