Representations, Cohomology, and Geometry in Tensor Triangulated Categories

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

Representation theory emerged over 100 years ago with the pioneering work of Frobenius and Schur. A representation is a realization of an algebraic structure (i.e., a group, Lie algebra, quantum group, or Lie superalgebra) by matrices. Such realizations have been successfully applied to provide deep insights into these complicated algebraic objects. Over the course of the last century, representation theory has emerged as a central area of modern mathematics with connections to combinatorics, algebraic geometry, topology, number theory, along with applications to physics. More recent developments have included using the representation theory of Lie groups to better understand signal processing and the translation of the P versus NP problem in computer science via geometric complexity theory into questions about decomposing representations for the general linear group. The PI will investigate the representation theory of algebraic groups, Lie algebras, and Lie superalgebras via categorical and geometric methods. In recent years, these algebro-geometric methods have played a predominant role in understanding the relationships between representations in various algebraic categories. Cohomological methods will be used to answer questions about the characters and existence of filtrations for representations. Geometric constructions and cohomology in triangulated categories will be employed to analyze the behavior of the representations for the aforementioned algebraic objects. The PI will also determine the relationship between a tensor triangulated category and its underlying categorical spectrum.