Radon transforms: geometric combinatorics, regularity, and applications

该项目的目标是解决数学领域中一系列有趣的开放性问题，这些问题与几何积分变换（如Radon和X射线变换）和振荡积分有关。 对这些和相关对象有更强的理论理解是必要的，因为它们在科学、工程和技术创新中有许多应用。医学成像问题，包括CT、SPECT和NMR，以及雷达和声纳应用，都依赖于对Radon变换的深入理解。光声层析成像，散射理论，甚至运动检测算法也在很大程度上依赖于数学，将进一步发展作为该项目的一部分。此外，该项目的一个主要组成部分将涉及对数学教育的贡献和赠款机构研究生的培训。这些学生将把作为本项目的一部分开发的知识和技能与他们进入劳动力，以进一步推进最先进的国家。该项目有两个主要的技术目标。首先是开发几何和组合的方法，用于证明统一估计的一类线性和多线性算子推广Radon样变换，子水平集算子，振荡积分算子。 这些方法可以追溯到布尔甘、沃尔夫、基督和许多其他数学家的工作。这里要研究的问题和相关的推广都与现代数学分析中的一些最重要的猜想有关，包括Kakeya猜想、Bochner-Riesz猜想、限制猜想和Sogge的局部平滑猜想。 第二个主要目标是应用这些想法和结果的重要公开问题领域的偏微分方程。这些应用已经对玻尔兹曼方程和Gross-Pitaevskii族的研究做出了重大贡献，在本项目中，这些工作将继续进行并扩大到包括对Ablowitz-Kaup-Newell-Segur族散射理论和振荡Riemann-Hilbert问题的新的数学理解。这些问题中的每一个都可以从这个项目中开发的工具中受益匪浅，这些工具将用更强大的几何工具和想法取代传统的傅立叶方法。

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.