Geometric Harmonic Analysis: Affine and Frobenius-Hormander Geometry for Multilinear Operators

几何调和分析：多线性算子的仿射和 Frobenius-Hormander 几何

• 批准号: 1764143
• 负责人: Philip Gressman
• 金额: $ 15万
• 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: 2018
• 资助国家: 美国
• 起止时间: 2018-08-01 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1764143&HistoricalAwards=false
• 关键词: Geometric; Harmonic; Analysis; Affine; Frobenius

The mathematics of geometric averages, also known as Radon-like operators, is of fundamental importance in a host of technological applications related to imaging: 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 proposes to study a family of questions in the area of geometric averages which, roughly speaking, correspond to quantifying the relationship between small changes in the imaged objects and the expected changes in measured data (which in practice is processed computationally to recover an approximate picture of the original object). 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.More precisely, the project will focus on several topics in mathematical analysis related to the development of new geometric methods for multilinear operators, with specific emphasis on the establishment of uniform estimates. The specific classes of operators to be studied include multilinear Radon-like averaging operators as well as related nonsingular oscillatory problems of the sort first studied by Christ, Li, Tao, and Thiele. 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, Phong-Stein operator van der Corput methods, and multilinear oscillatory integrals of convolution and related types. A secondary goal of the project is to apply these ideas to PDEs. A geometric and combinatorial approach will be used as the main toolkit since these methods have a consistent record of success, and the principal investigator has, in particular, made major progress in this direction in recent years. These methods incorporate ideas from a wide variety of areas in mathematics, including Geometric Invariant Theory, Geometric Measure Theory, Convex Geometry, and Signal Processing.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.

几何平均的数学，也被称为Radon-like算子，在许多与成像相关的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，相关的想法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。该项目拟研究几何平均数领域的一系列问题，粗略地说，几何平均数相当于量化成像物体的微小变化与测量数据的预期变化之间的关系（实际上，测量数据经过计算处理，以恢复原始物体的近似图像）。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，更准确地说，该项目将集中在数学分析中的几个主题有关的发展新的几何方法的多线性算子，特别强调建立统一的估计。具体类别的运营商进行研究，包括多线性Radon平均运营商以及相关的非奇异振荡问题的排序首先研究基督，李，陶，和Thiele。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，Phong-Stein运营商货车德Corput方法，和多线性振荡积分的卷积和相关类型。该项目的第二个目标是将这些想法应用于PDE。几何和组合的方法将被用作主要的工具包，因为这些方法有一贯的成功记录，特别是首席研究员，近年来在这方面取得了重大进展。这些方法融合了数学领域的各种思想，包括几何不变理论、几何测度理论、凸几何和信号处理。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。

项目成果

On the Oberlin affine curvature condition

关于 Oberlin 仿射曲率条件

• DOI: 10.1215/00127094-2019-0010
• 发表时间: 2019
• 期刊: Duke Mathematical Journal
• 影响因子: 2.5
• 作者: Gressman, Philip T.

Generalized curvature for certain Radon-like operators of intermediate dimension

某些中间维度类 Radon 算子的广义曲率

• DOI: 10.1512/iumj.2019.68.7562
• 发表时间: 2019
• 期刊: Indiana University Mathematics Journal
• 影响因子: 1.1
• 作者: Gressman, Philip

Reversing a Philosophy: From Counting to Square Functions and Decoupling

逆转哲学：从计数到平方函数和解耦

• DOI: 10.1007/s12220-020-00593-x
• 发表时间: 2021
• 期刊: The Journal of Geometric Analysis
• 作者: Gressman, Philip T.;Guo, Shaoming;Pierce, Lillian B.;Roos, Joris;Yung, Po-Lam
• 通讯作者: Yung, Po-Lam