Complexity of the objects of complex analysis and holomorphic mapping problems

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AbstractAward: DMS-0072197Principal Investigator: Steven R. BellProf. Bell has shown that the Bergman, Szego, and Poisson kernelsassociated to a finitely connected region in the plane areelementary combinations of only three, and sometimes even two,analytic functions of one complex variable related to geometricconstructions associated to the domain. Bell will study deeperquestions about complexity in potential theory posed by hisrecent findings and he will extend these results to finiteRiemann surfaces. Bell has also formulated a unique continuationproperty for the inhomogeneous Cauchy-Riemann equations that hehas shown yields information about the behavior of holomorphicmappings between domains in complex space. He will attempt toverify the property on important classes of domains such as thestrictly pseudoconvex domains.The mathematical objects of potential theory and conformalmapping are ubiquitous in Science, Mathematics, and Engineering.They carry encoded within them a vast amount of information aboutgeometric properties of regions in the plane. Although theseobjects are familiar and well studied, they continue to be asource of interesting and applicable new mathematics. ProfessorBell will express the classical objects of potential theoryassociated to a two dimensional region with holes in terms ofmuch simpler analytic objects. These results will give rise tonew and practical methods for understanding the solutions to manyclassical problems in differential equations, conformal mapping,and potential theory that should be of interest to scientists andengineers. Bell will explore applications of his ideas to morecomplicated constructions in the subject and he and his studentswill test the efficacy of the numerical methods stemming from thework. Because humans best perceive higher dimensional objects bytaking a series of two dimensional slices, the tools developed byBell could find many applications.