Mathematical Sciences: Riemann Surfaces and Analytic Torsion

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John Fay will carry out research into recent developments relating theta functions and Riemann surfaces. This is an area which has flourished because of its connections with the Kadomtsev-Petviashvili hierarchy of differential equations and with string theory. It also has great intrinsic interest for mathematics since it will contribute towards a theory of higher rank vector bundles on Riemann surfaces. The research will center on Ray-Singer analytic torsion as defined on the moduli spaces of vector bundles on Riemann surfaces. This will include study of three aspects. The first concerns the holomorphic factorization and vanishing properties of torsion on the Jacobi variety. The second involves investigation of the explicit dependence of torsion on the moduli of the Riemann surface. The final aspect will be concerned with the properties of the generalized Szego kernel and theta- functions defined by torsion for rank-two bundles.