# Course Identification

Algebraic theory of D-modules 2
20214032

## Lecturers and Teaching Assistants

Prof. Dmitry Gourevitch, Prof. Joseph Bernstein
N/A

## Course Schedule and Location

2021
Second Semester
Monday, 10:15 - 13:00
22/03/2021
31/08/2021

## Field of Study, Course Type and Credit Points

Mathematics and Computer Science: Lecture; Elective; Regular; 3.00 points

See the course website for possible changes to the syllabus

## Prerequisites

Algebraic theory of D-modules 1.

30

English

## Attendance and participation

Expected and Recommended

Pass / Fail

80%
20%

Take-home exam

N/A
N/A
-
N/A

6

## Syllabus

In the second part of the course, in the Spring, we will study D-modules on general
algebraic varieties.
This would require from the beginning to use language of sheaves and use more
sophisticated homological technique (derived categories). We will spend some time
discussing these subjects.
Of course at this stage we will have to assume that people know some basic properties
of coherent sheaves on algebraic varieties (or at least have heard about them).
In other words, we will try to make my lectures formally self contained, but a prior
knowledge of basic facts of algebraic geometry would help.
We are also planning to try to describe how the language of D-module theory allows
to give a geometric interpretation of representations of reductive Lie groups and Lie
algebras (localization of g-modules).
This localization technique is a very powerful tool that allows to prove many highly non-trivial results in representation theory. It also connects Representation Theory with other areas of Mathematics (D-modules. perverse sheaves, weight filtrations and so on).
If time permits we will discuss some of these subjects in more details.

## Learning Outcomes

Upon successful completion of this course students should be able to:

[1] Learn many advanced algebraic notions - like derived categories, perverse sheaves and D-modules on general algebraic varieties.

[2] Learn the localization techniques, that allows to introduce geometry into complicated algebraic problems.