Schedule, Notes/Handouts & Videos

We will introduce the current state of the art in categorified representation theory and low-dimensional topology.  We will aim to highlight the current techniques and applications while highlighting the open problems that could benefit from new higher categorical tools.

Exercise Worksheet

These lectures will serve as an introduction to a class of 4-dimensional topological field theories arising from ribbon categories. I will explain their relationship to the Crane-Yetter state sum, Chern-Simons theory, geometric Langlands program and skein modules.

In the series of talks I will explain some ideas behind categorifications arising from representation theory. We will start by
looking first at basic examples of Heisenberg categorifications due to Khovanov. The goal hereby is to get a feeling for the involved 2-categories and the background of the graphical calculus. The goal is to give some indications how representation theorists would get natural candidates for the space of 2-morphisms.

In the second talk I will make the connection with Delignes universal tensor categories and their representations and indicate who categorifications can be used to get information back to representation theory.

In the third talk I might explain some more recent applications to knot theory.

In the series of talks I will explain some ideas behind categorifications arising from representation theory. We will start by looking first at basic examples of Heisenberg categorifications due to Khovanov. The goal hereby is to get a feeling for the involved 2-categories and the background of the graphical calculus. The goal is to give some indications how representation theorists would get natural candidates for the space of 2-morphisms.

In the second talk I will make the connection with Delignes universal tensor categories and their representations and indicate who categorifications can be used to get information back to representation theory.

In the third talk I might explain some more recent applications to knot theory.

Abstract

I will describe the theory of traces in higher categories and describe some recent applications of this theory in geometric representation theory and the geometric Langlands program.