### Graduate seminar on category O and Soergel bimodules, Fall 2017

This is Season 8 of a joint MIT-NEU (mostly) Representation theory graduate seminar.
Running weekly on Tuesdays
alternating between MIT and Northeastern:

MIT, 2-139, (4.30-7.30pm w. pizza break).

NEU, Richards 231 -- for September and October (5-8pm w. pizza break).
Richards is building 42 on this map.

The seminar is a part of the Northeastern RTG activities.

Organized by Roman Bezrukavnikov,
Pavel Etingof, and
Ivan Losev.
As usual, Pavel Etingof is a Boss.

**Important update:** Prof. Etingof has been promoted to a Supreme Leader.

**Description**: The seminar roughly consists of two (related) parts. In the first part, we will cover the Bernstein-Gelfand-Gelfand
category O, one of the most important and fundamental objects in Representation theory. We will start from basics and get to Soergel's
theory connecting the category O to Soergel bimodules and bimodules that are a very hot current subject in Representation theory.
We will then systematically study the Soergel bimodules with the ultimate goal of covering the Elias-Williamson algebraic proof of
the Soergel conjecture on the K_0-classes of the indecomposable Soergel bimodules that implies the Kazhdan-Lusztig conjecture on the
characters of simple modules in the BGG categories O.

**List of topics (preliminary) **:

1) Introduction to universal enveloping algebras and Verma modules. By Aleksei Pakharev (NEU).

2) Category O and its basic properties. By Daniil Kalinov (MIT).

3) Tensoring with finite dimensional representations. By Chris Ryba (MIT).

4) Soergel's theorems and Soergel bimodules. By Dmytro Matvieievskyi (NEU).

5) Soergel bimodules, Hecke algebras, and Kazhdan-Lusztig basis. By Boris Tsvelikhovsky (NEU).

6) Classical Hodge theory and the Decomposition theorem via Hodge theory. By Xiaolei Zhao (NEU).

7) Hodge theory of Soergel bimodules.

8) Rouquier complexes and Khovanov homology.

9) Hodge Riemann bilinear relations for Soergel bimodules.

10) Hard Lefschetz for Soergel bimodules.

**Prerequisites**: the structure and finite dimensional representation theory of semisimple Lie
algebras. Healthy portions of intellectual curiosity and bravery.

**Schedule **:
Tuesday, Sept 12, MIT, 2-139, 4-7pm. Aleksei. Hand-written notes.

Tuesday, Sept 19, NEU, Richards 231, 5-8pm. Danill. Notes.

Tuesday, Sept 26, MIT, 2-139, 4.30-7.30pm. Chris.

**References **: TBA.