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.

在与成像和数据分析有关的许多技术应用中，几何平均值的数学平均值至关重要，CT，SPECT和NMR以及雷达和声纳的应用，都取决于对radon转换的深刻理解，以及相关的思想在光学上的声学层造影中出现，甚至在散射的理论，甚至一些运动中，以及一些散射的Motiontection和一些运动。令人惊讶的是，尽管该领域已经取得了许多令人难以置信的成功，但在这一数学领域中存在许多基本的理论问题。该项目研究了几何平均地区的一个问题家族，例如，该系列与量化成像对象的小变化与所测量数据的预期变化之间的关系相对应（实际上将在计算上对其进行处理以恢复原始对象的近似图片）。诸如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之间建立基本确切的关系。得益于PI了解这些对象的最新进展，该项目的位置良好，可以产生重要的结果。实现该项目的主要目标将导致在许多相关数学领域的进步，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，他们可以在劳动力中一次将这些技能转移到其他批判性需求的其他领域。这项工作包括各种特殊案例，这些特殊案例既包括sublevel set和振荡性积分问题。值得一提的主要特殊案例包括多参数级的估计值，中间维度的ra样变换的最大曲率，低编构象，傅立叶限制和相关的广义确定性功能的退化ra变换以及卷积和相关类型的多线性振荡性积分。 PI对这些方法的方法涉及过去5年中开发的各种新工具，这些工具结合了几何不变理论，几何测量理论，解耦理论和其他领域的技术。这些新工具中有PI的最新结果，它提供了一种全新的方式，可以从几何数量方面估算Raad-Brascamp-Lieb不平等的规范，这可以理解为类似于Lieb的Brascamp-Lieb常数公式。该项目的主要目标是了解局部几何标准，该标准隐含地控制了在radon-Brascamp-lieb条件下出现的非本地积分的有限性。该项目在谐波分析，几何措施理论和发病率几何形状的交集中对其他感兴趣的问题有许多潜在的应用。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估来通过评估来支持的。