# Geometric Langlands Seminar, 2022--2023

This is the website for the 2022--2023 Geometric Langlands Seminar at UChicago.

Please email me at amathew (at) math (dot) uchicago (dot) edu for the Zoom coordinates. The lecture videos work best if they are first downloaded (or they sometimes stop after an hour).

During October 2022, Pavel Etingof and Victor Ostrik will give a series of lectures.

Title for the series: Symmetric tensor categories and modular representation theory.

Relevant material includes:

Title for Lecture 1 (10/3): Algebra and representation theory without vector spaces.
Abstract for lecture 1: A modern view of representation theory is that it is a study not just of individual representations (say, finite dimensional representations of an affine group or, more generally, supergroup scheme G over an algebraically closed field k) but also of the category Rep(G) formed by them. The properties of Rep(G) can be summarized by saying that it is a symmetric tensor category (shortly, STC) which uniquely determines G. A STC is a natural home for studying any kind of linear algebraic structures (commutative algebras, Lie algebras, Hopf algebras, modules over them, etc.); for instance, doing so in Rep(G) amounts to studying such structures with a G-symmetry. It is therefore natural to ask: does the study of STC reduce to (super)group representation theory, or is it more general? In other words, do there exist STC other than Rep(G)? If so, this would be interesting, since algebra in such STC would be a new kind of algebra, one "without vector spaces". Luckily, the answer turns out to be “yes". We will discuss examples in characteristic zero and p>0, and also Deligne's theorem, which puts restrictions on the kind of examples one can have.

Notes and video for Lecture 1.

Title for Lecture 2 (10/10): Symmetric tensor categories of moderate growth and modular representation theory.
Abstract for lecture 2: Symmetric tensor categories of moderate growth and modular representation theory. There is an almost obvious necessary condition for a symmetric tensor category (STC) to have any realization by finite dimensional vector spaces (and in particular to be of the form Rep(G)): for each object X the length of the n-th tensor power of X grows at most exponentially with n. We call this property ``moderate growth". Not all STC have this property - so called Deligne categories provide counterexamples. Thus it is natural to ask if there exist STC of moderate growth other than Rep(G). In characteristic zero, the negative answer is given by the remarkable theorem of Deligne (2002). Namely Deligne's theorem says that a STC of moderate growth can always be realized in supervector spaces. However, in characteristic p the situation is much more interesting. Namely, Deligne's theorem is known to fail in any characteristic p>0. The simplest exotic symmetric tensor category of moderate growth (i.e., not of the form Rep(G)) for p>3 is the semisimplification of the category of representations of Z/p, called the Verlinde category. For example, for p=5, this category has an object X such that the tensor square of X is isomorphic to X+1, so X cannot be realized by a vector space (as its dimension would have to equal the golden ratio). We will discuss some aspects of algebra in these categories, in particular failure of the PBW theorem for Lie algebras (and how to fix it) and a generalization of Deligne's theorem in characteristic p by Coulembier, Etingof and Ostrik. This generalization allows one to prove new properties of modular representations of finite groups (and, more generally, affine group schemes) which were previously out of reach. We will also discuss a family of non-semisimple exotic categories in characteristic p constructed in the work of Benson, Etingof and Ostrik, and their relation to the representation theory of groups (Z/p)^n over a field of characteristic p.

Notes and video for Lecture 2.

Ostrik's notes and Etingof's notes and video for Lecture 3.