Counteracting flatness with affine measures and related problems in harmonic analysis

谐波分析领域源于对自然信号的研究和编码，方法是将其分解为基本部分。在这个项目中，首席研究员将努力以定量的方式理解曲率对这种分解的影响。例如，在医学成像中，x射线穿过人体，通过测量x射线被吸收或散射的多少来重建人体的密度函数。数学上，这个过程是由一个算子表示的，被称为x射线变换，它沿着直线路径平均函数。首席研究员正在进行一个项目，试图了解当平均值沿着曲线路径移动时会发生什么。众所周知，路径的足够曲率会导致输出的更大稳定性，该项目旨在量化路径是弯曲的，但可能有平坦区域的中间情况下的稳定效果。作为另一个例子，傅里叶变换将自然信号表示为等速波的叠加。如果速度在一个平面上，那么它们都是对齐的，并且，就像由稳定的微风形成的海浪一样，信号不会衰减。但是如果波速位于曲面上，比如球的表面，那么就会有一些衰减。精确测量这种衰减是谐波分析中的一个重要问题，其答案是未知的。首席研究员的研究是在这些情况之间的界面上，当速度在某些区域接近平面而在其他区域接近弯曲的表面上时，她试图精确地量化衰减速率。这些问题对量子力学研究中出现的偏微分方程有潜在的影响。首席研究员将研究欧几里得调和分析中出现的曲率相关问题，以及色散偏微分方程的一些应用。这项工作将包括三个方向。一个是证明了曲率沿非空集消失的流形的傅里叶变换的限制的新的曲率无关的边界；另一个是证明平均算子的类似结果；最后，她将学习极值器问题和集中紧致技术，其中一些应用于偏微分方程。作为该项目的一部分，首席研究员将组织会议，以促进数学知识的传播，并将致力于改进谐波分析方面的研究生培训，并增加代表性不足的群体，特别是妇女的参与。

The field of harmonic analysis grew out of an effort to study and encode natural signals by breaking them into their fundamental parts. In this project, the principal investigator will endeavor to understand, in a quantitative way, the effects of curvature on such decompositions. For example, in medical imaging, X-rays are passed through a body, and one reconstructs the density function of the body by measuring how much of the X-radiation is absorbed or scattered. Mathematically, this process is represented by an operator, known as the X-ray transform, which averages functions along straight-line paths. The principal investigator is working on a project that seeks to understand what happens when averages are taken along curved paths. It is known that sufficient curvature of the paths leads to greater stability of the output, and the project aims to quantify the stabilizing effect in an intermediate case where the paths are curved, but may have flat regions. As another example, the Fourier transform expresses a natural signal as a superposition of constant velocity waves. If the velocities lie on a plane, then they are all aligned, and, like ocean waves formed by a steady breeze, the signal does not decay. But if the wave velocities lie on a curved surface, such as the surface of a ball, then there is some decay. Precisely measuring this decay is an important question in harmonic analysis, the answer to which is unknown. The principal investigator's research lies at the interface between these situations, when the velocities lie on a surface that is nearly planar in some regions and curved in others, and she seeks to precisely quantify the rate of decay. Such questions have potential implications to partial differential equations that arise in the study of quantum mechanics. The principal investigator will study curvature-related problems arising in Euclidean harmonic analysis, as well as some applications to dispersive partial differential equations. This work will encompass three directions. One is to prove new, curvature-independent bounds for the restriction of the Fourier transform to manifolds whose curvature vanishes along some nonempty set; another is to prove analogous results for averaging operators; finally, she will study extremizer problems and concentration compactness techniques, some having applications to partial differential equations. As part of this project, the principal investigator will organize conferences to facilitate the dissemination of mathematical knowledge and will make a dedicated effort to improve graduate training in harmonic analysis and to increase the participation of underrepresented groups, especially women.