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The Geometry of Measures and Analytic Properties of Associated Operators

测度几何和关联算子的解析性质

基本信息

批准号：
2103534
负责人：
Benjamin Jaye
金额：
$ 9.39万
依托单位：
Georgia Tech Research Corporation
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-15 至 2024-04-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2103534&HistoricalAwards=false
关键词：
Geometry Measures Analytic Properties Associated

项目摘要

This project concerns a study of the some of the mathematics behind two basic physical questions. The first question considered is to what extent can the geometry of a body be determined from information about a force field associated to the body (for instance its gravitational field)? Such inverse problems in potential theory have a rich history, but the mathematical tools needed to answer this question are currently underdeveloped. The investigation will focus especially on what can be said if one only knows that the field has bounded magnitude, which is of particular interest in applications. A second question to be explored is how, and to what degree of accuracy, can one determine the asymptotic (or long term) behavior of a random function that evolves with time, based on certain empirical measurements? More specifically, the principal investigator proposes to research several questions concerning the relationship between the geometrical properties of a measure and the regularity properties of an associated operator. The primary question of interest is the following: What can be deduced about a measure from the knowledge that singular integral operator associated to it has good regularity properties? Under these circumstances, can the measure have a fractal structure, or must its support be contained in (a countable number of) Lipschitz sub-manifolds of appropriate dimension? Attempts to solve this problem has led to theory which has found recent applications in the calculus of variations, the study of free boundary problems, and the geometry of harmonic measure. The principal investigator intends to further develop tools that serve as a bridge from the analytic condition on the singular integral operator to the geometric structure of the measure. A second topic of research concerns understanding the long-term behavior of a stationary Gaussian process given information about its spectral measure, building upon recent work involving the principal investigator.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.

这个项目涉及两个基本物理问题背后的一些数学研究。首先要考虑的问题是，在多大程度上可以根据与物体有关的力场（例如它的引力场）的信息来确定物体的几何形状？在势理论中，这样的逆问题有着丰富的历史，但回答这个问题所需的数学工具目前还不发达。调查将特别集中在什么可以说，如果一个人只知道，该领域有界的幅度，这是特别感兴趣的应用。第二个要探讨的问题是，如何以及在何种程度上的准确性，可以确定的渐近（或长期）行为的随机函数，随着时间的推移，根据一定的经验测量？更具体地说，主要研究者提出了研究几个问题的几何性质之间的关系的措施和相关的运营商的正则性。感兴趣的主要问题是：什么可以推导出的知识，奇异积分算子相关的措施，它具有良好的正则性？在这种情况下，测度是否具有分形结构，或者它的支集是否必须包含在适当维数的Lipschitz子流形中（可数个）？试图解决这个问题导致了理论发现最近的应用在变分法，研究自由边界问题，几何调和措施。主要研究者打算进一步开发工具，作为桥梁，从奇异积分算子的分析条件的几何结构的措施。第二个研究课题是基于主要研究者最近的工作，在给定其光谱测量信息的情况下，了解平稳高斯过程的长期行为。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。