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

几何平均数的数学运算被称为Radon-like算子，在与成像和数据分析相关的许多技术应用中具有根本的重要性：CT，SPECT和NMR，以及RADAR和SONAR应用，都依赖于对Radon变换的深入理解，相关的思想出现在光声层析成像，散射理论，甚至一些运动检测算法中。有些令人惊讶的是，有许多基本的理论问题，在这一领域的数学仍然没有解决，尽管许多令人难以置信的成功领域已经取得了。该项目研究几何平均数领域的一系列问题，例如，几何平均数相当于量化成像物体的微小变化与测量数据（实际上将通过计算处理恢复原始物体的近似图像）的预期变化之间的关系。这样一个问题的理论挑战是精确地量化变化的概念，并建立投入和产出变化幅度之间的基本精确关系。由于PI在理解这些物体方面的最新进展，该项目处于有利地位，将产生重要成果。实现该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。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，他们可以将这些技能转移到其他急需的领域。PI研究数学分析中的主题，与Radon样变换，振荡积分和傅立叶限制问题的新几何方法的发展有关。这项工作包括各种特殊情况下的子水平集和振荡积分问题。值得一提的主要特殊情况包括多参数子水平集估计，最大曲率的Radon变换的中间尺寸，退化的Radon变换在低余维，傅立叶限制和相关的广义行列式泛函，和多线性振荡积分的卷积和相关类型。PI的方法，这些涉及到各种新的工具，在过去的5年内，其中包括技术，从几何不变理论，几何测量理论，解耦理论，和其他领域的发展。在这些新的工具是最近的结果PI提供了一个全新的方式来估计规范的Radon-Brascamp-Lieb不等式的几何数量可以理解为类似于Lieb的公式Brascamp-Lieb常数。这个项目的一个主要目标是了解当地的几何准则，隐含地管理有限的非局部积分出现在Radon-Brascamp-Lieb条件。该项目在调和分析、几何测量理论和入射几何学的交叉点上对其他感兴趣的问题有许多潜在的应用。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.