# Small Lecture Series

## Mikado braids, Soergel bimodules, and positivity in Hecke and Temperley-Lieb algebras

**Thomas Gobet **(Université de Lorraine)

**Lecture I: Tuesday, October 10, 14:15 - 15:45**

**Lecture II: Wednesday, October 11, 14:15 - 15:45**

**Lecture III: Thursday, October 12, 14:15 - 15:45**

**Abstract**

The aim of the lecture is to present some interesting elements of the braid group called Mikado braids, which admit a very simple topological definition, and their categorifications. These elements can be used to introduce generalizations of Kazhdan-Lusztig polynomials and to prove related positivity properties generalizing Kazhdan-Lusztig positivity conjecture using the framework of Soergel bimodules (and their bounded homotopy category). In the first lecture, we will discuss algebraic characterizations of Mikado braids, which will allow a generalization to Artin-Tits groups attached to Coxeter groups (Artin-Tits groups are generalizations of the classical braid group, but with no known topological description in general). In the second lecture, we will recall the definition of Kazhdan-Lusztig polynomials and their combinatorics, and consider categorifications of Mikado braids in terms of complexes of Soergel bimodules; using the nice homological properties of these complexes, we will deduce the above mentioned positivity properties of generalized Kazhdan-Lusztig polynomials (conjectured by Dyer). In the last lecture, we will discuss applications to dual Garside structures of Artin-Tits groups (in the case where the Coxeter group is finite) and to positivity properties involving base change matrices in the (classical) Temperley-Lieb algebra.

## Slides I

## Slides II

## Slides III

## An Algebraic Introduction to Kapustin-Witten Theory

**Christopher Elliott** (Institut des Hautes Études Scientifiques)

**Lecture I: Tuesday, October 24, 14:15 - 15:45**

**Lecture II: Wednesday, October 25, 14:15 - 15:45**

**Lecture III: Thursday, October 26, 14:15 - 15:45**

**Abstract**

Kapustin and Witten constructed a family of "twisted N=4 gauge theories" in four dimensions in order to build a bridge between gauge theory and the geometric Langlands correspondence. In these lectures I'll introduce N=4 theories and explain how to derive their twists in a purely algebraic way. I'll discuss some aspects of the quantification of these theories, and explain an application to the theory of singular supports in geometric Langlands. This is joint work with Philsang Yoo.

See also https://www.mpim-bonn.mpg.de/node/7654 for a closely related talk of the speaker on Monday.

## Chern-Simons theory & quantum groups approached through factorization algebras

**Owen Gwilliam** (Max Planck Institute for Mathematics Bonn)

**Lecture I: Tuesday, November 7, 14:15 - 15:45**

**Lecture II: Wednesday, November 8, 14:15 - 15:45**

**Lecture III: Thursday, November 9, 14:15 - 15:45**

**Abstract**

The three-dimensional topological field theory called Chern-Simons theory has actively received the attention of various mathematical communities over the last few decades. Recently, ideas and methods from higher algebra have been applied to it. These lectures will give an introduction to such work, using it as an excuse to explain some sophisticated ideas, like E_{n }algebras, factorization homology, or filtered Koszul duality. (We will not assume comfort with higher algebra on the part of the audience.) The focus will be on joint work with John Francis and Kevin Costello, which directly connects the Feynman diagrammatic approach (think Vassiliev invariants of knots) to the quantum group approach (think HOMFLYPT polynomials). We will also relate our approach to the recent work of Ben-Zvi, Brochier, and Jordan on "integrating quantum groups" and of Calaque, Pantev, Toen, Vaquie, and Vezzosi on categorical deformation quantization.

## Categorification of Verma modules

**Pedro Vaz **(Université catholique de Louvain)

**Lecture I: Wednesday, November 15, 14:15 - 15:45**

**Lecture II: Thursday, November 16, 14:15 - 15:45**

**Lecture III: Friday, November 17, 14:15 - 15:45**

**Abstract**

In this series I will explain my joint work with Grégoire Naisse on categorification of Verma modules for quantum Kac-Moody algebras. The plan is to give a detailed overview on the categorification of the finite-dimensional irreducible representations of quantum sl_{2} using categories of modules for cohomologies of finite-dimensional Grassmannians and partial flag varieties. This is due to Frenkel-Khovanov-Stroppel and indepently to Chuang-Rouquier, and is an example of how such categorifications arive naturally. By working with infinite Grasmannians and adding a bit more structure we are able to categorify Verma modules for quantum sl_{2}. One natural sub-product of the geometric approach to categorification of Verma modules is a superalgebra extending the well-known NilHecke algebra, one of the fundamental ingredients in the categorification of quantum sl_{2}. In the last part I will explain the case of categorification of Verma modules for Kac-Moody algebras. This requires a generalization of KLR algebras, the latter being the main ingredient in the categorification of quantum groups by Khovanov-Lauda-Rouquier.