Rectifiability and Fine Geometry of Sets, Radon Measures, Harmonic Functions, and Temperatures

集合的可整流性和精细几何、氡气测量、调和函数和温度

• 批准号: 2154047
• 负责人: Matthew Badger
• 金额: $ 27.85万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154047&HistoricalAwards=false
• 关键词: Rectifiability; Fine; Geometry; Sets; Radon

Geometric measure theory provides an analytical toolkit to detect hidden structure in high-dimensional data sets and to describe non-smooth phenomena such as the formation of corners and singularities in soap bubble clusters. The term ‘measure’ refers to an abstract generalization of length, area, and volume, which assigns a notion of size to each mathematical set. The pervasiveness of measures within contemporary mathematics notwithstanding, there are presently only a few tools available to analyze measures in regimes with low regularity. The proposed investigation seeks to develop novel and robust quantitative methods to study the geometry of general sets and measures in the absence of traditional simplifying hypotheses. The project will explore applications of these tools and methods to the analysis of harmonic functions and temperatures (solutions to the heat equation) in domains with rough boundary, both in ideal settings and inside non-homogenous media. The project will also provide training to graduate research assistants at the University of Connecticut and to visiting Ph.D. students working in geometric measure theory, harmonic analysis, and partial differential equations.The project will develop four interrelated threads of research on the fine geometry of sets, the structure and regularity of measures, and the solutions of partial differential equations. The first line of inquiry pursues questions about subsets of rectifiable curves in infinite dimensional Banach spaces, with a concrete goal of solving the Analyst's Traveling Salesman Problem in a non-Hilbert setting. A second line of inquiry concerns the parameterization problem for Lipschitz images of the plane and the classification of 2-rectifiable Radon measures. Additional work will focus on the structure of measures supported on the graphs of Hölder continuous functions. A third direction of study will involve improved upper bounds on the Hausdorff dimension of harmonic and caloric measures on general domains, along with the relationship between the static and time-dependent cases. The final strand of the research program will confront contemporary challenges and initiate new inquiries in the theory of non-variational free boundary problems for harmonic and elliptic measures with two or more phases.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.

几何测量理论提供了一个分析工具包，可在高维数据集中检测隐藏的结构，并描述非平滑现象，例如肥皂泡簇中的角落和奇异性的形成。 “度量”一词是指长度，面积和音量的抽象概括，该概括为每个数学集合分配了大小的概念。尽管如此，当代数学中的措施的普遍性目前只有一些工具可用于分析规律性较低的制度的措施。拟议的调查旨在开发新颖和鲁棒的定量方法，以研究一般集合和度量的几何形状，而没有传统的简化假设。该项目将探索这些工具和方法在具有粗糙边界的域函数和温度（对热方程的解决方案）分析的应用，无论是在理想的设置还是内部的非叙源介质中。该项目还将为康涅狄格大学和访问博士学位的研究生研究助理提供培训。从几何测量理论，谐波分析和部分微分方程中工作的学生。该项目将开发四个相互关联的研究线程，以了解集合的精细几何形状，测量的结构和规则性以及部分微分方程的解决方案。第一线询问提出了有关无限尺寸Banach空间中可矫正曲线子集的问题，其目的是在非希尔伯特环境中解决分析师的旅行人员问题。第二线查询涉及平面Lipschitz图像的参数化问题以及2个可调的rad测量结果的分类。额外的工作将集中在霍尔德持续功能图上支持的测量结构上。第三个研究方向将涉及在通用域的谐波和热量测量的Hausdorff维度上的上限，以及静态和时间依赖性病例之间的关系。研究计划的最终链将面临当代挑战，并在非不同的自由边界问题中提起新的询问，并具有两个或多个阶段的谐波和椭圆度措施。这项奖项反映了NSF的法定任务，并通过使用该基金会的知识分子和更广泛的影响来评估NSF的法定任务，并被视为珍贵的支持。