Complexity of the objects of complex analysis and holomorphic mapping problems

Purdue Research Foundation, West Lafayette IN

Investigators

Abstract

Abstract Award: DMS-0072197 Principal Investigator: Steven R. Bell Prof. Bell has shown that the Bergman, Szego, and Poisson kernels associated to a finitely connected region in the plane are elementary combinations of only three, and sometimes even two, analytic functions of one complex variable related to geometric constructions associated to the domain. Bell will study deeper questions about complexity in potential theory posed by his recent findings and he will extend these results to finite Riemann surfaces. Bell has also formulated a unique continuation property for the inhomogeneous Cauchy-Riemann equations that he has shown yields information about the behavior of holomorphic mappings between domains in complex space. He will attempt to verify the property on important classes of domains such as the strictly pseudoconvex domains. The mathematical objects of potential theory and conformal mapping are ubiquitous in Science, Mathematics, and Engineering. They carry encoded within them a vast amount of information about geometric properties of regions in the plane. Although these objects are familiar and well studied, they continue to be a source of interesting and applicable new mathematics. Professor Bell will express the classical objects of potential theory associated to a two dimensional region with holes in terms of much simpler analytic objects. These results will give rise to new and practical methods for understanding the solutions to many classical problems in differential equations, conformal mapping, and potential theory that should be of interest to scientists and engineers. Bell will explore applications of his ideas to more complicated constructions in the subject and he and his students will test the efficacy of the numerical methods stemming from the work. Because humans best perceive higher dimensional objects by taking a series of two dimensional slices, the tools developed by Bell could find many applications.