Mathematical Sciences: Classical Complex Analysis

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Work supported by this award focuses on problems arising in the mathematical theory of complex function theory. Three primary research themes will be followed. The first concerns angular derivatives of conformal maps, that is, mappings of domains in the complex plane which are univalent and analytic. It has been a long standing problem to determine whether or not a conformal mapping of a simply connected region onto the unit disk has an angular derivative. Although many special cases are known to be true, only recently has the techniques of extremal length been developed to the point where a general existence result is now possible. It has been possible to obtain a precise statement about the existence for the heretofore unknown case of strip domains. More work remains. A second area of work will focus on the continued refinement of numerical algorithms for conformal maps. A fast, reliable program has been developed and used primarily for finding experimental properties of maps. Work will now be done to understand convergence properties and use them to study the classical corona problem for triply-connected domains. Finally, efforts will be made to determine whether the class of interpolating Blaschke products generate the entire space of bounded holomorphic functions on the disk. It was shown in 1976 that if one uses all Blaschke products, the statement is true. The interpolating products are much less likely to occur, yet have more regularity and therefore are a better class to use as approximates. Complex analysis, the study of differentiable functions of a complex variable, lies at the heart of vast areas of mathematics stretching from number theory through potential theory and on to linear algebra and numerical analysis. This particular project combines some long-standing problems and methods with new applications of computational geometry and graph theory.