Wall-Crossing and Dualities in Representation Theory

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Representation theory is a branch of algebra studying symmetries, especially symmetries of linear mathematical structures. The incredible richness of the subject is partly due to the fundamental role it plays in quantum physics, as well as the way it enters formulations of the so-called higher (non-abelian) reciprocity laws of number theory in the Langlands duality program, which is building bridges connecting number theory, analysis, and theoretical physics. The present project is related to both these topics: one of its aims is to use geometry of infinite dimensional spaces to answer questions motivated by Langlands duality; another one is to exploit more classical, finite dimensional geometric objects and their quantization to study representations.The project will advance the representation theory of quantized symplectic resolutions by applying ideas of mirror symmetry and Langlands duality. Another goal is to further develop l-adic sheaf methods in harmonic analysis on p-adic groups. The investigator will also continue to explore D-module methods in representation theory, concentrating on applications of Hodge D-modules and generalizations of the derived localization theorem in positive characteristic.