There is a funded PhD position available with a starting date of September 2024 in the area of representation theory and tensor categories. See here for details on the application procedure and the position. The application deadline is January 31 2024.
Overview: Typically the category of finite dimensional representations of a group, or an affine group scheme, a supergroup or a quantum group, is a tensor category. One would like to understand the fusion rules that describe the decomposition of tensor products or one seeks qualitative information like the classification of tensor ideals, the connection to other categories via tensor functors or the use of the tensor structure to perform constructions such as the Reshetikhin-Turaev 3-manifold invariants built from modular tensor categories. While the most important examples of tensor categories, or more generally, monoidal categories, arise from representation theory, the theory has outgrown its origins and has emerged into a vast and complex theory on its own, providing a unified language for many phenomena in different fields. Monoidal categories are now ubiquitious in areas such as representation theory, invariants of links and $3$-manifolds, algebraic geometry, quantum computing and mathematical physics.
I am particularly interested in the following two research areas.
1) Algebraic supergroups (such as $GL(m|n)$ and $OSp(m|2n)$) generalize algebraic groups. First motivated by questions in mathematical physics, they have become a thriving area in mathematics. I am interested in tensor product decompositions, the Duflo-Serganova cohomology functor, character/dimension formulae, quantized versions of Lie superalgebras and possible applications to mathematical physics. In a project in this area the successful candidate would study various ways to understand the tensor structures via categorical techniques, character formulas or homotopical algebra.
2) Deligne categories are families of universal tensor categories that interpolate other representation categories (e.g. representations of the symmetric group $S_n$ can be extended to complex parameters $S_t$, $t \in \mathbb{C}$). Quantized versions of these categories can be found in the works of Turaev. Other mathematicians - Knop, Etingof, Flake, Maassen, Meir, Khovanov, Kononov, Ostrik - have defined a vast number of interpolating categories which capture the phenomenon of stabilization with respect to rank. In a project in this area the successful candidate would further develop this theory, in particular by comparing them to other categories which capture the phenomenon of stabilization with respect to rank.
If this sounds interesting and you would like to have more information, please write me an email.