Geometric Harmonic Analysis: Advances in Radon-like Transforms and Related Topics

几何平均的数学被称为类氡算子，在与成像和数据分析相关的许多技术应用中具有重要意义：CT、SPECT和NMR，以及雷达和声纳应用，都依赖于对氡变换的深刻理解，相关思想出现在光学声层析成像、散射理论，甚至一些运动检测算法中。有些令人惊讶的是，尽管这个领域已经取得了许多令人难以置信的成功，但在这个数学领域仍有许多基本的理论问题没有得到解决。该项目研究几何平均领域的一系列问题，例如，对应于量化成像对象的微小变化与测量数据的预期变化之间的关系（在实践中，将对其进行计算处理以恢复原始对象的近似图像）。在这类问题中，理论上的挑战是精确地量化变化的概念，并在输入和输出变化的幅度之间建立本质上精确的关系。由于PI最近在了解这些天体方面的工作取得了进展，该项目已经做好了产生重要结果的准备。实现这个项目的主要目标将导致数学的一些相关领域的进步，并可能影响未来的成像技术。此外，该项目还为本科生和博士生的高级数学培训提供了独特的机会，他们可以在就业后将这些技能转移到其他关键领域。PI研究数学分析中的主题，涉及到类氡变换、振荡积分和傅立叶限制问题的新几何方法的发展。这项工作包括子水平集和振荡积分问题的各种特殊情况。值得一提的主要特殊情况包括多参数子水平集估计、中维类Radon变换的最大曲率、低余维简并Radon变换、傅里叶限制及相关的广义行列式泛函、卷积及相关类型的多线性振荡积分。PI的方法涉及到过去5年中开发的各种新工具，这些工具结合了几何不变量理论、几何度量理论、解耦理论和其他领域的技术。在这些新工具中，PI的最新结果提供了一种全新的方法来估计Radon-Brascamp-Lieb不等式在几何量方面的规范，可以理解为类似于Brascamp-Lieb常数的Lieb公式。本项目的一个主要目标是理解在Radon-Brascamp-Lieb条件下隐式控制非局部积分有限的局部几何准则。该项目在谐波分析、几何测量理论和入射几何的交叉领域有许多潜在的应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The mathematics of geometric averages known as Radon-like operators is of fundamental importance in a host of technological applications related to imaging and data analysis: CT, SPECT, and NMR, as well as RADAR and SONAR applications, all depend on a deep understanding of the Radon transform, and related ideas appear in optical-acoustic tomography, scattering theory, and even some motion-detection algorithms. Somewhat surprisingly, there are many basic theoretical problems in this area of mathematics which remain unsolved despite the many incredible successes the field has already achieved. This project studies a family of questions in the area of geometric averages which, for example, correspond to quantifying the relationship between small changes in the imaged objects and the expected changes in measured data (which in practice would be processed computationally to recover an approximate picture of the original object). The theoretical challenge in a problem such as this is to precisely quantify the notion of change and to establish essentially exact relationships between the magnitude of input and output changes. Thanks to recent advances in the PI's work to understand these objects, the project is well-positioned to yield important results. Achieving the main goals of this project would lead to advances in a number of related areas of mathematics and may influence future imaging technologies. The project furthermore provides unique opportunities for the advanced mathematical training of both undergraduate and PhD students, who can transfer these skills to other areas of critical need once in the workforce.The PI studies topics in mathematical analysis related to the development of new geometric approaches to Radon-like transforms, oscillatory integrals, and Fourier restriction problems. This work includes various special cases of both sublevel set and oscillatory integral problems. Major special cases deserving mention include multiparameter sublevel set estimates, maximal curvature for Radon-like transforms of intermediate dimension, degenerate Radon transforms in low codimension, Fourier restriction and related generalized determinant functionals, and multilinear oscillatory integrals of convolution and related types. The PI's approach to these involves a variety of new tools developed within the last 5 years which incorporate techniques from Geometric Invariant Theory, geometric measure theory, decoupling theory, and other areas. Among these new tools is a recent result of the PI which provides an entirely new way to estimate norms of Radon-Brascamp-Lieb inequalities in terms of geometric quantities which can be understood as analogous to Lieb's formula for the Brascamp-Lieb constant. A major goal of this project is to understand the local geometric criteria which implicitly govern the finiteness of the nonlocal integrals appearing in the Radon-Brascamp-Lieb condition. The project has numerous potential applications to other problems of interest at the intersection of harmonic analysis, geometric measure theory, and incidence geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.