Geometry of Sets and Measures

几何测量理论是20世纪20年代和30年代发展起来的一个数学领域，它是出于描述非光滑现象的实际需要而发展起来的，例如肥皂泡沫团中角落的形成。术语“测量”指的是长度、面积或体积的抽象概括，它为每个数学集合分配一个尺寸值。几何测量理论的传统渠道，如变分和几何分析，在最近几十年已经扩展到包括偏微分方程组和调和分析。几何测量理论在不同分析领域的广泛应用和目前的使用，证明了它的继续发展是合理的。关于集合和测度几何的研究旨在开发新的技术，以扩展几何测度论为相邻领域的分析和几何研究人员提供的工具箱。本项目主要研究欧氏空间中关于集合和测度几何的两组问题。第一组问题涉及几何测度论的核心研究对象之一--可纠错测度。具体地说，这些问题的目的是在没有过去假定的长期规律性假设的情况下，增进对可纠正措施的理解。主要方法需要采用琼斯和大卫-塞姆斯在20世纪90年代开发的定量技术来研究衡量标准的定性可纠正性。第二组问题旨在研究Reifenberg型集合的几何，Reifenberg型集合是可以在所有位置近似并可由一种或多种类型的模型集缩放的集合。出现Reifenberg型集合的实例包括几何极小化问题和椭圆型偏微分方程组的自由边界问题。这项研究的一般目标是确定理想模型(光滑设置)中的问题的解在什么情况下以及在多大程度上在受控扰动(弱正则性)下保持良好的性质。

Geometric measure theory is a field of mathematics that developed starting in the 1920s and 1930s, growing out of a practical need to describe nonsmooth phenomena such as the formation of corners in soap bubble clusters. The term "measure" refers to an abstract generalization of length, area, or volume, which assigns a size value to every mathematical set. Traditional outlets for geometric measure theory, such as the calculus of variations and geometric analysis, have expanded in recent decades to include partial differential equations and harmonic analysis. The widespread utility and current use of geometric measure theory in different areas of analysis justifies its continued development. The proposed investigation on the geometry of sets and measures seeks to develop new techniques that will expand the toolbox that geometric measure theory provides for researchers in adjacent areas in analysis and geometry.This project focuses on two groups of questions about the geometry of sets and measures in Euclidean space. The first group of questions concerns rectifiable measures, one of the core objects of study in geometric measure theory. Specifically, these questions are aimed at increased understanding of rectifiable measures in the absence of a standing regularity assumption that has been assumed in the past. The main approach entails adapting quantitative techniques developed in the 1990s by Jones and David-Semmes to study the qualitative rectifiability of measures. The second group of questions are designed to examine the geometry of Reifenberg-type sets, which are sets that can be approximated at all locations and scales by one or more kinds of model sets. Instances where Reifenberg-type sets occur include geometric minimization problems and free boundary problems for elliptic partial differential equations. A general goal of this inquiry is to determine in what situations and to what extent good properties of solutions to problems in ideal models (smooth settings) persist under controlled perturbation (weak regularity).