Complex Analysis, Potential Theory and Applications

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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.