Workshop

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