Complexity in Complex Analysis

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ABSTRACT Complexity of the objects of complex analysis (DMS-0305958) Steven R. BellProf. Bell has shown that the most widely used kernelfunctions and metrics of complex analysis associated tocertain finite Riemann surface are elementary combinationsof only three, and sometimes even two, analytic functions ofone complex variable related to special conformal mappingsof the domain. Bell will study deeper questions aboutcomplexity in complex analysis and potential theory posed byhis recent findings and he will extend his results in theplane to quadrature domains and general finite Riemannsurfaces. Bell has formulated a unique continuationprinciple for the inhomogeneous Cauchy-Riemann equationsthat he has shown yields information about the behavior ofholomorphic mappings between domains in complex space. Hewill apply this principle to some of the questions mentionedabove and he will also use the property to try to gain amore geometric understanding of the Bergman projection inseveral complex variables.The mathematical objects of potential theory and conformalmapping are ubiquitous in Science, Mathematics, andEngineering. They carry encoded within them a vast amountof information about geometric properties of regions in theplane. Although these objects are familiar and wellstudied, they continue to be a source of interesting andapplicable new mathematics. Professor Bell will express theclassical objects of potential theory associated to a twodimensional surface with holes in terms of much simpleranalytic objects. These results will give rise to new andpractical methods for understanding and zipping the solutionsto many problems in differential equations, conformal mapping,and potential theory that should be of interest, not only tomathematicians, but to scientists and engineers as well.Because humans best perceive higher dimensional objects bytaking a series of two dimensional slices, the toolsdeveloped could find many applications.