The Principal Investigator (PI) proposes to address questions in the representation theory of algebraic (super)groups, Lie (super)algebras, quantum (super)groups, finite groups, and related algebras. The purpose of this proposal is to integrate the study of these algebraic objects with tools from low-dimensional topology, cohomology, algebraic geometry, and algebraic combinatorics. Building on results already obtained by the PI and collaborators, the PI will use support varieties and cohomology to study the representations of complex Lie superalgebras. In particular, the PI will investigate complexity, the Balmer spectrum, and related questions to shed light in this poorly understood area. In a separate project the PI will use modified trace and dimension functions arising in low-dimensional topology to prove the generalized Kac-Wakimoto conjecture for classical Lie superalgebras, and to investigate the tilting modules of algebraic and quantum groups. Although these tools have individually proven to be fundamental to new developments in the field, relatively little work has been done using them in conjunction. This project is in the area of mathematics known as representation theory.
Algebraic structures such as groups and Lie algebras arise in nature as the symmetries of some object. The ``super'' versions of these are the ones which involve both symmetries and anti-symmetries. These so-called supersymmetries play a fundamental role in physics and mathematics. Representation theory is devoted to understanding how these structures interact with other objects. Because of the intricate nature of these objects, it is productive to use geometric and topological tools to extract new information about these systems. This is the approach taken in this proposal. It is expected that the research conducted in this proposal will shed new light on these areas. Because of the insight representation theory provides regarding the underlying structure and symmetries of an object, it has proven valuable in other areas of mathematics, physics, chemistry, biology, cryptography, quantum computing, computer graphics, and art. Gains in understanding in representation theory can be expected to pay dividends in these other fields. In terms of broader impacts, the PI has been active in the promotion of integrating research and education. The PI will continue to lead a vertically integrated group of undergraduate and early-career graduate students on research related to this proposal. He will also continue to take a leadership role in engaging younger students in mathematics. This includes co-organizing the University of Oklahoma Math Day, leading the departmental Math Club, and authorship of the OU Math Club Blog. It is noteworthy that the Blog has over 100 visitors per day from around the world. He will also continue his efforts to mentor junior mathematicians through formal and informal collaborations. The PI will promote the development of algebraic, geometric, and topological tools in representation theory as an invited speaker in the U.S. and abroad and as an organizer of conferences in the area.