The Geometry of Measures and Analytic Properties of Associated Operators

点击翻译按钮获取中文摘要

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.