Complexity in Complex Analysis

Purdue University, West Lafayette IN

Investigators

Abstract

ABSTRACT Complexity of the objects of complex analysis (DMS-0305958) Steven R. Bell Prof. Bell has shown that the most widely used kernel functions and metrics of complex analysis associated to certain finite Riemann surface are elementary combinations of only three, and sometimes even two, analytic functions of one complex variable related to special conformal mappings of the domain. Bell will study deeper questions about complexity in complex analysis and potential theory posed by his recent findings and he will extend his results in the plane to quadrature domains and general finite Riemann surfaces. Bell has formulated a unique continuation principle 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 apply this principle to some of the questions mentioned above and he will also use the property to try to gain a more geometric understanding of the Bergman projection in several complex variables. 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 surface with holes in terms of much simpler analytic objects. These results will give rise to new and practical methods for understanding and zipping the solutions to many problems in differential equations, conformal mapping, and potential theory that should be of interest, not only to mathematicians, but to scientists and engineers as well. Because humans best perceive higher dimensional objects by taking a series of two dimensional slices, the tools developed could find many applications.