# Mathematical Sciences Research Institute

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# Workshop

Geometric Aspects of the Langlands Program March 18, 2002 - March 22, 2002
 Registration Deadline: March 22, 2002 about 13 years ago December 18, 2001 over 13 years ago
Parent Program: Infinite-Dimensional Algebras and Mathematical Physics
Organizers E. Frenkel, V. Ginzburg, G. Laumon and K. Vilonen
Speaker(s)

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Description

The Langlands Program has emerged in the late 60's in the form of a series of far-reaching conjectures tying together seemingly unrelated objects in number theory, algebraic geometry, and the theory of automorphic forms (such as Galois representations, motives, and automorphic forms). In recent years it was realized that the Langlands conjectures (in the function field case) may be formulated geometrically, thereby allowing one to state them over an arbitrary field (e.g., the field of complex numbers). This approach has led to Drinfeld's proof of the Langlands conjecture for GL(2) in the function field case. More recently, A. Beilinson and V. Drinfeld have proved a variant of the geometric Langlands correspondence over complex field. It relates Hecke eigensheaves on the moduli stack of G-bundles over a complex curve X and local systems for the Langlands dual group of G. They construct this correspondence via quantization of an integrable system on the cotangent bundle to the moduli space of G-bundles defined by Hitchin. Their work uses in an essential way representation theory of affine Kac-Moody algebras. On the other hand, L. Lafforgue has proved the Langlands conjecture for the case of the field of functions on a curve over a finite field. Geometry of bundles on curves with additional structures (shtukas) also plays an important role in his proof. In this workshop, we would like to bring together people working in different parts of this diverse area to enable them to learn from each other and to find points of contact between different directions. We are planning to concentrate in particular on the following topics: 1. Works of Beilinson and Drinfeld on the geometric Langlands correspondence. 2. Lafforgue's proof of the global Langlands conjecture for GL(n) in the function field case. 3. Local geometric problems, such as local models for Shimura varieties, fundamental lemma, and geometric realizations of Hecke algebras. Confirmed participants include: A. Beilinson, R. Bezrukavnikov, V. Drinfeld, G. Faltings, D. Gaitsgory, T. Haines, M. Harris, M. Kapranov, L. Lafforgue, S. Lysenko, R. MacPherson, I. Mirkovic, M. Rapoport. Group photo of participants

Funding & Logistics

## Show Funding

To apply for funding, you must register by the funding application deadline displayed above.

Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Funding awards are typically made 6 weeks before the workshop begins. Requests received after the funding deadline are considered only if additional funds become available.

## Show Lodging

A block of rooms has been reserved at the Rose Garden Inn. Reservations may be made by calling 1-800-992-9005 OR directly on their website. Click on Corporate at the bottom of the screen and when prompted enter code MATH (this code is not case sensitive). By using this code a new calendar will appear and will show MSRI rate on all room types available.

A block of rooms has been reserved at the Hotel Durant. Reservations may be made by calling 1-800-238-7268. When making reservations, guests must request the MSRI preferred rate. If you are making your reservations on line, please go to this link and enter the promo/corporate code MSRI123. Our preferred rate is \$129 per night for a Deluxe Queen/King, based on availability.

Additional lodging options (short term housing page - Short Term Housing

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Schedule
Mar 18, 2002
Monday
 09:15 AM - 09:45 AM Morning Tea 09:45 AM - 10:00 AM Welcoming Remarks Loa Nowina-Sapinski 10:00 AM - 11:00 AM On Langlands correspondence in the de Rham setting, part I Alexander Beilinson (University of Chicago) 11:00 AM - 11:30 AM Break 11:30 AM - 12:30 PM Perverse sheaves on loop Grassmannians Ivan Mirkovic (University of Massachusetts, Amherst) 12:30 PM - 02:00 PM Lunch 02:00 PM - 03:00 PM Central sheaves on he affine flag variety: A prerequisite for Bezrukavnikov's work Dennis Gaitsgory (Harvard University) 03:00 PM - 03:30 PM Afternoon Tea 04:10 PM - 05:10 PM Arthur-Selberg trace formula, Drinfeld shtukas and Langlands' correspondence Laurent Lafforgue
Mar 19, 2002
Tuesday
 10:00 AM - 11:00 AM Introduction to Shimura varieties Gerd Faltings 11:30 AM - 12:30 PM Local models of Shimura varieties in the ramified case Michael Rapoport (Universität Bonn) 02:00 PM - 03:00 PM Combinatorial patterns in the weight filtration on the nearby cycles for some Shimura varieties with bad reduction Thomas Haines (University of Maryland) 03:30 PM - 04:30 PM Geometric realization of local Langlands correspondences Michael Harris (Institut de Mathématiques de Jussieu)
Mar 20, 2002
Wednesday
 10:00 AM - 11:00 AM Pavings of polyhedra, glueing of Schubert cells and compactification of configuration spaces, part I Laurent Lafforgue 11:30 AM - 12:30 PM Line-bundles on the moduli-space of G-torsors Gerd Faltings 02:00 PM - 03:00 PM Sheaves on affine flags, and modular representations of the Langlands dual Lie algebra Roman Bezrukavnikov 03:30 PM - 04:30 PM p-Adic automorphic representations Matthew Emerton (University of Chicago)
Mar 21, 2002
Thursday
 10:00 AM - 11:00 AM On Langlands correspondence in the de Rham setting, part Ii Alexander Beilinson (University of Chicago) 11:30 AM - 12:30 PM Pavings of polyhedra, glueing of Schubert cells and compactification of configuration spaces, part II Laurent Lafforgue 02:00 PM - 03:00 PM Uhlenbeck spaces for A^2 and affine Lie algebra $$\hat{sl}_n$$ Michael Finkelberg 03:30 PM - 04:30 PM An introduction to the Drinfeld-Langlands program Gerard Laumon (Université de Paris XI)
Mar 22, 2002
Friday
 10:00 AM - 11:00 AM Modular representations of p-adic groups and q-affine Schur algebras Marie-France Vigneras (Université de Paris VII (Denis Diderot)) 11:30 AM - 12:30 PM Geometric Rankin-Selberg for GL(n) Sergey Lysenko 02:00 PM - 03:00 PM Formal arcs to algebraic semi-groups and automorphic L-functions Amy Braverman 03:30 PM - 04:30 PM Fourier transform for quantized completely integrable systems Dima Arinkin (University of Wisconsin)