Radon-like Transforms: Possible Applications to Partial Differential Equations and Inverse Problems

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ABSTRACT:In my thesis and in extensions I have made over the past year and a half, I have studied variants of Radon transforms known as singular Radontransforms, generalizing earlier work of Christ, Nagel, Stein and Wainger. In my proposed research, I intend to extend my methods to include a broaderclass of operators, which would include fractional and more traditional smooth Radon transforms, proving boundedness properties on Sobolev and L^pspaces. I also intend to research possible applications to partial differentialequations; there is a similarity between concepts that I use and results ofFefferman and Phong in subelliptic PDE theory that might lead to interestingresults in that direction. Further, I would like to generalize my work in moreapplied directions. For example, tomography, an important tool in medical imaging and related areas, requires the inversion of Radon transforms. Due to the close relation between PDE's and inverse problems, I hope to be able to apply my methods to help solve inverse problems such as those in tomography.In recent work, I have studied mathematical objects known as singular Radontransforms. They are variants of standard Radon transforms, which haveimportant applications in both mathematics as well as the sciences. Forexample, in medical imaging one frequently uses X-ray tomography to generatean image of an internal organ that one can not easily analyze through otherimaging techniques. X-ray tomography involves inverting a Radon transformoperator, in other words, the determination of an image from the Radontransform of this image. In my proposed research, in addition to extending myearlier results to strictly mathematical questions, I hope to be able to extend my research to help tackle more applied problems such as these. Ideas from disparate areas of mathematics and science might be needed.