Given a modular category C, the irreducible characters of its fusion ring are in one-to-one correspondence with the set Irr(C) of isomorphism classes of simple objects of C. Consequently, the action of the absolute Galois group on these characters induces a permutation action on Irr(C). The analysis of this action is essential to the classification of modular categories, whether by rank or by other characteristics. Another essential property of a modular category is its associated (projective) SL(2,Z)-representation, whose kernel is a congruence subgroup.  This gives another connection between the theory of modular categories and classical number theory.

I will talk about the recent progress on the classification of modular categories using representation and number theoretic methods. Then I will discuss the results regarding the classification of modular categories according to the number of Galois orbits. The talk is based on joint work with Siu-Hung Ng and Yilong Wang; and separate joint work with Julia Plavnik, Andrew Schopieray, and Zhiqiang Yu.

Speaker’s webpage: https://zhangqing2513.github.io/

Location: Jointly in person and virtually on Zoom. SAS 2225 for in-person participation. The Zoom link is sent out to the Algebra and Combinatorics mailing list, please contact Corey Jones at cmjones6@ncsu.edu to be added.