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

几何平均的数学，也称为类Radon算子，在许多与成像相关的技术应用中是至关重要的：CT、SPECT和核磁共振，以及雷达和声纳应用，都依赖于对Radon变换的深入理解，相关概念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，在这个数学领域有许多基本的理论问题仍然没有解决，尽管这个领域已经取得了许多令人难以置信的成功。这个项目建议研究几何平均领域的一系列问题，粗略地说，几何平均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，通过计算处理这些变化以恢复原始对象的近似图像)。实现这个项目的主要目标将导致一些相关数学领域的进步，并可能影响未来的成像技术。更准确地说，该项目将侧重于数学分析中的几个主题，这些主题与发展多线性算子的新几何方法有关，并特别强调建立统一估计。要研究的特定类型的算子包括多线性类Radon平均算子以及相关的非奇异振荡问题，这类问题首先由Christian、Li、Tao和Thiele研究。值得一提的主要特例包括多参数次水平集估计、中维Radon类变换的最大曲率、低余维退化Radon变换、傅立叶限制及相关的广义行列式泛函、Phong-Stein算子van der Corput方法、卷积及相关类型的多线性振荡积分。该项目的次要目标是将这些想法应用于PDE。几何和组合方法将被用作主要工具包，因为这些方法的成功记录始终如一，特别是首席研究员近年来在这一方向上取得了重大进展。这些方法吸收了数学中广泛领域的思想，包括几何不变理论、几何测量理论、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优势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, 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.