Complexity of the objects of complex analysis and holomorphic mapping problems

项目编号：dms -0072197Bell已经证明，与平面上有限连通区域相关的Bergman、Szego和Poisson核是与该区域相关的几何结构相关的一个复变量的解析函数的三个（有时甚至两个）初等组合。贝尔将研究他最近的发现所提出的关于势理论复杂性的更深层次的问题，他将把这些结果扩展到有限黎曼曲面。Bell还为非齐次Cauchy-Riemann方程制定了一个独特的连续性质，他已经展示了关于复空间中域之间全纯映射行为的信息。他将尝试在一些重要的领域（如严格伪凸领域）上验证其性质。势理论和共形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。虽然这些对象是熟悉的，并且研究得很好，但它们仍然是有趣和适用的新数学的来源。贝尔教授将用更简单的解析对象来表达与有洞的二维区域有关的势理论的经典对象。这些结果将产生新的和实用的方法来理解许多经典问题的解在微分方程，保角映射，和势能理论，应该是科学家和工程师感兴趣的。贝尔将探索将他的想法应用于该学科中更复杂的结构，他和他的学生将测试由此产生的数值方法的有效性。由于人类最好地感知高维物体是通过拍摄一系列二维切片，因此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.