Radon transforms: geometric combinatorics, regularity, and applications

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The goal of this project is to address a series of interesting open questions in the mathematical field of relating to what are known as geometric integral transforms (like the Radon and X-ray transforms) and oscillatory integrals. Stronger theoretical understanding of these and related objects is necessary, for they have many applications in science, engineering, and technological innovation. Imaging problems from medicine, including CT, SPECT, and NMR, as well as RADAR and SONAR applications, all depend on a deep understanding of the Radon transform. Optical-acoustic tomography, scattering theory, and even motion-detection algorithms also rely heavily on the mathematics that will be further developed as a part of this project. In addition, a major component of this project will involve contributions to mathematical education and training of graduate students at the grant institution. These students will take the knowledge and skills developed as a part of this project with them into the workforce to further advance the state of the art.The project has two major technical objectives. The first is to develop geometric and combinatorial methods useful for proving uniform estimates for a class of linear and multilinear operators generalizing Radon-like transforms, sublevel set operators, and oscillatory integral operators. The approaches to be developed trace their roots to work of Bourgain, Wolf, Christ, and many other mathematicians. The problems to be studied here and related generalizations are connected to some of the most important conjectures in modern mathematical analysis, including the Kakeya conjecture, the Bochner-Riesz conjecture, the Restriction conjecture, and Sogge's local smoothing conjecture. The second major objective is the application of these ideas and results to important open questions in the field of partial differential equations. Such applications have already made significant contributions to the study of the Boltzmann equation and the Gross-Pitaevskii hierarchy, and during this project, such work will be continued and expanded to include new mathematical understandings of the scattering theory of the Ablowitz-Kaup-Newell-Segur hierarchy and oscillatory Riemann-Hilbert problems. Each of these problems can benefit greatly from the tools to be developed in this project, which will replace traditional Fourier methods with more robust, geometric tools and ideas.