Mathematical Sciences: Classical Complex Analysis

该奖项支持的工作集中在复数函数理论的数学理论中出现的问题。将遵循三个主要研究主题。第一类是共形映射的角导数，即复平面上一元解析域的映射。确定单连通区域在单位圆盘上的保角映射是否有角导数是一个长期存在的问题。虽然已知许多特殊情况是正确的，但直到最近，极限长度技术才发展到现在可能得出一般存在结果的程度。对于迄今为止未知的条形域的存在性，已经有可能得到一个精确的表述。还有更多的工作要做。第二个工作领域将集中于持续改进保角映射的数值算法。开发了一个快速、可靠的程序，主要用于寻找地图的实验性质。现在要做的工作是理解收敛性质，并用它们来研究三连通域的经典电晕问题。最后，将努力确定一类插值Blaschke积是否生成磁盘上有界全纯函数的整个空间。1976年的研究表明，如果一个人使用了所有的Blaschke产品，这个说法是正确的。插值乘积发生的可能性要小得多，但有更多的规律性，因此是一个更好的类，可以用作近似值。复分析，即对复变量的可微函数的研究，是数论、势论、线性代数和数值分析等众多数学领域的核心。这个特殊的项目将一些长期存在的问题和方法与计算几何和图论的新应用相结合。

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.