Mathematical Sciences: Complex Analysis: Computation and Control.

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This project will focus on four areas of mathematical analysis. Work will be done on estimating two-dimensional harmonic measure which is used in potential theory and in measuring growth of analytic functions. At the present time estimates are known for the harmonic measure of curves in a disc meeting all radii in some sector. The width of the maximal sector remains to be determined. Theoretical and numerical efforts will be combined in this process. Work will continue on problems of optimization using bounded holomorphic functions. These investigations have been underway for several years. They arise from certain engineering problems in optimal control and involve finding best approximations to given functions by bounded or continuous functions defined on a circle which have analytic extensions to the interior. A third, and more recent, goal of this project will involve methods for the computation of conformal maps. There are many techniques extant for such computations and mappings of domains with smooth boundaries that generally have numerical approximants which converge rapidly. The present work is concerned with approximation involving maps of domains with corners. Recent progress has been encouraging although repetitions of the approximation process shows degradation after several passes. Efforts will be made to single out the best boundary curves for which current methods are practical. It is likely that these curves will fall into the class known as chord-arc curves which have restricted bending. Finally, in a more theoretical vein, studies will be made into the existence and nature of inner functions which are constant on Gleason parts of the maximal ideal space of a disc. While general theory indicates existence of such inner functions there is little known as to whether there are Blaschke products in this class. Related to this work is a characterization of the zero-distribution of such products. In addition to applications to potential theory, this work adds to fundamental knowledge regarding numerical approximation schemes for partial differential equations.