CAREER: Degeneracies of Curvature in Harmonic Analysis

谐波分析领域试图通过将数学对象分解成更简单的部分来理解物理信号和现象。最初，该领域的工具针对这些对象的基本成分是线性的情况进行了优化：傅里叶变换将信号（如声音）分解为恒频波（音符）的叠加；x射线变换是一种了解物体密度的方法，方法是让（直线）辐射光束穿过物体，并测量另一侧发射的光束的强度。然而，近年来，谐波分析工具在曲率起作用的情况下也变得越来越有用和重要。例如，Schrödinger方程描述了一个量子系统的时间演化，最近对这个方程解的理解取得了进展，这得益于50年前人们认识到Schrödinger方程解的所有傅立叶数据都位于抛物线物体上。曲率具有局域效应：由于所有的波都以不同的方向传播，它们的叠加只能在一个小的时空区域内大；相反，如果傅里叶数据位于一个平面上，那么在垂直于该平面的方向上，叠加将是恒定的。其他物理信号的傅里叶数据可能被限制在不同的表面上，该项目的一个方面是精确量化这些信号的定位，以一种仅依赖于底层物体曲率的方式。另一个例子是，在曲面上平均信号的运算符出现在各种情况下，包括三维几何光学。在垂直于平面的方向上，固定平面平移的平均值固有地不稳定，而在曲面上的平均值在所有方向上都使原始信号平滑。该项目将精确量化表面的平滑程度，可能有一些弯曲的部分和一些平坦的部分。最后，近年来，人们对反问题的兴趣激增，其中试图描述由上述衰减和平滑效应非常弱的数学信号组成的极端情况；该项目的一部分是研究这个方向的一些基本问题。这些逆问题与工程、物理和医学成像的潜在应用有关。这些科学努力与研究者帮助培养下一代数学家的努力密不可分。这种劳动力发展包括三个主要方向：为数学博士生提供建议，为本科生创造夏季研究机会，以及组织一系列促进各级数学家之间互动的专题讨论会。在过去的五十年中，调和分析中的一个重要主题是一些底层流形的曲率导致算子的行为比预期的要好。本课题的主要部分考虑了与变曲率流形相关的傅里叶限制和平均算子。曲率使这些算子的行为比简单地计算流形的维度所预测的要好，而该活动的主要目标是通过给流形配备一个度量来证明这些算子的统一边界，该度量给曲率较小的区域赋予较小的权重。这些曲率无关的边界本质上是所考虑的算子的最强可能。此外，这些结果将精确地量化曲率在相关算子中的作用。研究者还将致力于对这种类型的某些算子饱和勒贝格空间不等式的函数进行表征。作为该项目的一个组成部分，研究者将通过担任论文顾问、指导本科生研究人员以及组织包括各级数学家在内的研讨会来培养初级数学家。

The field of harmonic analysis attempts to understand physical signals and phenomena by decomposing mathematical objects into simpler parts. Initially, the tools in the field were optimized for the case when the fundamental components of these objects are linear: The Fourier Transform decomposes a signal (such as a sound) as a superposition of constant-frequency waves (notes); the X-ray transform is a way of understanding the density of a body by passing (straight line) beams of radiation through the body and measuring the strength of the beam emitted on the other side. In recent years, however, it has become increasingly apparent that harmonic analysis tools are also useful and important in situations where curvature plays a role. As an example, the Schrödinger equation describes the time evolution of a quantum system, and recent advances toward understanding solutions to this equation were facilitated by the fifty-year-old realization that all of the Fourier data of a solution to the Schrödinger equation lies on a parabolic object. Curvature has a localizing effect: since the waves all propagate in different directions, their superposition can only be large in a small region of spacetime; by contrast, if the Fourier data were to lie on a flat plane, the superposition would be constant in directions perpendicular to that plane. Other physical signals may have Fourier data constrained to different surfaces, and one aspect of the project is to precisely quantify the localization of these signals, in a way that depends only on the curvature of the underlying object. As another example, operators that average signals over curved surfaces arise in a variety of contexts, including three-dimensional geometric optics. Whereas averages over translates of a fixed plane are inherently unstable in directions perpendicular to the plane, averages over curved surfaces smooth out the original signal in all directions. The project will precisely quantify the degree of smoothing for surfaces that may have some curved parts and some flat parts. Finally, in recent years, there has been an explosion of interest in inverse problems wherein an attempt is made to characterize extreme cases consisting of mathematical signals for which the decay and smoothing effects described above are very weak; part of the project is to study some basic questions in this direction. These inverse problems are connected with potential applications in engineering, physics, and medical imaging. These scientific endeavors are inextricably linked with the investigator's efforts to help train the next generation of mathematicians. This workforce development encompasses three main directions: advising Ph.D. students in mathematics, creating summer research opportunities for undergraduate students, and organizing a series of symposia that foster interactions among mathematicians at all levels. During the past five decades, an important theme in harmonic analysis has been problems wherein the curvature of some underlying manifold causes operators to behave better than expected. The main part of this project considers Fourier restriction and averaging operators associated to manifolds with varying curvature. Curvature causes these operators to behave better than would be predicted by simply counting the dimension of the manifold, and the chief goal of the activity is to prove uniform bounds for these operators by equipping the manifold with a measure that gives small weight to regions where the curvature is small. These curvature-independent bounds are essentially the strongest possible for the operators considered. Moreover, these results would precisely quantify the role of curvature in the associated operators. The investigator will also work toward a characterization of functions that saturate the Lebesgue space inequalities for certain operators of this type. As an integral part of this project, the investigator will work to train junior mathematicians by serving as a dissertation advisor, by mentoring undergraduate researchers, and by organizing symposia that will include mathematicians at all levels.