Mathematical Sciences: Problems in Function Theory

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Garnett 9401269 The research supported by this award focuses on problems arising in the mathematical theory of functions of one complex variable and one-dimensional Schrodinger operators. The work continues investigations in four problem areas. In the first, work will be done investigating the density of Blaschke products whose roots are restricted to interpolating sequences for the algebra of bounded analytic functions. It is expected that all Blaschke products are uniform limits of the restricted ones and that the closed convex hull of these Blaschke products is the unit ball for the algebra. A second line of investigation concerns analytic capacity to help understand the nature of sets having positive Hausdorff dimension but which have zero capacity. Work will also be done on determining for which powers the derivative of a conformal mapping is integrable along straight lines in simply connected domains. Related investigations will consider the same question when the integrals are taken with respect to area measure. Efforts to understand the gap sequences (bands) in one-dimensional periodic Schrodinger operators will be taken up for operators with complex potentials. Recent work has shown that the spectrum can be realized in terms of Riemann surfaces. The purpose of this work is to obtain a more intrinsic description of the spectrum. Complex function theory is the study of the analytic and geometric properties of differentiable functions of a complex variable. The subject has grown in time to include ancillary lines of investigation such as quasiconformal mapping, quasiregular mapping and potential theory. Work in the field is motivated by problems in geometry, differential equations, fluid dynamics and control theory. Recent work also involves studies of computational conformal mapping through new applications of circle packing. ***