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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发出的基本音调。类似的现象也出现在流形上，该项目的目的是精确地研究它们的形状，如它们是如何弯曲的，如何影响由此产生的特征函数。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。这些特征函数是类似于波动方程的微分方程解，PI也希望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。PI将在与拟议研究相关的领域指导研究生。在需要研究的具体问题中，该项目旨在获得衡量特征函数大小和集中度的改进估计。为了做到这一点，PI将发展所谓的“全局调和分析”，这是经典调和分析、微观局部分析和几何学技术的混合。要考虑的基本估计是本征函数和准模的Lp估计，以及相关的对浓度敏感的高度局部化的L2估计。主要问题围绕着测地线流的几何形状和全球动力学如何影响估计以及使估计饱和的函数种类。后一个问题与谱渐近中的模和准模的集中、振荡和尺寸性质等问题密切相关(但还不是很清楚)。这些问题也自然地与波动方程、薛定谔方程的解算子的长期性质以及来自度规拉普拉斯的预解估计有关。PI对高频解决方案和在几何假设下改进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优势和更广泛的影响审查标准进行评估，被认为值得支持。