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.
The stated goal of this project was to address a series of interesting open problems in mathematical analysis relating to what are known as geometric integral transforms (like the Radon and X-ray transforms) and oscillatory integrals. As expected, the project was extremely successful, and several deep new theoretical insights into these fundamentally important objects were made. Several of these insights were quite suprising to the community of mathematicians specializing in these areas, highlighting ways in which our earlier understandings fall short and will not be able to keep pace with further developments and applications. The project was also able to demonstrate critically-important continuity properties and asymptotic behaviors in a number of important situations which were, until now, poorly understood. Some of the more tangible products of the project include: * Eight articles were published or submitted for publication in high quality, peer-reviewed mathematical journals. * Various aspects of this project were presented at over 13 mathematical conferences and workshops, including conferences, seminars, and courses at the international level in Mexico, Spain, Switzerland, and the United Kingdom. * Significant contributions to American mathematical workforce training were made through the supervision of two undergraduate honors projects, three Master's degree projects, one (ongoing) PhD student, and joint supervision of two postdoctoral mathematicians. While the primary goals of the project were highly theoretical and technical in nature, there is significant hope that ongoing work in this area will yield fruit in applied fields relating to medical imaging (CT and SPECT scans, NMR imaging), RADAR, SONAR, optical-acoustic tomography, scattering theory, and even motion-detection. Work on the Boltzmann equation that was completed as a part of this project was recently selected as a focus of attention by a team of applied mathematicans related to fluid and gas dynamics as well as aeronautics. The hope is that this work will lead to breakthroughs in next-generation computer aerodynamics simulations.
