Talbot 2005

Mentored by David Ben-Zvi of University of Texas, Austin

Monday:

• From Langlands to geometric Langlands [Scott Carnahan]

• Hecke and Iwahori-Hecke algebras [Elizabeth Mann]

• Moduli of bundles and algebraic stacks [Chris Douglas]

• Perverse sheaves [Andre Henriques]

Tuesday:

• Langlands & Deligne's proof for GL(1) [Mike Hill]

• D-modules [Travis Schedler]

• Vertex algebras [Reimundo Heluani]

• VA & Langlands [David Ben-Zvi]

Wednesday:

• Geometric Satake isomorphism [Joel Kamnitzer]

• Beilinson-Bernstein localization [A.J. Tolland]

• Hitchin's integrable system [Alexei Oblomkov]

• Quantization of Hitchin's system [David Ben-Zvi]

Thursday:

• Fourier-Mukai transform [John Francis]

• Fourier-Mukai transform and Langlands over C [David Ben-Zvi]

• GL(2), Eichler-Shimura [Mark Behrens]

• New developments [David Ben-Zvi]

Moduli of vector bundles on surfaces. Algebraic stacks.

• Sorger: Lectures on moduli of principal G-bundles over algebraic curves.

Tannakian categories. Reconstructing an algebraic group from its tensor category of finite-dimensional representations.

• Breen: Tannakian categories (in the big Motives books).

• Deligne: Categories tannakeniennes.

• Deligne and Milne: Tannakian categories.

and as Tannakian categories relate to geometric Langlands for reductive groups other than GL(n):

• Mirkovic and Vilonen: Geometric Langlands duality and representations of algebraic groups over commutative rings.

• Ginzburg: Perverse sheaves on a loop group and Langlands duality.

D-modules. Beilinson-Bernstein localization (giving an equivalence between Harish-Chandra modules and sheaves of D-modules on the flag variety).

• Bernstein: Course on D-modules

• Milicic: Algebraic D-modules and representation theory of semisimple Lie groups. Available from Milicic's webpage.

• Milicic: Localization and represention theory of reductive Lie groups. Available from Milicic's webpage.

• Borel: Algebraic D-modules.

Perverse sheaves. Riemann-Hilbert correspondence with D-modules. Intersection cohomology. l-adic perverse sheaves.

• Goresky and MacPherson: various

• Rietsch: An introduction to perverse sheaves.

• Massey: Notes on perverse sheaves and vanishing cycles.

• Kietag and Friehl: Weil Conjectures, Perverse Sheaves and l'adic Fourier Transform.

• Borel: Intersection cohomology.

• Kashiwara and Schapira: Sheaves on manifolds.

• Beilinson, Bernstein and Deligne: Faisceaux Pervers.

Fourier-Mukai transform. Equivalences of derived categories

• Mukai: derived categories of sheaves on abelian varieties.

• Rothstein: Sheaves with connection on abelian varieties.

• Bondal and Orlov: Derived categories of coherent sheaves.

Representations of affine Kac-Moody algebras.

• Ben-Zvi and Frenkel: Vertex algebras and algebraic curves.

• Beilinson and Drinfeld: Chiral algebras.

Integrable systems, quantizations of, and Hitchin's integrable system.

• Hitchin: Stable bundles and integrable systems.

• Donagi and Markman: Spectral curves, algebraically completely integrable Hamiltonian systems, and moduli of bundles.

• Semenov-Tian-Shansky: Quantum and classical integrable systems.

• Beilinson and Drinfeld: Quantization of Hitchin's integrable system and Hecke eigensheaves.

Langlands program. Hecke algebras. There are surveys by Gelbart and by Knapp.

• Haines, Kottwitz and Prasad: Iwahori-Hecke algebras.

• Kazhdan and Lusztig: Representations of Coxeter groups and Hecke algebras.

• Schubert varieties and Poincare duality. (Proc Symp AMS...)

• Lusztig: Singularities, character formulas and a q-analogue for weight multiplicities. (Asterisque 1982)

Opers.

• Beilinson and Drinfeld: Opers.

• Quantization of Hitchin's integrable system and Hecke eigensheaves.

Other:

• Frenkel: Recent advances in the Langlands program.

• Gaitsgory, Frenkel and Vilonen: On the geometric Langlands conjecture.

• Bernstein and Gelbart (eds.): An introduction to the Langlands program.

Other references may be found from the geometric Langlands webpages of Vilonen and of Arinkin.