Complex Analysis, Potential Theory and Applications

该项目的智力重点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与最近解决的问题有关的问题，该问题涉及当来自远处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.Neumann证明了天体物理学家S.Rhie的一个猜想，即这个数字与n线性相关，而不是二次依赖于n。(这一结果大大减少了大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更现实的情况，当透镜效应是由一个气体质量分布等温的椭圆星系产生的。这个项目的另一个主题进一步发展了PI与Bell、Ebenfit和Shapiro共同获得的最新结果，以及PI的学生Lundberg最近获得的关于势论中边值问题解的代数性质的结果。该项目正在开发的主要新颖工具之一是将Arnold和Kolmogorov在20世纪50年代在解决关于函数叠加的第13个希尔伯特问题时率先提出的“闪电”概念扩展到复杂空间。复数闪电恰恰是阻碍二维复数空间中调和函数整体解析连续的障碍。该项目还解决了复杂分析和势能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强烈的跨学科色彩。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，是引力透镜。特别是，该项目解决了皮埃尔？S最近与天体物理学家法斯纳赫特和基顿的工作中出现的问题。PI正在继续普及他的研究的某些方面，并计划针对广泛的受众撰写几篇文章。PI过去曾举办过，并将继续根据项目的研究主题为本科生举办受欢迎的讲座。PI正在与当地一名高中生(M.Rabinovich)合作，他最近被选为2009年英特尔人才搜索大赛决赛入围者。国际和平研究所将继续努力传播他的研究成果，同时继续监督博士生。PI现在有两名学生：一名高级学生，一名二年级学生。一名初学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。国际和平协会继续与来自代表性不足群体的研究人员在项目的某些部分进行富有成效的合作。国际和平研究所还积极参与组织各种活动(会议、讲习班等)，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，以保持就该项目中的几个问题开展合作的强劲势头。

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.