Radon transforms: geometric combinatorics, regularity, and extensions

这个项目的目标是解决数学分析中与几何积分变换(如Radon变换和X射线变换)和振荡积分有关的一系列有趣的公开问题。其中一些目标包括确定某些多线性泛函(Holder-BrasCamp-Lieb不等式的非线性模拟)在勒贝格空间乘积上的有界性，以及理解平均算子(在标准和超定情况下)在勒贝格和勒贝格平方可积Sobolev空间上的正则性。这些问题和相关的推广与现代数学中一些最重要的猜想密切相关，包括Kakeya猜想、Bochner-Riesz猜想、限制猜想和Sogge局部光滑猜想。这里要研究的这些问题的智力价值在于，它们的解决方案需要基本的新见解，并有望成为解决其中一些深刻猜想的潜在重要步骤。该项目工作的更广泛影响可能通过医学成像感受到：CT和SPECT扫描、核磁共振成像、雷达和声纳应用都依赖于对Radon变换的深入理论和实践理解。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。与玻尔兹曼方程有关的更令人兴奋和意想不到的影响也在预料之中。玻尔兹曼方程是一个有140年历史的描述稀薄气体动力学的方程。最近与R.M.应变的联合工作揭示了统计力学的这个基本方程与与该项目相关的几何和分析之间的深层次联系。这些联系，以及由此产生的新见解所产生的想法和构造，有望对数学家理解这个方程的方式产生深刻和持久的影响。

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.