Counteracting flatness with affine measures and related problems in harmonic analysis

谐波分析领域的发展源于通过将自然信号分解为基本部分来研究和编码自然信号的努力。在这个项目中，主要研究者将奋进以定量的方式了解曲率对这种分解的影响。 例如，在医学成像中，X射线穿过身体，并且通过测量吸收或散射多少X辐射来重建身体的密度函数。 在数学上，这个过程是由一个运算符表示的，称为X射线变换，它沿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沿着曲线路径取时会发生什么。 众所周知，路径的足够曲率导致输出的更大稳定性，并且该项目旨在量化路径弯曲但可能具有平坦区域的中间情况下的稳定效果。 作为另一示例，傅里叶变换将自然信号表示为恒定速度波的叠加。 如果速度位于一个平面上，那么它们都是对齐的，就像稳定微风形成的海浪一样，信号不会衰减。 但如果波的速度位于一个曲面上，比如一个球的表面，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。 首席研究员的研究位于这些情况之间的界面，当速度位于一个在某些区域几乎是平面而在其他区域是弯曲的表面上时，她试图精确地量化衰变率。 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。主要研究者将研究欧几里德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。 一个是证明新的，曲率独立的界限限制的傅立叶变换流形的曲率消失沿着一些非空集;另一个是证明类似的结果平均运营商;最后，她将研究极值问题和浓度紧技术，一些应用偏微分方程。 作为该项目的一部分，主要研究员将组织会议，以促进数学知识的传播，并将致力于改善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.