Graduate Seminar. Geometric Representation theory. Fall 2009

Announcements & Schedule

Announcements

• On Thurs., Nov. 19, Jacob will continue his talk on crystals. Note that the seminar will beging at 6pm, and not at 5.30pm as usual .
• Notes from the talk of Jacob Lurie on the approach to D-modules and D-schemes via crystals have been uploaded.
• Notes by Dustin Clausen have been updated and now include the material from his second talk as well.
• There will be no seminar during the Thanksgiving week.
• The last talk this term will be by David Kazhdan on Dec. 8: Classical vs. Geometric Langlands

Future talks

• Nov. 19 (MIT), 6pm (!). Jacob Lurie. Crystals of Schemes.
• Dec. 1 (Harvard). Dustin Clausen. Quantization!
• Dec. 8 (MIT). David Kazhdan. Classical vs. Geomteric Langlands

Seminar Notes

The link to the text of the Beilinson-Drinfeld book

• Pages 1-100
• Pages 101-200
• Pages 201-300
• Pages 301-384

Notes from the current seminar

• Sept. 8: The Introductory talk (Dennis Gaitsgory)
• Sept. 15: Stacks (Dennis Gaitsgory and Toly Preygel)
• Sept. 15-17: Stacks (Toly Preygel)
• Sept. 17: Bun(G) (Dennis Gaitsgory)
• Sept. 22: D-modules on stacks-I: tangent complex and cotangent space (Sam Raskin)
• Sept. 24: D-modules on stacks-II: equivariant and twisted D-modules (Sam Raskin)
• Oct. 1: Why Bun(G) is an algebraic stack (Nir Avni)
• Oct. 6: D-modules on stacks-III: equivariant and twisted D-modules (Sam Raskin)
• Oct. 13: Affine Grassmannian and the Loop group (Dennis Gaitsgory, notes by D.G. and N. Rozenblyum)
• Oct. 20: Hitchin map: global vs. local (Andrei Negut)
• Oct. 22: Global nilpotent cone and flatness of the Hitchin map (Xinwen Zhu)
• Oct. 27: Higgs bundles, Kostant section, and local triviality of G-bundles (Dennis Gaitsgory)
• Oct. 29: Higgs bundles and affine Springer fibers (Roman Bezrukavnikov)
• Nov. 3: Infinite-dimensional linear algebra, determinant line bundle and Kac-Moody extension (Dustin Clausen)
• Nov. 12: D-schemes, jets, and conformal blocks (Andrei Negut)
• Nov. 17-19: D-modules and D-schemes via crystals (Jacob Lurie)

Other notes

• Notes by Tomas Gomez

Suggested Background Reading

Homological algebra

Introduction to derived categories

• "Methods of Homological Algebra" by S. Gelfand and Yu. Manin
• The same thing as a .djvu
• Lectures by A. Beilinson
• Lectures notes by D. Milicic

DG categories

• Introduction and appendices to "DG quotients of DG categories" by V. Drinfeld
• Section 1 of "DG guide to Voevodsky's motives" by A. Beilinson and V. Vologodsky

D-Modules

General theory

• Joseph Bernstein's lectures on D-modules
• Sasha Braverman's lectures on D-modules
• Notes from Roman Bezrukavnikov's course at MIT
• Notes of a course by V. Ginzburg at U of C
• Notes of a course by D. Milicic
• "Algebraic D-modules" by A. Borel et al.
• "D-modules, Perverse Sheaves, and Representation theory", by R. Hotta, K. Takeuchi and T. Tanisaki

Nearby and Vanishing cycles

• A transcript of Beilinson's paper "How to glue perverse sheaves"
• Notes on "How to glue perverse sheaves" by Ryan Reich
• Sam Lichtenstein's senior thesis on vanishing cycles and "How to glue perverse sheaves"
• arXiv/0810.1504, by A. Beilinson and D.G.
• "A proof of Jantzen conjectures" by A. Beilinson and J. Bernstein, Sect. 4.1-4.2

Twisted differential operators (TDO) and D-modules in the equivariant setting

• "A proof of Jantzen conjectures" by A. Beilinson and J. Bernstein, Sect. 1 & 2

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

• Section III.8 in "Methods of Homological algebra" by S. Gelfand and Yu. Manin
• Background section from "Equivariant sheaves and functors" by J. Bernstein and V. Luntz, LNM 1578

Etale cohomology

• "Etale cohomology" by J. Milne
• Lecture notes by J. Milne
• "Etale cohomology and Weil conjectures" by E. Freitag and R. Kiehl

Constructible sheaves in the l-adic setting

• SGA 4.5, Arcata
• "Weil conjectures, perverse sheaves and l'adic Fourier transform" by R. Kiehl and R. Wiessauer, Chapter 2 and Appendix
• "La conjecture de Weil-II", D. Deligne, Sect. 1.1

Perverse sheaves

• "Faisceaux pervers", a.k.a. [BBD], by A. Beilinson, J. Bernstein and P. Deligne
• "Weil conjectures, perverse sheaves and l'adic Fourier transform" by R. Kiehl and R. Wiessauer, Chapter 3

Algebraic stacks

Descent theory

• Chapter 4 by A. Vistoli

• Descent-I

Why do certain moduli problems admit solutions? Quot schemes, Hilbert schemes, Picard schemes, etc.

• Chapter 5 by N. Nitsure & Chapter 6 by S. Kleiman

• A. Grothendieck, Quot schemes
• A. Grothendieck, Hilbert schemes
• A. Grothendieck, Picard schemes

Definition of stacks

• Section 4 of "Irreducibility of the space of curves of given genus" by P. Deligne and D. Mumford
• "Champs Algebriques" by G. Laumon and L. Moret-Bailly
• TBL's minor thesis
• A. Vistoli's article
• Book in progress by multiple authors

The stack of G-bundles

• Lectures notes by C. Sorger

Category O

• "Structure of representations that are generated by vectors of higher weight"
• "A certain category of g-modules"

• A book by J. Humphreys
• Notes by D.G.

Number theory

Local and global fields, adeles

• Cassels-Frohlich, which still seems to be the best rerefence.

Automorphic functions

• Volume 6 of "Generalized functions" by Gelfand, Graev and Piatetskii-Shapiro.

Class Field theory

• Beilinson's lectures