Variable Coefficient Fourier Analysis

在这个项目中，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称为二维或二维以上的几何流形的对象。 与给定流形相关联的是称为其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，主要研究者希望精确地研究它们的形状，比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。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，或者背景是否变得非常接近接近无穷大的真空。在具体问题中，首席研究员将研究中，他试图获得改进的估计，衡量的大小和浓度的特征函数。 为了做到这一点，他将开发他所谓的“全球谐波分析”，这是一个混合物的经典谐波分析，微观局部分析，和技术，从几何。 人们心中的基本估计是本征函数和准模的估计，以及对浓度敏感的相关高度局部化的估计。 主要的问题集中在如何几何和全球动态的测地线流影响的估计和饱和的功能，他们的种类。 后一个问题是密切相关的研究（但仍然没有很好地理解）的浓度，振荡和尺寸特性的模式和准模式的谱渐近的问题。 这些问题也自然地与波动方程的解算子的长时间性质和来自度量拉普拉斯算子的预解估计有关。 首席研究员也有兴趣了解谐波分析和频谱理论的运营商，产生的边界跟踪。

In this project the principal investigator will study several problems in geometric Fourier analysis. The settings for these problems involve objects known as geometric manifolds of dimension two or greater. Associated to a given manifold are fundamental objects called its 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 the principal investigator wishes 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 the principal investigator will study, he seeks to obtain improved estimates that measure the size and concentration of eigenfunctions. In order to do this he will develop what he calls "global harmonic analysis," which is a mixture of classical harmonic analysis, microlocal analysis, and techniques from geometry. The basic estimates that one has in mind are estimates for eigenfunctions and quasimodes, and related highly localized 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 and resolvent estimates coming from the metric Laplacian. The principal investigator is also interested in understanding the harmonic analysis and spectral theory of operators that arise from boundary traces.