Classical and Quantum Geometric Langlands Correspondence

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For a smooth complete algebraic curve X over an algebraically closed field k, and a reductive group G, there is a fundamental object called the moduli space of G-bundles on X, and denoted Bun(G). The theory of automorphic sheaves, which is a geometrization of the classical theory of automorphic functions, studies various categories of sheaves on Bun(G). Most fundamentally, we study the (derived) category of quasi-coherent sheaves QCoh(Bun(G)), as well as various categories of twisted D-modules. The goal of Geometric Langlands Correspondence is to study the remarkable relation of this category to the similarly defined category for another reductive group, called the Langlands dual of G. As was discovered in the middle of the 20th century, one can give a list of all fundamental laws of symmetry that occur in nature--these are classified by Dynkin diagrams, also known as root systems. Our fundamental object of study, the theory of automorphic sheaves, couples these symmetry laws with another fundamental object in modern mathematics, namely algebraic curves (also known in their incarnation as Riemann surfaces). The discovery of Langlands correspondence indicates that the theory of automorphic sheaves admits additional symmetries of its own. The current project aims to study these new symmetries at their most fundamental.