Mathematical Sciences: Problems in Function Theory

该奖项支持的研究集中在一元复变函数和一维薛定谔算子的数学理论中出现的问题。这项工作继续在四个问题领域进行调查。在第一章中，我们将研究Blaschke积的密度，这些积的根被限制为有界解析函数代数的插值序列。期望所有Blaschke积都是受限积的一致极限，并且这些Blaschke积的闭合凸包是代数的单位球。第二条研究线涉及分析能力，以帮助理解具有正Hausdorff维但容量为零的集合的性质。我们还将讨论在单连通域中，保角映射的导数沿直线可积的幂。有关的研究在对面积进行积分时也将考虑同样的问题。对一维周期薛定谔算子的间隙序列（带）的理解将用于具有复杂势的算子。最近的研究表明，光谱可以用黎曼曲面来实现。这项工作的目的是获得光谱的更内在的描述。复变函数理论是研究复变函数的解析性质和几何性质的学科。随着时间的推移，该学科已经发展到包括诸如拟共形映射、拟正则映射和势理论等辅助研究线。在该领域的工作是由几何问题，微分方程，流体动力学和控制理论的动机。最近的工作还包括通过圆填充的新应用来研究计算保角映射。＊＊＊

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. ***