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 geometric integral transforms (like the Radon and X-ray transforms) and oscillatory integrals. Some of the goals include determining the boundedness of certain multilinear functionals (nonlinear analogues of the Holder-Brascamp-Lieb inequalities) on products of Lebesgue spaces and understanding of the regularity of averaging operators (in both the standard and overdetermined cases) on Lebesgue and Lebesgue square integrable Sobolev spaces. These problems and related generalizations are deeply connected to some of the most important conjectures in modern mathematics, including the Kakeya conjecture, the Bochner-Riesz conjecture, the Restriction conjecture, and Sogge's local smoothing conjecture. The intellectual merit of these problems to be studied here is that their solutions require fundamental new insights and hold the promise of being potentially significant steps on the road to the resolution of some of these deep conjectures.The broader impacts of the work in this project may be felt throughout medical imaging: 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. More exciting and unexpected impacts are also anticipated in connection with the Boltzmann equation - a 140-year-old equation describing the dynamics of a dilute gas. Recent joint work with R. M. Strain has uncovered deep connections between this fundamental equation of statistical mechanics and the geometry and analysis connected to this project. These connections and the ideas and constructions arising from this new insight promise to have a deep and lasting impact on the way that this equation is understood by mathematicians.