Variable Coefficient Fourier Analysis

变系数傅立叶分析

基本信息

批准号：
1953413
负责人：
Christopher Sogge
金额：
$ 26.3万
依托单位：
Johns Hopkins University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1953413&HistoricalAwards=false
关键词：
Variable Coefficient Fourier Analysis

项目摘要

The PI 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 emits. Similar phenomena arise for manifolds, and the project aims 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 PI also wishes to 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. The PI will be supervising graduate students in domains related to the proposed research. Such activities will be supported by the award.Among the specific problems to be studied, the project aims to obtain improved estimates that measure the size and concentration of eigenfunctions. In order to do this, the PI will develop what could be called "global harmonic analysis", which is a mixture of classical harmonic analysis, microlocal analysis and techniques from geometry. The basic estimates to have in mind are Lp-estimates for eigenfunctions and quasimodes, and related highly localized L2 estimates that are sensitive to concentration. The main questions center around how the geometry and the global dynamics of the geodesic flow affects 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, Schrodinger equation and resolvent estimates coming from the metric Laplacian. The PI is particularly interested in high frequency solutions and improving existing results under geometric assumptions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

PI 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。与给定流形相关的是称为特征函数的基本对象。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。乐器设计师很清楚，鼓或弦乐器音板的形状会影响其发出的基本音调。流形也会出现类似的现象，该项目旨在精确研究它们的形状（例如它们的弯曲方式）如何影响所得的本征函数。正如在音乐中一样，随着频率变得越来越大，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。这些特征函数是类似于波动方程的微分方程的解，PI也希望研究涉及它的类似问题。总体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。 PI 将指导与拟议研究相关领域的研究生。此类活动将得到该奖项的支持。在要研究的具体问题中，该项目旨在获得改进的估计，以测量特征函数的大小和集中度。为了做到这一点，PI 将开发所谓的“全局调和分析”，它是经典调和分析、微局部分析和几何技术的结合。要记住的基本估计是本征函数和准模态的 Lp 估计，以及对浓度敏感的相关高度局部化的 L2 估计。主要问题集中在测地流的几何形状和全局动力学如何影响估计值以及使估计值饱和的函数类型。后一个问题与谱渐进中模态和准模态的浓度、振荡和尺寸特性等经过大量研究（但尚未充分理解）的问题密切相关。这些问题也自然地与波动方程、薛定谔方程和来自度量拉普拉斯算子的求解估计的解算子的长期性质相关。 PI 对高频解决方案和改进几何假设下的现有结果特别感兴趣。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。