Equivariant methods in representation theory (V5A6)

First lecture: Wednesday, Oct 9.

Instructor: Alexandre Minets

Mentor: Catharina Stroppel

Schedule: Wednesdays, 4-6 pm, N 0.007 Neubau (behind the main math building in Endenicher Allee 60)

Exam: Exams will be oral. Details will be announced in the course.

Prerequisites: Algebraic Topology (singular homology/cohomology), Algebraic Geometry (complex algebraic varieties). Some familiarity with representation theory is helpful, but not required.

Course description:
One of the major advances in representation theory of algebras in the late 20th century was the realization that for many algebras, their finite dimensional modules can be found in cohomology groups of various topological spaces. Consequently, one can use geometric techniques to study purely algebraic question, and this line of thought has proved to be extremely fruitful. The topological spaces one considers often come equipped with an action of a big group of symmetries. One is therefore naturally led to look for a cohomology theory which remembers this action. Enter equivariant cohomology.

In this course, we will review the basic constructions of equivariant cohomology, with particular focus on the localization theorem, which will be our main computational tool. Having set up the foundations, we will proceed to study the geometry of various spaces, and see how their geometry reflects in the study of algebras of interest. Our main example will be flag varieties and Hecke algebras.

References:

• Anderson, Fulton: Equivariant cohomology in algebraic geometry
• Chriss, Ginzburg: Representation theory and complex geometry

A preliminary plan can be found here.

Notes will be made available as they are written. Current version: Lecture notes