Mathematical Sciences: Theta Functions and Moduli Spaces

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This project will focus on problems arising in the field of analytic torsion and the classical analytic torsion of Riemann surfaces. The work centers on classical moduli problems in theta-functions, analytic torsion and vector bundles on Riemann surfaces. Studies of tau-theta functions constructed from analytic torsion, from the loop-group approach using the Szego-kernel and from matrix singular integral operators using abelian Cauchy kernels will be carried out. The Szego-kernel and theta-functions will be developed for monodromy groups on punctured Riemann surfaces; this will help in the further understanding of isomonodromy deformations on higher-genus Riemann surfaces and the spectral curve (double-cover) approach to the moduli space of rank-two bundles. There will also be research on the spin-zero torsion divisor and the meromorphic extension of the classical Bergman and Szego kernel-functions to the moduli space of flat bundles with connections. This work is related with both classical mathematical function theory as well as deep questions arising in mathematical physics concerned with monodromy preserving deformations.