Complex Analysis, Potential Theory and Applications

复分析、势理论及应用

• 批准号: 0855597
• 负责人: Dmitry Khavinson
• 金额: $ 12.23万
• 依托单位: University of South Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0855597&HistoricalAwards=false
• 关键词: Complex; Analysis; Potential; Theory; Applications

The intellectual thrust of the projectfocuses on problems related to a recent solution by methods of complex analysis of a problem in astrophysics concerning the maximal number of images one may observe when a light from a distant object is deflected by n co-planar masses before reaching the observer. The PI and G. Neumann proved a conjecture by the astrophysicist S. Rhie that this number depends linearly, rather than quadratically; on n. (This result reduces substantially the number of relevant calculations for large n.) The PI, jointly with his student E. Lundberg, is planning to extend these ideas to a more realistic situation when the lensing effect is produced by an elliptical galaxy with an isothermal mass distribution of gas. Another main theme of this project develops further recent results of the PI obtained jointly with Bell, Ebenfelt and Shapiro, and, more recently, by the PI's student Lundberg, dealing with algebraic properties of solutions of the boundary value problems in potential theory. One of the main novel tools being developed in the project is the extension to the complex space of a notion of a "lightning bolt" pioneered by Arnold and Kolmogorov in the 1950s in their solution of the 13th Hilbert problem on superpositions of functions. Complex lightning bolts turn out to be precisely the obstacles preventing global analytic continuation of harmonic functions in two-dimensional complex space. The project also addresses several other fundamental long standing questions in complex analysis and potential theory. A large part of the project has a strong interdisciplinary flavor. This research continues a deeper study of some problems in astrophysics, more precisely, in gravitational lensing. In particular, the project deals with the problems that have arisen from the PI?s recent work with the astrophysicists Fassnacht and Keeton. The PI is continuing popularization of some aspects of his research and is planning several articles directed at a wide audience. The PI has given in the past and will continue to give popular lectures for undergraduates based on the research topics of the project. The PI is working with a local high school student (M. Rabinovich), who was recently selected as a 2009 Intel Talent Search Competition finalist. The PI will be continuing his efforts in dissemination of his research and, at the same time, continuing to supervise Ph D students. The PI now has two students: one advanced and one in his second year. A beginning female graduate student has recently expressed interest in working with the PI on some of the topics in the project. The PI continues his fruitful collaborations on some parts of the project with researchers from underrepresented groups. The PI is also actively involved at various levels in organizing multiple events (conferences, workshops, etc.), on the research topics discussed in the proposal and in bringing together mathematicians and physicists in order to uphold the strong momentum of collaboration on several problems in this project.

该项目的智力推力集中在与最近的解决方案有关的问题上，该解决方案是通过对天体物理学中的一个问题进行复杂分析的方法来解决的，该问题涉及当来自遥远物体的光在到达观察者之前被n个共面质量偏转时可以观察到的最大图像数量。PI和G。诺伊曼证明了天体物理学家S.这个数与n成线性关系，而不是二次关系。(This结果大大减少了对于大N的相关计算的数量）。PI和他的学生E.伦德伯格计划将这些想法扩展到更现实的情况，即具有等温气体质量分布的椭圆星系产生透镜效应。这个项目的另一个主题是进一步发展最近的结果PI共同获得贝尔，Ebenfelt和夏皮罗，并在最近，由PI的学生伦德伯格，处理代数性质的解决方案的边界值问题的潜在理论。该项目开发的主要新工具之一是将阿诺德和科尔莫戈罗夫在20世纪50年代解决第13希尔伯特问题时开创的“闪电”概念扩展到复杂空间。复杂的闪电恰恰是阻碍二维复空间中调和函数整体解析延拓的障碍。该项目还解决了复杂分析和潜在理论中的其他几个长期存在的基本问题。该项目的很大一部分具有强烈的跨学科色彩。这项研究继续深入研究天体物理学中的一些问题，更准确地说，在引力透镜。特别是，该项目涉及的问题，已出现的PI？他最近与天体物理学家法斯纳赫特和基顿一起工作。PI正在继续推广他的研究的某些方面，并计划针对广泛的受众发表几篇文章。PI过去已经并将继续根据该项目的研究主题为本科生提供受欢迎的讲座。PI正在与当地的一名高中生（M。Rabinovich），他最近被选为2009年英特尔人才搜索竞赛决赛选手。首席研究员将继续努力传播他的研究成果，同时继续监督博士生。PI现在有两个学生：一个先进的，一个在他的第二年。一名刚开始的女研究生最近表示有兴趣与PI就该项目中的一些主题进行合作。PI继续在该项目的某些部分与来自代表性不足群体的研究人员进行富有成效的合作。PI还在各级积极参与组织多种活动（会议、讲习班等），在提案中讨论的研究主题上，并将数学家和物理学家聚集在一起，以保持在该项目中几个问题上合作的强劲势头。