# Geometric and Modular Representation Theory

May 22-28, 2021

Uhlenbeck Lecturer:  Catharina Stroppel, University of Bonn

Terng Lecturer:  Monica Vazirani, University of California, Davis

Uhlenbeck Lecture: Representation theory and categorification

Abstract: In modern representation theory we often study the category of modules over an algebra, in particular its intrinsic and combinatorial structures.  Vice versa one can ask the question: which categories have a given combinatorics? This is the basic insight into the concept of categorification.   Categorification is a recent development which often provides finer invariants and is used in different areas of mathematics (such as knot theory). We will explain  a crucial ingredient  in categorification: diagrammatics of tensor categories.  We will focus on  a few examples that arise in current research, such as Deligne categories.

This course provides a categorical framework for the structures studied in the elementary course.

Terng Lecture:  Representation theory and combinatorics of the symmetric group and related structures

Abstract: With an eye toward coordinating with the advanced course, we will start with the representation theory of the symmetric group and related combinatorics. We will  focus on the functors of induction and restriction.  We will then consider related algebraic structures such as Hecke algebras. We can use diagrammatics to depict various elements, objects, morphisms, and so on, even in the setting of de-categorification.

References for both courses: Etingof’s lecture notes http://www-math.mit.edu/~etingof/reprbook.pdf;

Gruson and Serganova: A Journey Through Representation Theory;

Chapters 1, 2, 3 (particularly 3.1, 3.2, 3.3)  of Benson’s book D. J. Benson: Representations and Cohomology I: Basic Representation Theory of Finite Groups and Associative Algebras;

J.P. Serre: Linear Representations of Finite Groups;

James and Liebeck : Representations and Characters of Groups;

flipping through Fulton and Harris Representation Theory gives a nice sense of representation theory.