Radon transforms: geometric combinatorics, regularity, and applications

这个项目的目标是解决数学领域中的一系列有趣的公开问题，这些问题与所谓的几何积分变换(如Radon变换和X射线变换)和振荡积分有关。对这些对象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。医学上的成像问题，包括CT、SPECT和核磁共振，以及雷达和声纳的应用，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，这些数学将作为该项目的一部分进一步发展。此外，该项目的一个主要组成部分将涉及对资助机构的数学教育和研究生培训的贡献。这些学生将把作为这个项目的一部分开发的知识和技能带入劳动力市场，以进一步推动艺术的发展。该项目有两个主要的技术目标。第一个是发展几何和组合方法，用于证明一类线性和多线性算子的一致估计，这些算子推广了Radon-like变换、次水平集算子和振荡积分算子。要开发的方法可以追溯到布尔格纳、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局部光滑猜想。第二个主要目标是将这些想法和结果应用于偏微分方程领域中的重要开放问题。这些应用已经对Boltzmann方程和Gross-Pitaevskii方程的研究做出了重大贡献，在这个项目中，这些工作将继续并扩展到包括对Ablowitz-Kaup-Newell-Segur方程的散射理论和振荡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，它将用更健壮的几何工具和思想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.