Geometry of Sets and Measures

集合和测度的几何

• 批准号: 1500382
• 负责人: Matthew Badger
• 金额: $ 12万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500382&HistoricalAwards=false
• 关键词: Geometry; Sets; Measures

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).

几何测量理论是从1920年代和1930年代开始发展的数学领域，它源于描述非平滑现象的实际需求，例如肥皂泡簇中的角落形成。 术语“度量”是指长度，面积或音量的抽象概括，该概括为每个数学集为大小值分配。 几何理论的传统渠道，例如变化和几何分析的计算，在近几十年来扩展，包括部分微分方程和谐波分析。在不同分析领域的几何措施理论的广泛效用和当前使用的目前使用证明了其持续发展。 对集合和措施的几何形状进行的拟议调查旨在开发新技术，该技术将扩大工具箱的几何测量理论为分析和几何形状的相邻领域的研究人员提供的工具箱。本项目侧重于有关欧几里得空间中集合和措施的几组问题的两组问题。 第一组问题涉及可纠正的措施，这是几何测量理论中研究的核心对象之一。 具体而言，这些问题的目的是在没有过去假定的常规规律性假设的情况下，增加对可纠正措施的理解。 主要方法需要调整琼斯和戴维·塞姆斯（Jones and David-Semmes）于1990年代开发的定量技术，以研究措施的定性重新可及性。 第二组问题旨在检查Reifenberg型集合的几何形状，这些集合可以通过一种或多种模型集在所有位置和尺度上近似。 发生Reifenberg型集合的实例包括几何最小化问题和椭圆形偏微分方程的自由边界问题。 此询问的一个总体目标是确定理想模型（平滑设置）在受控扰动（弱规则性）下，解决方案解决方案的良好特性以及在何种程度上的良好特性。