The Geometry of Measures and Regularity of Associated Operators

该项目涉及以下基本物理问题背后的一些数学研究：在多大程度上可以从与物体相关的力场（例如，其引力场）的信息确定物体的几何形状？ 这种反问题在位势理论中有着丰富的历史，但正确回答这个问题所需的数学工具，特别是在将力场与物体的质量分布联系起来的算子对长程相互作用敏感的情况下，目前还不发达。 在这个项目中，主要研究者将开发工具，以进一步了解这个问题，特别是集中在什么可以说，如果一个人只知道，该领域有界的大小。更具体地说，该项目主要关注的是一个措施的几何形状和相关的微分或奇异积分算子的正则性之间的关系。 这是一个问题，吸引了数学家自从柯西和Riesz变换被引入作为工具来研究的行为分析和调和函数，分别。 一个综合的方法来解决这些问题，提出了通过无反射措施的研究。 这种方法最近产生了一些新的结果，并可能解决一些开放的问题，特别是那些关于光滑的支持措施，有界Riesz变换。 在这里，新的工具，定量几何和高阶偏微分方程需要开发，以取得进展。此外，首席研究员试图建立在最近的创新理论的准线性微分方程，考虑类似的问题，为广泛的非线性微分算子，其中没有积分表示。

This project concerns a study of the some of the mathematics behind the following basic physical question: 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 properly answer this question, especially in the case when the operator relating the force field to the mass distribution of the body is sensitive to long-range interactions, are currently underdeveloped. In this project, the principal investigator will develop tools to further understand this problem, concentrating especially on what can be said if one knows only that the field has bounded magnitude.More specifically, the project primarily concerns the relationship between the geometry of a measure and the regularity of an associated differential or singular integral operator. This is is a question that has attracted mathematicians ever since the Cauchy and Riesz transforms were introduced as tools to study the behavior of analytic and harmonic functions, respectively. An integrated approach to such problems is proposed that goes through the study of reflectionless measures. This approach has recently yielded several new results and could potentially address a number of open problems, especially those concerning the smoothness of the support of a measure that has a bounded Riesz transform. Here new tools in quantitative geometry and higher order partial differential equations need to be developed in order to make progress. Furthermore, the principal investigator seeks to build upon recent innovations in the theory of quasilinear differential equations to consider analogous problems for a wide range of nonlinear differential operators, where no integral representation is available.