Mathematical Sciences: Partial Differential Equations and Complex Analysis

摘要提案：DMS-9623098）PI：Bell 微分方程和复分析领域有着悠久的历史，富有成效的互动。 贝尔教授的研究有两个目标。他将致力于复杂的分析问题，如几何映射问题，其中偏微分方程方法是适用的，他将运用复变量技术，以获得有关偏微分方程问题和建设的经典分析的信息。 势理论和保角映射的数学对象在科学、数学和工程中无处不在。 它们携带着大量的关于平面区域几何特性的编码信息。 虽然这些对象是熟悉的和良好的研究，他们仍然是一个有趣的和有用的新数学的来源。 应用科学家经常需要计算它们。 例如，为数值计算机翼的跨音速绕流而建立适当的网格，通常作二维切片，并在 切片来生成网格。贝尔最近完成了一个程序来表达的经典对象的潜在的理论方面的Szego投影和内核。 他发现的公式表明，这些对象都是一个变量的许多基本功能的初等组合。 他的结果产生了新的和实用的方法，数值计算解决经典问题， 微分方程，保角映射，和潜在的理论，这将感兴趣的科学家和工程师。 贝尔教授将探讨应用他的想法，以其他问题的这类，他将试图扩大他的结果，以更复杂的结构的主题。 他和他的学生将测试源自他的工作的数值方法的有效性。 有趣的是，贝尔在飞机上研究的实际结果可以追溯到他的一些“好奇心” 在更高的维度上，“驱动”的工作似乎是理论上的，而不是实际意义。 贝尔将继续研究高维复分析中的相关问题。

ABSTRACT Proposal: DMS-9623098) PI: Bell The areas of Differential Equations and Complex Analysis have a long history of fruitful interaction. Professor Bell's research has two goals. He will work on problems in complex analysis, such as geometric mapping problems, to which partial differential equations methods are applicable, and he will apply complex variable techniques to gain information about the partial differential equations problems and constructions of classical analysis. The mathematical objects of potential theory and conformal mapping are ubiquitous in Science, Mathematics, and Engineering. They carry encoded within them a vast amount of information about geometric properties of regions in the plane. Although these objects are familiar and well studied, they continue to be a source of interesting and useful new mathematics. Applied scientists frequently need to compute them. For example, to create an appropriate grid for numerical calculations of transonic flow past a wing, it is common to make two dimensional slices and to use conformal mappings on the slices to generate the grid. Bell has recently completed a program to express the classical objects of potential theory in terms of the Szego projection and kernel. He has discovered formulas that reveal that these objects are all elementary combinations of finitely many basic functions of one variable. His results give rise to new and practical methods for numerically computing solutions to classical problems in differential equations, conformal mapping, and potential theory which will be of interest to scientists and engineers. Professor Bell will explore applications of his ideas to other problems of this kind and he will attempt to extend his results to more complicated constructions in the subject. He and his students will test the efficacy of the numerical methods stemming from his work. It is interesting to note that the practical results of Bell's research in the plane can trace their origins back to s ome of his "curiosity driven" work in higher dimensions that would appear to be of theoretical, rather than of practical significance. Bell will continue to work on related questions in higher dimensional complex analysis.