Introduction to Geometric Langlands

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The field of «Geometric Langlands Correspondence» was started by Drinfeld in the early 1980's as a way to prove the Galois => Automorphic direction in the classical Langlands program in the case of function fields. The initial observation is that if our function field F is the field of rational functions on a curve X over F_q, then the automorphic space attached to F and a group G can be thought of as the set of F_q-points of the moduli space Bun_G(X) that classifies G-bundles on X. Drinfeld's idea is that one can use Grothendieck's sheaf => functions dictionary to construct automorphic functions starting from l-adic sheaves on Bun_G(X). In the talk we will give an introduction to these ideas, and how the various familiar objects in the theory of automorphic functions play out in this enhanced geometric context.