Mathematical Sciences: Differential Geometry and Riemann Surfaces

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Two senior investigators and two postdocs will study various aspects of differential geometry and Riemann surfaces. One will establish the finiteness of string theories to all orders of perturbation theory. This will require the development of holomorphic supergeometry and new parametrizations of Teichmuller space suggested by Feynman diagrams. The second investigator will work on problems in complex differential geometry. This will include a study of Hermitian symmetric domains, holomorphic mappings between them, and their classification. One postdoc will analyze homomorphic immersions between compact complex hyperbolic space forms. The second will derive further estimates for eigenvalues of manifolds with conelike singularities. Four investigators will study various aspects of differential geometry. Several of the problems to be studied have their origins in string theory. This theory has been used to describe complex interactions between elementary particles. In this theory, each particle is represented by a string which vibrates in ten dimensional space. These investigations are part of a continuing effort by mathematicians and physicists to use this new theory to explain the results of high-energy particle experiments.