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.

该项目的智力主旨集中于与最近通过复杂分析方法解决天体物理学问题相关的问题，该问题涉及当来自远处物体的光在到达物体之前被 n 个共面质量偏转时，人们可以观察到的图像的最大数量。观察者。 PI 和 G. Neumann 证明了天体物理学家 S. Rhie 的猜想，即这个数字与线性相关，而不是二次相关；在 n. （这个结果大大减少了大 n 的相关计算数量。）PI 与他的学生 E. Lundberg 一起，计划将这些想法扩展到更现实的情况，当透镜效应是由具有等温线的椭圆星系产生时气体的质量分布。该项目的另一个主题进一步发展了 PI 与 Bell、Ebenfelt 和 Shapiro 联合​​获得的最新成果，以及最近由 PI 的学生 Lundberg 获得的成果，处理势论中边值问题解的代数性质。该项目中开发的主要新颖工具之一是将阿诺德和柯尔莫哥洛夫在 20 世纪 50 年代解决函数叠加的第 13 希尔伯特问题时首创的“闪电”概念扩展到复杂空间。事实证明，复杂闪电恰恰是二维复空间中调和函数全局解析连续性的障碍。该项目还解决了复杂分析和势理论中其他几个长期存在的基本问题。该项目的很大一部分具有浓厚的跨学科气息。这项研究继续对天体物理学，更准确地说，引力透镜中的一些问题进行更深入的研究。特别是，该项目解决了 PI 最近与天体物理学家 Fassnacht 和 Keeton 合作中出现的问题。 PI 正在继续普及他的研究的某些方面，并计划发表几篇针对广大受众的文章。 PI过去已经做过并将继续根据项目的研究主题为本科生进行热门讲座。 PI 正在与当地一名高中生 (M. Rabinovich) 合作，他最近被选为 2009 年英特尔人才搜寻大赛决赛入围者。 PI将继续努力传播他的研究成果，同时继续指导博士生。 PI 现在有两名学生：一名是高年级学生，一名是二年级学生。一名新晋女研究生最近表示有兴趣与 PI 就该项目的某些主题进行合作。 PI 继续与来自代表性不足群体的研究人员在该项目的某些部分进行富有成效的合作。 PI还积极参与组织多个活动（会议、研讨会等），讨论提案中讨论的研究主题，并将数学家和物理学家聚集在一起，以保持在多个问题上合作的强劲势头。这个项目。