Radon transforms: geometric combinatorics, regularity, and extensions

这个项目的目标是解决一系列有趣的数学分析中的开放问题，涉及几何积分变换（如Radon和X射线变换）和振荡积分。 一些目标包括确定某些多线性泛函（Holder-Brascamp-Lieb不等式的非线性类似物）在勒贝格空间的乘积上的有界性，以及理解勒贝格和勒贝格平方可积Sobolev空间上的平均算子（在标准和超定情况下）的正则性。 这些问题和相关的推广与现代数学中一些最重要的猜想有着深刻的联系，包括Kakeya猜想，Bochner-Riesz猜想，限制猜想和Sogge的局部平滑猜想。 这些问题的智力价值在这里要研究的是，他们的解决方案需要基本的新见解，并持有的承诺，是潜在的重要步骤的道路上的决议，其中一些深刻的缺陷。更广泛的影响，在这个项目的工作可能会感觉到整个医学成像：CT和SPECT扫描，NMR成像，雷达和声纳应用都依赖于对Radon变换的深刻理论和实践理解。光声层析成像，散射理论，甚至运动检测算法也依赖于Radon变换。 更令人兴奋和意想不到的影响也预计与玻尔兹曼方程-一个140岁的方程描述了稀释气体的动力学。 最近与R. M.应变揭示了统计力学的基本方程与几何和分析之间的深层联系。 这些联系以及由此产生的思想和结构将对数学家理解这个方程的方式产生深远的影响。

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.