Complexity of the objects of complex analysis and holomorphic mapping problems

摘要奖：DMS-0072197首席研究员：史蒂文·R·贝尔教授。贝尔已经证明，与平面上有限连通区域相关的Bergman核、Szego核和Poisson核是只有三个，有时甚至是两个的一个复变量的解析函数的基本组合，这些函数与与该区域相关的几何结构有关。贝尔将研究他最近的发现提出的关于位势理论复杂性的深入问题，他将把这些结果推广到有限黎曼曲面。Bell还给出了非齐次Cauchy-Riemann方程的一个唯一的连续性质，它给出了关于复空间中区域间全纯映射行为的信息。他将尝试证明重要的区域类上的性质，如严格伪凸域。位势理论和构象映射的数学对象在科学、数学和工程中是普遍存在的。它们携带着大量关于平面区域的几何性质的信息。虽然这些对象是熟悉的和研究得很好的，但它们仍然是有趣和适用的新数学的来源。贝尔教授将用简单得多的解析对象来表示与带有空洞的二维区域相关的势论的经典对象。这些结果将提供一种新的、实用的方法来理解许多经典问题的解，这些问题应该引起科学家和工程师的兴趣。贝尔将探索他的想法在更复杂的结构中的应用，他和他的学生将测试从这项工作中产生的数值方法的有效性。由于人类通过获取一系列二维切片来感知更高维度的物体最好，贝尔开发的工具可以找到许多应用。

AbstractAward: DMS-0072197Principal Investigator: Steven R. BellProf. Bell has shown that the Bergman, Szego, and Poisson kernelsassociated to a finitely connected region in the plane areelementary combinations of only three, and sometimes even two,analytic functions of one complex variable related to geometricconstructions associated to the domain. Bell will study deeperquestions about complexity in potential theory posed by hisrecent findings and he will extend these results to finiteRiemann surfaces. Bell has also formulated a unique continuationproperty for the inhomogeneous Cauchy-Riemann equations that hehas shown yields information about the behavior of holomorphicmappings between domains in complex space. He will attempt toverify the property on important classes of domains such as thestrictly pseudoconvex domains.The mathematical objects of potential theory and conformalmapping are ubiquitous in Science, Mathematics, and Engineering.They carry encoded within them a vast amount of information aboutgeometric properties of regions in the plane. Although theseobjects are familiar and well studied, they continue to be asource of interesting and applicable new mathematics. ProfessorBell will express the classical objects of potential theoryassociated to a two dimensional region with holes in terms ofmuch simpler analytic objects. These results will give rise tonew and practical methods for understanding the solutions to manyclassical problems in differential equations, conformal mapping,and potential theory that should be of interest to scientists andengineers. Bell will explore applications of his ideas to morecomplicated constructions in the subject and he and his studentswill test the efficacy of the numerical methods stemming from thework. Because humans best perceive higher dimensional objects bytaking a series of two dimensional slices, the tools developed byBell could find many applications.