Representation Theory, Geometry, and Cohomology in Tensor Triangulated Categories

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Algebraic structures like groups, Lie algebras, and Lie superalgebras have internal symmetries that have been used in many different physical applications in biology, chemistry, and physics. Representation theory, a method to codify information about groups and algebras through matrix realizations, has played a fundamental role in mathematics over the past one hundred years. Modern trends in representation theory have involved using tools from topology (i.e., cohomology) and algebraic geometry to package the representations of an algebraic object via the tensor product structure to connect the representation theory to ambient geometric structures. This project will use these deep interactions between the representations, the geometry, and the cohomology as a powerful device to prove new results about these algebraic structures and to answer important questions pertaining to geometric and homological invariants (like the complexity and atypicality) for these representations. The PI will organize conferences in algebra with an emphasis toward the development of junior mathematicians, and he will disseminate working knowledge of modern methods in representation theory through lectures at seminars, workshops, and summer schools.This project explores central problems of interest in the representation theory of reductive algebraic groups, Lie algebras, and Lie superalgebras. This study includes discovering new conjectures about the characters of simple modules for reductive algebraic groups in positive characteristic, and classical Lie superalgebras in characteristic zero. The determination of formulas for extensions between irreducible representations for finite Chevalley groups and loop algebras will also be explored. In this investigation, the PI will make use of tensor triangulated geometry and other geometric constructions in triangulated categories to compute cohomological invariants (e.g. cohomology groups, module varieties) of representations for the aforementioned algebraic objects.