Geometric Harmonic Analysis: Affine and Frobenius-Hörmander Geometry

几何平均的数学，也被称为类氡算子，在与成像相关的许多技术应用中具有重要意义：CT， SPECT和NMR，以及雷达和声纳应用，都依赖于对氡变换的深刻理解，相关思想出现在光学声断层扫描，散射理论，甚至一些运动检测算法中。尽管该领域已经取得了许多令人难以置信的成功，但在这一数学领域仍有许多基本的理论开放性问题尚未解决。在这个项目中，将研究几何平均领域的一系列问题。这些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（在实践中，计算处理以恢复原始对象的近似图像）。该项目的主要目标将推动数学的一些相关领域，并可能影响未来的成像技术。研究生也参与了这个项目。PI将专注于数学分析中的几个主题，这些主题与发展新的几何方法有关，这些方法涉及与类氡算子的映射性质，振荡积分和傅立叶限制算子有关的一系列问题。要研究的算子的具体类别包括多线性类radon平均算子以及其他研究者首先研究的一类相关的非奇异振荡问题。值得一提的主要特殊情况包括多参数子水平集估计、中维类Radon变换的最大曲率、低余维简并Radon变换、傅里叶限制及相关的广义行列式泛函数、Phong-Stein算子van der Corput方法、卷积及相关类型的多线性振荡积分。PI将使用几何和组合方法作为主要工具，其中包括过去5年开发的各种新工具，结合几何不变量理论、几何测量理论、解耦理论和其他领域的技术。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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, 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. There are many basic theoretical open questions in this area of mathematics which remain unsolved despite the many incredible successes the field has already achieved. In this project a family of questions will be studied in the area of geometric averages. These 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). The main goals of this project will advance a number of related areas of mathematics and may influence future imaging technologies. Graduate students are involved in the project.PI will focus on several topics in mathematical analysis related to the development of new geometric methods for a family of questions relating to the mapping properties of Radon-like operators, oscillatory integrals, and Fourier restriction operators. The specific classes of operators to be studied include multilinear Radon-like averaging operators as well as related nonsingular oscillatory questions of the sort first studied by other researchers. 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. PI will use a geometric and combinatorial approach as the main toolkit, which includes a variety of new tools developed within the last 5 years incorporating techniques from geometric invariant theory, geometric measure theory, decoupling theory, and other areas.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.