Geometric Harmonic Analysis: Affine and Frobenius-Hörmander Geometry

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。有许多基本的理论开放的问题，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。在这个项目中，将研究几何平均数领域的一系列问题。这些对应于量化成像对象中的小变化与测量数据中的预期变化之间的关系（实际上，测量数据被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进数学的许多相关领域，并可能影响未来的成像技术。研究生将参与该项目。PI将专注于数学分析中的几个主题，这些主题与Radon类算子，振荡积分和傅立叶限制算子的映射特性相关的一系列问题的新几何方法的发展有关。要研究的具体类别的运营商，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由其他研究人员。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。PI将使用几何和组合方法作为主要工具包，其中包括在过去5年内开发的各种新工具，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。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而被认为值得支持。

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.