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.

该项目涉及两个基本物理问题背后的一些数学研究。首先考虑的问题是，在多大程度上可以根据与物体相关的力场（例如其引力场）信息来确定物体的几何形状？势论中的此类反问题有着悠久的历史，但回答这个问题所需的数学工具目前还不发达。这项研究将特别关注如果人们只知道该场具有有限大小的话可以说些什么，这在应用中特别令人感兴趣。要探讨的第二个问题是，如何基于某些经验测量来确定随时间演化的随机函数的渐近（或长期）行为以及精确度如何？更具体地说，主要研究者建议研究有关测度的几何特性与相关算子的正则性特性之间关系的几个问题。主要感兴趣的问题如下：根据与测量相关的奇异积分算子具有良好的正则性性质的知识，可以推断出什么关于测量的信息？在这种情况下，测度是否可以具有分形结构，或者它的支持必须包含在（可数个）适当维数的 Lipschitz 子流形中？解决这个问题的尝试催生了一些理论，该理论最近在变分法、自由边界问题的研究以及调和测度的几何学中得到了应用。首席研究员打算进一步开发工具，作为从奇异积分算子的解析条件到测度的几何结构的桥梁。研究的第二个主题涉及在首席研究员最近的工作基础上，在给定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智力价值和更广泛的影响审查标准。