Categorification in Representation Theory (MIEMIETZV_U26EMP)

Categorification has led to major advances in representation theory in the last 25 years, such as the proof of Broué’s abelian defect group conjecture for symmetric groups and the proof of the Kazhdan—Lusztig conjectures for all Coxeter types. Classically, a representation of an algebra is an action of the latter on a vector space via linear maps. This is categorified to an action of a monoidal category on another category, or more generally of a 2-category on a collection of categories, via (nice) functors.

By studying categorifications, one can obtain new information about both the original algebra and the categories one is acting on. There are many open questions about the general theory of such higher representations, as well as about specific examples. This project will focus on understanding specific examples.

Prerequisites are a good understanding of algebras and their representations. Additional knowledge of some category theory would useful. The standard minimum entry requirement is 2:1 in Mathematics.