Counteracting flatness with affine measures and related problems in harmonic analysis

调和分析领域起源于通过将自然信号分解为基本部分来对其进行研究和编码的努力。在这个项目中，首席研究员将努力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射线穿过人体，人们通过测量X射线被吸收或散射的量来重建人体的密度函数。在数学上，这个过程由一个称为X-射线变换的运算符表示，它沿着直线路径对函数求平均。首席研究人员正在进行一个项目，试图了解当平均值沿着弯曲的路径进行时会发生什么。众所周知，路径的足够曲率会导致更大的输出稳定性，该项目的目的是在路径是弯曲的、但可能具有平坦区域的中间情况下量化稳定效果。作为另一个例子，傅里叶变换将自然信号表示为恒速波的叠加。如果速度在平面上，那么它们都是对齐的，就像稳定的微风形成的海浪一样，信号不会衰减。但如果波的速度位于曲面上，例如球的表面，那么就会有一些衰减。准确测量这种衰减是调和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两种情况之间的交界处，即速度位于一个表面上，该表面在一些区域几乎是平面的，在另一些区域是弯曲的，她试图精确地量化衰减率。这些问题对量子力学研究中出现的偏微分方程式有潜在的影响。主要研究人员将研究欧几里德调和分析中出现的与曲率有关的问题，以及在色散偏微分方程中的一些应用。这项工作将包括三个方向。一个是证明傅里叶变换对曲率沿非空集消失的流形的限制的新的、与曲率无关的界；另一个是证明平均算子的类似结果；最后，她将研究极值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，首席调查员将组织会议，以促进数学知识的传播，并将致力于改进调和分析方面的研究生培训，增加代表不足的群体，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.