Mathematical Sciences: Partial Differential Equations and Complex Analysis

PI: Bell （DMS-9623098）微分方程和复分析领域有着悠久的相互作用的历史。贝尔教授的研究有两个目标。他将研究复杂分析中的问题，如偏微分方程方法适用的几何映射问题，他将应用复变量技术来获得关于偏微分方程问题和经典分析结构的信息。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。它们携带着大量关于平面区域几何特性的编码信息。虽然这些对象很熟悉，而且研究得很好，但它们仍然是有趣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，要为机翼跨音速流动的数值计算创建合适的网格，通常会制作二维切片，并在切片上使用保角映射来生成网格。贝尔最近完成了一个程序来表达经典对象的势理论在Szego投影和核。他发现了一些公式，这些公式揭示了这些物体都是一个变量的有限多个基本函数的初等组合。他的结果为微分方程、保角映射和势理论等经典问题的数值计算提供了新的实用方法，这将引起科学家和工程师的兴趣。贝尔教授将探索将他的思想应用于其他这类问题，并尝试将他的成果扩展到这门学科中更复杂的结构。他和他的学生将测试他的工作所产生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在更高维度上的一些“好奇心驱使”的工作，这些工作似乎是理论的，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.