Mathematical Sciences: Polynomially Convex Hulls and their Applications

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9412392 Slodkowski The problems in this proposal are concerned with polynomial convex hulls and their applications. The proposer intends to pursue his long term interest in polynomial hulls fibered over the unit circle. he will concentrate especially on analyzing the structure of polynomial hulls with non-connected fibers and finding their new applications. He also plans to study weak solutions of Levi equations in two or more dimensions and their applications to description of polynomial hulls and envelopes of holomorphy of submanifolds of n-dimensional complex Euclidean space. The role of mathematical analysis is to provide qualitative, quantitative and geometric information concerning the solutions of equations. When the equations are defined over domains in several complex variables the approximation of functions and solutions is complicated by the geometry of the domain. This project studies this interplay between the complex geometry and approximation. In particular, the project will have application to the location of extreme values of solutions to complex equations. ***