CAREER: Representation Theory on Curves

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The geometric Langlands program proposes an extraordinary analog of harmonic analysis in the algebraic geometry of bundles on curves, inspired by the Langlands philosophy which binds harmonic analysis and Galois theory over number fields. In this geometric harmonic analysis, function spaces are replaced with categories of sheaves, and the spectral theory of operators on these sheaves is analyzed using geometric Fourier transforms. The investigator is developing applications of these cutting edge ideas to classical questions in representation theory. In particular, in collaboration with D. Nadler, he proposes an enhancement of the geometric Langlands program which performs a spectral decomposition of the categories of representations of real Lie groups. This significantly enhances the Langlands classification of irreducible representations (part of the classical Langlands program) and binds it to the geometric (Borel-Weil) realizations of representations. The investigator also spearheads efforts to make this material available to a far broader audience, in particular through the organization of GRASP (Geometry Representations and Some Physics), electronically distributed expository lecture series on the fundamental ideas and underlying currents in this rapidly developing area.A fundamental theme of modern mathematics is the exploitation of symmetry as an organizing principle, linking diverse and potentially baffling phenomena in an elegant overarching framework. Perhaps the prime example of this trend is the Langlands program, which identifies a general pattern in the appearance of symmetry in algebra and number theory, and counts among its successes the solution of Fermat's Last Theorem. In recent years, a new geometric setting for the applicationof the Langlands philosophy has emerged, which makes contact with new symmetry principles, in particular those underlying the exciting developments of string theory in physics. My CAREER proposal is aimed at advancing this geometric Langlands program in two ways. First, the GRASP (Geometry Representations And Some Physics) program will provide an electronic resource center for students interested in this exciting and varied but potentially intimidating and inaccessible area. At the center of GRASP is a series of expository lectures introducing the fundamental concepts underlying and relating to the Langlands program to a wide audience, via the web. In parallel, the research component of the proposal develops a novel program to apply the cutting edge geometric Langlands technology to more classical problems in algebra. In particular I expect these ideas to have a significant impact on the classification of symmetries arising in linear algebra with real numbers, a question with a distinguished history and origins in quantum mechanics.