Mathematical Sciences: Partial Differential Equations and Complex Analysis

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ABSTRACT Proposal: DMS-9623098) PI: Bell The areas of Differential Equations and Complex Analysis have a long history of fruitful interaction. Professor Bell's research has two goals. He will work on problems in complex analysis, such as geometric mapping problems, to which partial differential equations methods are applicable, and he will apply complex variable techniques to gain information about the partial differential equations problems and constructions of classical analysis. 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 useful new mathematics. Applied scientists frequently need to compute them. For example, to create an appropriate grid for numerical calculations of transonic flow past a wing, it is common to make two dimensional slices and to use conformal mappings on the slices to generate the grid. Bell has recently completed a program to express the classical objects of potential theory in terms of the Szego projection and kernel. He has discovered formulas that reveal that these objects are all elementary combinations of finitely many basic functions of one variable. His results give rise to new and practical methods for numerically computing solutions to classical problems in differential equations, conformal mapping, and potential theory which will be of interest to scientists and engineers. Professor Bell will explore applications of his ideas to other problems of this kind and he will attempt to extend his results to more complicated constructions in the subject. He and his students will test the efficacy of the numerical methods stemming from his work. It is interesting to note that the practical results of Bell's research in the plane can trace their origins back to s ome of his "curiosity driven" work in higher dimensions that would appear to be of theoretical, rather than of practical significance. Bell will continue to work on related questions in higher dimensional complex analysis.