This PhD project will investigate questions surrounding categorification in representation theory.
Categorification has led to many breakthroughs in representation theory in the last 15 years. On the most basic level (going in some sense from categorical level zero to level 1), vector spaces over a fixed field categorify natural numbers. By taking the dimension of a vector space, we can reproduce a natural number, but we have additional information for vector spaces when looking at linear maps from one to the other that are not visible purely on the level of dimensions. In representation theory, one typically looks at passing from categorical level one to categorical level 2, and roughly speaking, replaces algebras acting on vector spaces (in classical representation theory) by higher categorical objects, more specifically 2-categories, acting on categories. This gives rise to new methods to tackle long-standing problems and has led for example led to a proof of Broué's abelian defect group conjecture for symmetric groups, an algebraic proof of the Kazhdan--Lusztig conjectures for all Coxeter types, and counterexamples to James’ conjecture.
The PhD project will focus on both advancing the general theory and studying special classes of examples of categorification, e.g. coming from Soergel bimodules, which categorify Hecke algebras. The technical ingredients for the project will come from classical representation theory of finite-dimensional algebras, category theory and homological algebra.