Some results and conjectures in the theory of vertex operator algebras

Abstract: 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 theory of twisted affine Lie
algebras at non-integrable levels.

Parts of the talk will be based on joint works with my collaborators.