Complexity of the objects of complex analysis and holomorphic mapping problems

复分析对象的复杂性与全纯映射问题

• 批准号: 0072197
• 负责人: Steven Bell
• 金额: $ 26.13万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072197&HistoricalAwards=false
• 关键词: Complexity; objects; complex; analysis; holomorphic

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.

摘要奖：DMS-0072197主要研究者：Steven R.贝尔教授贝尔已经证明，与平面中有限连通区域相关的伯格曼核、塞戈核和泊松核是与该域相关的几何结构相关的一个复变量的仅三个、有时甚至两个解析函数的初等组合。 贝尔将研究更深层次的问题复杂性潜在的理论所提出的他最近的发现，他将这些结果扩展到有限黎曼曲面。 贝尔还制定了一个独特的连续性质的非齐次柯西-黎曼方程，他已经表明，产量的行为信息holomorphicmapping域之间的复空间。 他将尝试验证重要的域类的属性，如严格pseudoconvex domains.势理论和conformalmapping的数学对象是无处不在的科学，数学和工程.他们携带编码在他们大量的信息aboutgeometric属性的区域在平面上. 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适用的新数学的集合。 贝尔教授将用简单得多的分析对象来表示与二维空穴区域相关的势能理论的经典对象。 这些结果将产生一个新的和实用的方法来理解解微分方程，保角映射和潜在的理论，应该感兴趣的科学家和工程师的许多经典问题。 贝尔将探索应用他的想法，以更复杂的结构在主题，他和他的学生将测试的有效性，数值方法源于工作。 由于人类通过一系列二维切片来最好地感知更高维度的物体，贝尔开发的工具可以找到许多应用。