Complex Analysis, Potential Theory and Applications

该项目的智力重点是与天体物理学中一个问题的复杂分析方法最近的解决方案有关的问题，该问题涉及当来自遥远物体的光在到达观测者之前被n个共面质量偏转时，可以观察到的最大图像数。PI和G.诺伊曼证明了天体物理学家S. rihie的一个猜想，即这个数字是线性的，而不是二次的；（这个结果大大减少了大n的相关计算次数。）PI和他的学生E. Lundberg正计划将这些想法扩展到一个更现实的情况，即透镜效应是由具有等温气体质量分布的椭圆星系产生的。该项目的另一个主要主题是进一步发展PI最近与Bell， Ebenfelt和Shapiro共同获得的结果，以及最近由PI的学生Lundberg获得的结果，处理位势理论中边值问题解的代数性质。该项目开发的主要新工具之一是将“闪电”的概念扩展到复杂空间，该概念是由Arnold和Kolmogorov在20世纪50年代解决关于函数叠加的第13希尔伯特问题时提出的。复杂闪电恰恰是二维复空间中谐波函数全局解析延拓的障碍。该项目还解决了复杂分析和潜在理论中其他几个长期存在的基本问题。这个项目的很大一部分都有很强的跨学科的味道。这项研究继续对天体物理学中的一些问题进行更深入的研究，更准确地说，是对引力透镜的研究。特别是，该项目处理了PI？他最近与天体物理学家Fassnacht和Keeton合作。PI正在继续普及他研究的某些方面，并计划针对广大读者发表几篇文章。PI过去已经并将继续根据项目的研究主题为本科生提供受欢迎的讲座。PI正在与一名当地高中生（M. Rabinovich）合作，他最近被选为2009年英特尔人才搜索比赛的决赛选手。PI将继续努力传播他的研究成果，同时继续指导博士生。PI现在有两个学生：一个是高年级学生，一个是二年级学生。一位刚毕业的女研究生最近表示有兴趣与PI一起研究项目中的一些主题。PI继续在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还积极参与组织各种级别的活动（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，以保持在该项目中几个问题上合作的强劲势头。

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.