Critical level representations of affine Kac-Moody algebras and the geometric Langlands program

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The proposed research project is inspired by a very mysterious duality in mathematics that appears, in various disguises, in such diverse areas as number theory, representation theory, combinatorics or geometry. It was first discovered by Robert Langlands, and is nowadays known as the Langlands program. In recent years, Langlands’ philosophy was translated into a geometric context and provided a series of far reaching conjectures that transcend the boundaries of mathematical disciplines. Typically the first step in proving a Langlands-type conjecture is to study the corresponding “local” situation, where the problem often acquires an algebraic or representation theoretic flavour. The main objective of the research project is to answer some of the fundamental questions of the local geometric Langlands program. Our approach is distinctively different from those proposed in the existing literature.