Mathematical Sciences: Classical Complex Analysis

该奖项所支持的工作重点是以下方面出现的问题： 复变函数论的数学理论。 三个主要 研究主题将继续。 第一个是关于棱角的 保角映射的导数，即， 复平面是单叶的、解析的。 这是一个漫长 一个长期存在的问题是， 单位圆盘上的单连通区域具有角 衍生物 虽然许多特殊情况是已知的是真实的，只有 最近已经开发了极值长度技术， 这是一个普遍存在的结果现在是可能的点。 它有 有可能获得一个关于存在的确切声明， 这是迄今为止尚不为人知的带状域的情况。 还有更多的工作。 第二个工作领域将侧重于继续完善 共形映射的数值算法 快速可靠的程序 主要用于寻找实验性的 地图的属性 现在要做的工作是了解 收敛性质，并利用它们来研究经典日冕 三重连通域的问题最后， 为了确定插值Blaschke乘积类是否 上生成有界全纯函数的整个空间 磁盘. 1976年的研究表明，如果使用所有的Blaschke产品， 这个说法是正确的。插值积少得多 可能发生，但有更多的规律性，因此是一个 更好的类作为近似值。 复分析，研究可微函数的一种 复变量，是数学中广大领域的核心 从数论到位势论， 线性代数和数值分析。 这个特别的项目 将一些长期存在的问题和方法与新的 计算几何和图论的应用。

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.