Mathematical Sciences: Problems in Complex Analysis

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Robert Gunning will continue his research on theta functions and Riemann surfaces. The goal is the complete analysis of linear relations between partial derivatives of Riemannian second order theta functions at the origin, extending the known relations that provide explicit solutions of the Korteweg-deVries and Kadomtsev- Petviashvili equations, and the investigation of the role of these relations in the Schottky problem of characterizing Jacobi varieties. Fornaess will carry out research on function-theoretic properties of smoothly bounded domains in the space of several complex variables. The goals are to obtain estimates for solutions of certain differential equations in various classes of domains, extending results that are known for strongly pseudoconvex domains, and to study a notion of uniform extendability of domains that seems critical to the understanding of the asymptotic behaviour of the Bergman kernel function at the boundary of a domain.