Variable Coefficient Fourier Analysis

主要研究者将研究几何傅立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本征函数的基本对象。 这些是流形振动的基本模式，它们是熟悉的圆三角函数的高维类似物。 乐器的设计师们都很清楚，比如说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，如何影响所得的本征函数。 就像在音乐中一样，人们特别期望随着频率变得越来越大，不同的形状和几何形状在振动的基本模式的行为中变得更加明显。 这些本征函数是一个类似于波动方程的微分方程的解，该项目也将研究涉及波动方程的类似问题。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，例如黑洞是否存在或背景是否变得非常接近无穷大的真空。在项目的具体问题中，主要研究者希望获得本征函数的节点集（零集）的改进估计。 有一个猜想的丘断言，这余维一集的大小应该是可比的频率。 虽然它已经完全解决了在真实的分析设置，少是已知的光滑流形。 首席研究员已经表明，这个问题和勒贝格空间估计的本征函数之间存在联系，可以检测某些类型的浓度。 该项目中的几个问题涉及开发这一活跃领域。 主要研究者还希望在负曲率假设下获得所谓的本征函数周期积分的改进界限。 这个假设是必要的，这个问题测量了本征函数沿测地线沿着的随机抵消。 这个问题和解析数论有联系，但到目前为止，主要研究者最近使用调和分析获得的结果是最著名的。 他想将它们与数论技术联合收割机结合起来，试图获得改进的界限。 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（“哈茨估计”）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.