Radon transforms: geometric combinatorics, regularity, and extensions

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The goal of this project is to address a series of interesting open problems in mathematical analysis relating to the Radon transform and its generalizations. These problems include determining the boundedness of certain multilinear functionals (nonlinear analogues of the Holder-Brascamp-Lieb inequalities) on products of Lebesgue spaces, as well as the understanding of the regularity of averaging operators (in both the standard and overdetermined cases) on Lebesgue and Lebesgue square integrable Sobolev spaces.Radon transforms and their generalizations are intimately connected to some of the greatest outstanding problems in modern analysis, including the Kakeya conjecture, the Bochner-Riesz conjecture, the Restriction conjecture, and Sogge?s local smoothing conjecture. The intellectual merit of the particular problems to be studied in this project is that their solutions require significant new theoretical insight, and they are potentially significant steps on the road to solution of some of these broader outstanding problems.A better understanding the Radon transform and its generalizations also may have broader impacts on other fields within the scientific community. Medical imaging, including CT and SPECT scans, NMR imaging, RADAR, and SONAR applications all depend on a deep theoretical and practical understanding of the Radon transform. Optical-acoustic tomography, scattering theory, and even motion-detection algorithms also depend on the Radon transform. All of these fields and more could potentially benefit from insights produced by this project.