Sébastien Palcoux
Yanqi Lake Beijing Institute of Mathematical Sciences and Application, Beijing, China

A well-known open problem is whether there exists a finite quantum group which cannot be "cooked up" from finite groups, and more generally, an integral fusion category which is not weakly group-theoretical. In the simple case, we will see that it is equivalent to the existence of non-pointed simple integral modular category. A way to investigate this problem is to look for simple integral fusion rings, and see whether a non-grouplike one can be categorified, or at least a modular data.

In joint works with Max A. Alekseyev, Winfried Bruns, Sebastien Burciu, Huang Linzhe, Zhengwei Liu, Fedor V. Petrov, Yunxiang Ren and Jinsong Wu, we developed several categorification criteria, involving modular arithmetic, hypergroup theory, quantum Fourier analysis, localization strategies (of the pentagon equations), and we applied them as efficient filters for above investigation. We obtained classification results of the Grothendieck rings of simple integral fusion categories up to rank $8$, and of the modular data of integral modular fusion categories up to rank $12$.