Variable Coefficient Fourier Analysis

首席研究员将研究几何傅里叶分析中的几个问题。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。类似的现象也出现在流形中，我们希望精确地研究它们的形状，比如它们是如何弯曲的，是如何影响得到的特征函数的。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类似的微分方程的解，本项目也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。在项目的具体问题中，主要研究者希望获得特征函数的节点集（零集）的改进估计。丘有一个猜想，认为这个协维集的大小应该与其频率相当。虽然它已经完全解决了实际的解析设置，很少知道光滑流形。首席研究员已经表明，这个问题与勒贝格空间估计之间存在联系，特征函数可以检测某些类型的浓度。项目中的几个问题涉及到开发这一活跃领域。主要研究者还想在负曲率假设下得到所谓本征函数周期积分的改进界。这个假设已知是必要的，并且该问题测量沿测地线的特征函数的随机消去。这个问题与解析数论有联系，但迄今为止，首席研究员最近使用谐波分析获得的结果是最著名的。他想将它们与数论技术结合起来，试图获得改进的边界。这些问题与该项目将研究的其他问题之间存在联系，这些问题涉及波动方程的估计（“Strichartz估计”）和微局部分析，特别是奇点的传播。

The principal investigator will study several problems in geometric Fourier analysis. The settings for these problems involve geometric manifolds of dimension two or more. Associated to 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. Similar phenomena arise for manifolds, and we wish to study precisely how their shapes, such as how they are curved, affect the resulting 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 project will also 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.Among the specific problems in the project, the principal investigator desires to obtain improved estimates for the nodal sets (zero sets) of eigenfunctions. There is a conjecture of Yau asserting that the size of this codimension-one set should be comparable to its frequency. Although it has been fully settled in the real analytic setting, less is known for smooth manifolds. The principal investigator has already shown that there are connections between this problem and Lebesgue space estimates for eigenfunctions that can detect certain types of concentration. Several problems in the project involve developing this active field. The principal investigator would also like to obtain improved bounds for so-called period integrals of eigenfunctions under the assumption of negative curvature. This assumption is known to be necessary, and the problem measures the random cancellation of eigenfunctions along geodesics. There are connections with this problem and analytic number theory, but to date the results that the principal investigator has recently obtained using harmonic analysis are the best known. He would like to combine them with number theory techniques to try to obtain improved bounds. There are connections between these problems and other problems that the project will study involving estimates for wave equations ("Strichartz estimates") and microlocal analysis, especially propagation of singularities.