CAREER: Equivariant topological field theories and higher cluster categories

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This proposal is concerned with several extensions and applications of the theory of homotopical higher categories. Our first foundational objective is to develop equivariant versions of the many known models for homotopical higher categories and to establish equivalences between them. Our first proposed application is the development of homotopical approaches to equivariant extended topological field theories. The second application is the development of topological cluster categories arising from surfaces; extending to higher dimensions should allow for the development of new invariants of higher-dimensional manifolds analogous to ones for surfaces. This second application should also inform the first, with higher-dimensional cluster categories giving new information about topological field theories. The third application is concerned with connections between Hall algebras and algebraic K-theory. In one direction, constructions of homotopical Hall algebras are expected to give rise to K-theory spectra which should give new information about Hall algebras, especially those related to quantum groups. In another, variations of Hall algebra constructions have corresponding variants of algebraic K-theory which are worthy of further investigation. Broadly speaking, this proposal is concerned with incorporating algebraic information into categorical and topological structures which are currently being used in a wide range of kinds of mathematics. We then seek to apply these enhanced structures in mathematical physics, manifold theory, and representation theory. The educational component of this proposal consists of a series of four summer workshops for mathematics majors who are in the process of transferring to UC Riverside. The goal is to help twenty participants each year to make the transition to upper-level mathematics via introduction to proof techniques, more theoretical concepts, and a broad overview of the range of topics in higher-level mathematics courses. Students will be provided with some follow-up mentoring activities, including opportunities for participating in undergraduate research.