Variable Coefficient Fourier Analysis

PI将研究几何谐波分析中的几个问题。这些问题的设置涉及二维或二维以上的几何流形。与给定流形相关联的是称为特征函数的基本对象。这些是流形振动的基本模态，它们是圆的熟悉的三角函数的高维类比。乐器设计师很清楚，鼓或弦乐器的音板的形状会影响它所忽略的基本音调，以及音量。类似的现象也出现在流形中，PI将精确地研究它们的形状，比如它们是如何弯曲的，如何影响特征函数的性质。就像在音乐中一样，当频率变得越来越大时，人们特别期望不同的形状和几何形状在基本振动模式的行为中变得更加明显。这些特征函数是与波动方程相似的微分方程的解，PI将研究与之相关的类似问题。一般的主题是研究波动方程的解如何受到它们的物理背景的影响，比如黑洞是否存在，或者背景是否变得非常接近于接近无穷大的真空。本项目为研究生提供研究训练机会。在PI要研究的具体问题中，他们希望得到改进的估计，以衡量特征函数的大小和浓度。为了做到这一点，他们将发展所谓的“全局谐波分析”，它是经典谐波分析、微局部分析和几何技术的混合。基本估计是特征函数和准模的lp估计以及相关的对浓度敏感的高度局域L2估计。主要问题集中在测地线流的几何和全局动力学如何影响估计和使它们饱和的各种函数。后一个问题与谱渐近中模态和准模态的集中、振荡和大小性质密切相关（但仍未得到很好的理解）。这些问题也自然地与波动方程、薛定谔方程的解算符的长时间性质以及来自度量拉普拉斯算子的解算符估计联系在一起。高频解和几何假设下的尖锐结果特别有趣。他们还将研究以不同方式饱和估计的函数，这取决于流形的截面曲率的符号。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The PI will study several problems in Geometric Harmonic Analysis. The settings for these problems involve geometric manifolds of dimension two or more. Associated with a given manifold are fundamental objects called eigenfunctions. These are the fundamental modes of vibration of the manifold, and they are the higher dimensional analogs of the familiar trigonometric functions for the circle. Designers of musical instruments are well aware that the shape of, say, a drum or the soundboard of stringed instrument affects the basic tones that it omits, as well as the sound volume. Similar phenomena arise for manifolds, and the PI will study precisely how their shapes, such as how they are curved, affect the properties properties of eigenfunctions. Just as in music, one particularly expects different shapes and geometries to become more apparent in the behavior of the fundamental modes of vibration as the frequency becomes larger and larger. These eigenfunctions are solutions of a differential equation that is similar to the wave equation, and the PI will study similar problems involving it. The general theme is to study how solutions of wave equations are affected by their physical backgrounds, such as whether or not black holes are present or whether the background becomes very close to a vacuum near infinity. This project provides research training opportunities for graduate students.Among the specific problems the PI shall study, they wish to obtain improved estimates that measure the size and concentration of eigenfunctions. In order to do this, they will develop what is called ``global harmonic analysis’’, which is a mixture of classical harmonic analysis, microlocal analysis, and techniques from geometry. The basic estimates are Lp-estimates for eigenfunctions and quasimodes and related highly localized L2 estimates that are sensitive to concentration. The main questions center around how the geometry and the global dynamics of the geodesic flow affect the estimates and the kinds of functions that saturate them. The latter issue is closely related to the much-studied (but still not well-understood) questions of concentration, oscillation, and size properties of modes and quasimodes in spectral asymptotics. These questions are also naturally linked to the long-time properties of the solution operator for the wave equation, Schrodinger equation, and resolvent estimates coming from the metric Laplacian. High frequency solutions and obtaining sharp results under geometric assumptions are particularly interesting. They will also study functions that saturate the estimates in different ways depending on the sign of the sectional curvatures of the manifolds.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.