Colloquium abstracts

Shashank Kanade
University of Denver
December 9, 2021

Some results and conjectures in the theory of vertex operator algebras:  Individual vertex operators arose in the mathematical literature nearly four decades ago in Lepowsky-Wilson's Lie algebraic proof of the Rogers-Ramanujan identities. Vertex operator algebras (VOAs) were also central to Borcherds' proof of the moonshine conjecture -- the moonshine module constructed by Frenkel-Lepowsky-Meurman and used in Borcherds' proof is a VOA. Since their inception, the study of VOAs has seen a rapid growth guided by various conjectures in mathematics and physics. Most well-known VOAs are in some way connected to affine Lie algebras and their study is naturally related to representation theory, tensor categories, algebraic combinatorics and number theory. In this talk, I will survey a selection of results and conjectures pertaining to these topics. I will focus on (a subset of) -- 1. Rogers-Ramanujan-type identities related to affine Lie algebras, 2. Tensor categorical aspects related to conformal embeddings of VOAs, 3. Some problems in the representation