Variable Coefficient Fourier Analysis

在这个项目中，首席研究员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被称为二维或更大维度的几何流形的对象。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，首席研究员希望精确地研究它们的形状，如它们是如何弯曲的，如何影响所产生的特征函数。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。在主要研究人员将研究的具体问题中，他试图获得衡量特征函数大小和浓度的改进估计。为了做到这一点，他将发展他所说的“全局调和分析”，这是古典调和分析、微观局部分析和几何学技术的混合体。人们头脑中的基本估计是对本征函数和准模的估计，以及相关的对浓度敏感的高度局部化的估计。主要问题围绕着大地线流的几何和全球动力学如何影响估计以及使估计饱和的函数类型。后一个问题与谱渐近中的模和准模的集中、振荡和尺寸特性等问题密切相关(但仍未得到很好的理解)。这些问题也自然地与波动方程的解算子的长期性质和来自度规拉普拉斯的预解估计有关。首席研究人员还对理解由边界迹产生的算子的调和分析和谱理论感兴趣。

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.