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

几何调和分析：类氡变换及相关主题的进展

• 批准号: 2348384
• 负责人: Philip Gressman
• 金额: $ 23.91万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348384&HistoricalAwards=false
• 关键词: Geometric; Harmonic; Analysis; Advances; Radon

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.

被称为类氡算子的几何平均数学在与成像和数据分析相关的许多技术应用中至关重要：CT、SPECT 和 NMR，以及雷达和声纳应用，所有这些都依赖于对Radon 变换以及相关思想出现在光声断层扫描、散射理论，甚至一些运动检测算法中。令人有些惊讶的是，尽管该领域已经取得了许多令人难以置信的成功，但该数学领域仍有许多基本理论问题尚未解决。该项目研究几何平均值领域的一系列问题，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（实际上将通过计算处理以恢复近似图像）原始对象）。此类问题的理论挑战是精确量化变化的概念，并在输入和输出变化幅度之间建立本质上精确的关系。由于 PI 在理解这些物体方面的工作最近取得了进展，该项目处于有利位置，可以产生重要的成果。实现该项目的主要目标将导致数学相关领域的进步，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，他们一旦进入劳动力市场，就可以将这些技能转移到其他急需的领域。PI 研究与开发新几何方法相关的数学分析主题。类 Radon 变换、振荡积分和傅里叶限制问题。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、中间维度的类 Radon 变换的最大曲率、低余维的简并 Radon 变换、傅里叶限制和相关的广义行列式泛函，以及卷积和相关类型的多线性振荡积分。 PI 解决这些问题的方法涉及过去 5 年内开发的各种新工具，其中结合了几何不变量理论、几何测度理论、解耦理论和其他领域的技术。在这些新工具中，PI 的最新成果提供了一种全新的方法来估计 Radon-Brascamp-Lieb 不等式的几何量范数，可以将其理解为类似于 Brascamp-Lieb 常数的 Lieb 公式。该项目的一个主要目标是了解隐式控制 Radon-Brascamp-Lieb 条件中出现的非局部积分的有限性的局部几何准则。该项目在调和分析、几何测量理论和入射几何交叉方面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优点和更广泛的影响审查进行评估，被认为值得支持标准。