Variable Coefficient Fourier Analysis

主要研究人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两维或更多维的几何流形。与给定流形相关的是称为本征函数的基本对象。这些是流形的基本振动模式，它们是圆的常见三角函数的高维类似物。乐器的设计者很清楚，鼓或弦乐器的响板的形状会影响它省略的基本音调。类似的现象也出现在流形上，我们希望精确地研究它们的形状，例如它们是如何弯曲的，如何影响所产生的本征函数。就像在音乐中一样，当频率变得越来越大时，人们特别期待不同的形状和几何在基本振动模式的行为中变得更加明显。这些特征函数是一个类似于波动方程的微分方程的解，该项目还将研究涉及它的类似问题。总的主题是研究波动方程的解如何受到其物理背景的影响，例如是否存在黑洞，或者背景是否变得非常接近无穷大的真空。在项目中的具体问题中，主要研究人员希望获得对特征函数的节点集(零集)的改进估计。有一个Yau的猜想，断言这个余维的大小-一个集合的大小应该与它的频率相当。虽然它已经在真实的解析环境中完全解决了，但关于光滑流形却鲜为人知。首席研究员已经证明，这个问题与可以检测某些类型的集中度的本征函数的勒贝格空间估计之间存在联系。该项目中的几个问题涉及开发这一活跃的领域。主要研究者还希望在负曲率的假设下得到特征函数的所谓周期积分的改进的界。已知这一假设是必要的，该问题测量了特征函数沿测地线的随机抵消。这个问题与解析数论有联系，但到目前为止，主要研究者用调和分析得到的结果是最著名的。他想把它们与数论技术结合起来，试图获得更好的界。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”)和微局域分析，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.