The Geometry of Measures and Analytic Properties of Associated Operators

这个项目涉及两个基本物理问题背后的一些数学研究。考虑的第一个问题是，从与物体相关的力场（例如其引力场）的信息可以确定物体的几何形状到什么程度？势理论中的这类逆问题有着丰富的历史，但目前回答这一问题所需的数学工具还不发达。调查将特别集中在什么可以说，如果一个人只知道场有有限的大小，这是特别感兴趣的应用。第二个要探讨的问题是，基于某些经验测量，如何确定随时间演变的随机函数的渐近（或长期）行为，以及在多大程度上的准确性？更具体地说，主要研究者提出了几个关于测度的几何性质和关联算子的正则性之间的关系的问题。感兴趣的主要问题如下：从与测度相关的奇异积分算子具有良好正则性的知识中可以推导出什么？在这种情况下，度量是否具有分形结构，或者它的支持必须包含在适当维数的（可数的）Lipschitz子流形中？解决这个问题的尝试导致了最近在变分演算、自由边界问题的研究和调和测度的几何中得到应用的理论。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量的几何结构的桥梁。研究的第二个主题是理解平稳高斯过程的长期行为，给出其光谱测量的信息，建立在涉及首席研究员的最近工作的基础上。该奖项反映了美国国家科学基金会的法定使命，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。

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.