Mathematical Sciences: Research in Geometric Analysis

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9626427 Harvey This project lies in the interface of differential geometry and algebraic geometry. The investigator plans to further pursue the theory of characteristic currents, calibrations and their various applications. Of particular note is the research on the differentiable version of the Riemann-Roch theorem. Calibrations on a curved space give infinitesimal versions of volume-minimizing subspaces. There are certain 'exceptional' calibrations that are closely related to continuous symmetry groups. The Riemann-Roch theorem is an important theorem in algebraic geometry, but its applicability is restricted to the so-called holomorphic category. One of the stated goals of this project is to extend the Riemann-Roch theorem to a more general setting requiring only the differentiability assumption.