Geometry and Physics

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Proposal DMS-0305505Title: "Geometry and Physics"ABSTRACTThere are three individual researchers supported on this grant. ProfessorFreed and various coauthors study first twisted K-theory and loop groups,secondly differential K-theory and geometric invariants of Dirac operators,and finally anomalies in M-theory and string theory. Professor Hauselresearches new mathematics in quaternionic geometry which corresponds toknown mathematics in complex geometry, such as the moduli space of Higgsbundles, square integrable harmonic forms, integration on circle compactmanifolds and face vectors of hyperplane arrangements. Professor Uhlenbeckhas two projects on hyperbolic equations, one of finding fundamentalsolutions of covariant-derivative wave operators and one studying wave mapblow-up in two space dimensions. She also studies Virasoro actions in TCFT.The three supported researchers are part of a large geometry group whoseresearch projects span many topics in geometry which originate in high energytheoretical physics or engineering physics. In mathematics proper the workof the group is not limited to geometric analysis, but also involvesalgebraic geometry, geometric topology, algebraic topology, representationtheory, harmonic analysis, and mathematical physics. Essential to the groupresearch are collaborations among mathematicians with diverse backgrounds inmathematics, physics and biology.