Graduate Seminar. Geometric Representation theory. Fall 2009--Spring 2010
Announcements & Schedule
We shall study the unfinished book
"Quantization of Hitchin's integrable system and Hecke eigensheaves" by A. Beilinson and V. Drinfeld.
In the spring semester we'll meet mostly on Tuesdays 5.30-8pm (pizza break in the middle),
alternating between Harvard (Sci Cen 507) and MIT (2-139).
Most weeks on Thursdays, D.G. will hold office hours for this seminar in his office (Sci Cen 338, inside
the Birkhoff Library; 5-6.30pm).
Everybody is welcome at attend. We'll discuss whatever questions
people might have about the seminar talks or related topics.
Announcements
- The first talk this term will be on Feb. 9.
- Thurs., Feb. 11. (D.G.'s office; 5-6.30pm). Seminar Office hours.
Future talks
- Tue., Feb. 9. (Harvard). Dennis Gaitsgory. Overview and recap of Dustin's talk on quantization
- Tue., Feb. 16. (MIT). Ryan Reich. Affine Grassmannian-I: factorization, convolution, and fusion
- Tue., Feb. 23. (Harvard). Ryan Reich. Affine Grassmannian-II: geometric Satake equivalence
Seminar Notes
The link to the text of the Beilinson-Drinfeld book
Notes from the Spring Semester 2010
If you have any comments on these notes (mathematical, pedagogical or typos), please let me know!
Notes from the Fall Semester 2009
Other notes
Suggested Background Reading
If you are aware of additional/better references on the subjects listed below (especially, number theory), or can provide
URL's or .pdf files, please let me know!
Homological algebra
Introduction to derived categories
DG categories
D-Modules
General theory
Nearby and Vanishing cycles
Twisted differential operators (TDO) and D-modules in the equivariant setting
Constructible and perverse sheaves
Constructible sheaves on complex algebraic varieties
- "Sheaves on Manifolds" by M. Kashiwara and P. Shapira
- "Sheaves in Topology" by A. Dimca (.pdf is available)
- Notes by L. Nicolaescu
See also:
Etale cohomology
Constructible sheaves in the l-adic setting
Perverse sheaves
Algebraic stacks
Descent theory
See also the original article by Grothendieck:
Why do certain moduli problems admit solutions? Quot schemes, Hilbert schemes, Picard schemes, etc.
See also the original (wonderful) articles by Grothendieck:
Definition of stacks
The stack of G-bundles
A good intro to the kind of things we'll be doing is:
Category O
The original papers by Bernstein-Gelfand-Gelfand in Functional Analysis and Applications:
- "Structure of representations that are generated by vectors of higher weight"
- "A certain category of g-modules"
See also:
Number theory
Some familiarity with local and global fields, adeles, adele groups and basics of the theory of automorphic
functions and representation would be useful.
Local and global fields, adeles
Automorphic functions
- Volume 6 of "Generalized functions" by Gelfand, Graev and Piatetskii-Shapiro.
Class Field theory
There are numerous expositions. Below is the link to informal lectures by A. Beilinson at U of C: