Geometric Langlands Transform and Dualities

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The principal investigators will conduct research in the geometric Langlands program. Given a connected smooth projective algebraic curve X over a field of characteristic zero and a reductive group G, the goal of this program is to construct a canonical equivalence between the DG category of D-modules on the stack of G-bundles on X and a certain DG category related to O-modules on the stack of local systems on X with respect to the Langlands dual of G. The principal investigators will work on the construction of this equivalence for G=GL(n). Using methods of non-commutative geometry, they will study the duality on the DG category of D-modules that corresponds to Serre duality for O-modules. They will also study the classical automorphic counterpart of this duality, which is a non-standard scalar product on the space of automorphic forms.The Langlands program was formulated in 1967 as a series of conjectures in number theory and the theory of automorphic forms. Later the geometric Langlands program was formulated in terms of algebraic geometry over algebraically closed fields and interpreted in physical terms as a duality in quantum field theory. This project is aimed at proving the main conjecture of the geometric Langlands program in an important particular case. It combines ideas from the geometric and classical theory of automorphic forms as well as non-commutative geometry.