## **Algebra and Geometry lecture series.**

**Time: Th 4-6, location: zoom **
(ask Ivan for a password).

Organized by Ivan Losev.

This seminar features lecture series by participants who are usually Yale faculty and postdocs.

Videos are posted in Yale box.
In case they cannot access the Yale box, participants outside of Yale
should contact Ivan.

The first lecture series in the Spring is by Gurbir Dhillon
and the topic is *An informal introduction to categorical actions of groups*. The first lecture is on Feb 3.
For a prospective schedule, notes and videos, visit here.
**Abstract**: The theory of categorical actions of groups is an important actor in modern representation theory. It has undergone rapid development in the past twenty years, principally due to its applications in the geometric Langlands program, with decisive contributions by Beilinson, Drinfeld, Frenkel, and Gaitsgory, and more recently Beraldo, Raskin, and Yang. Our goal will be to give a motivated introduction to this circle of ideas.

Categorical actions are deeply intertwined with the representation theory of Lie algebras in their history and major applications, and we will emphasize these connections throughout. In particular, using ideas from categorical representation theory, we will give a very simple proof of Beilinson--Bernstein localization for semisimple Lie algebras. Moreover, using ideas from local geometric Langlands, we will meet a version of Beilinson--Bernstein localization for affine Lie algebras at critical level, as developed by the above authors.

Prerequisites and Material: It will be helpful to have some familiarity with semisimple Lie algebras and their representation theory, algebraic geomety, and homological algebra, but not much beyond a first course in any of these topics. We will largely follow a
survey of this area.

The first lecture series was by Ivan in Fall 2021 and the topic was *Quantizations in positive characteristic and applications*.
Click here for an abstract including a list of prerequisites and references.

Lecture 0: *Prerequisite lecture about basics of quantizations*. Includes the definition and examples
of Poisson algebras, the definition and examples of filtered quantizations, the definition of formal quantizations and their relation with filtered
ones. Finally, there is a discussion of quantizations of schemes.
Sept 8, 4-5pm: Office hour on Lecture 0.
Sept 9, Lecture 1: *Quantizations via Hamiltonian reduction*, Notes.
Sept 16, Lecture 2: *Quantization commutes with reduction*, Notes. *Differential operators in characteristic p*,
Notes.
Sept 22, office hours, 3-4pm.
Sept 23, Lecture 3: *Frobenius constant quantizations. Derived equivalences from quantizations.*
Notes.
Sept 30, Lecture 4:* Splitting bundles.* Notes.
Oct 7, Lecture 5: *The case of Springer resolution and connection to modular representations of semisimple
Lie algebras*. Notes (updated 10/8).
Oct 14, Lecture 6: *The case of Springer resolution and connection to modular representations of semisimple
Lie algebras, cont'd*, Notes.
Oct 21: no lecture.
Oct 28: Lecture 7: *Hilbert schemes and Procesi bundles. Hamiltonian reduction.* Notes.
Nov 4, Lecture 8: *Quantizations of symmetric powers and Hilbert schemes.* Notes.
Nov 11, Lecture 9: *Construction of Procesi bundles via quantizations in characteristic p.* Notes.
Nov 18, Lecture 10: *Construction of Procesi bundles, cont'd. Lifting to characteristic 0 and rational Cherednik algebras.*
Notes.
Nov 25, no lecture.
Dec 2, Lecture 11: *Macdonald positivity*, Notes.
THE END of this lecture series. Something to follow in the Spring semester.