The Geometry of Measures and Analytic Properties of Associated Operators

这个项目涉及对两个基本物理问题背后的一些数学问题的研究。考虑的第一个问题是，从与物体相关的力场(例如，它的引力场)的信息可以在多大程度上确定物体的几何形状？位势理论中的这种反问题有着丰富的历史，但目前回答这个问题所需的数学工具还不够发达。调查的重点将特别放在如果人们只知道磁场具有有界震级的情况下可以说些什么，这在应用中特别有意义。第二个要探索的问题是，基于某些经验测量，人们可以如何确定随时间演变的随机函数的渐近(或长期)行为，以及精确度达到什么程度？更具体地说，主要研究人员建议研究有关度量的几何性质和相关算子的正则性之间的关系的几个问题。我们感兴趣的主要问题是：从与之相关的奇异积分算子具有良好的正则性这一知识中可以得出关于度量的什么结论？在这种情况下，度量是否具有分形结构，或者它的支撑点是否必须包含在(可数个)适当维度的Lipschitz子流形中？解决这个问题的尝试导致了最近在变分、自由边界问题的研究和调和测量的几何中找到应用的理论。主要研究人员打算进一步开发工具，作为从奇异积分算子的分析条件到度量的几何结构的桥梁。第二个研究主题涉及在给定关于其光谱测量的信息的情况下，理解平稳高斯过程的长期行为，建立在涉及主要研究人员的最近工作的基础上。这一奖项反映了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。

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.