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

几何测量理论提供了一个分析工具来检测高维数据集中的隐藏结构，并描述非光滑现象，如肥皂泡簇中的角和奇点的形成。术语“度量”指的是长度、面积和体积的抽象概括，它为每个数学集分配了一个大小的概念。尽管测量在当代数学中的普及，但目前只有少数工具可用于分析具有低规律性的制度中的测量。提出的调查旨在发展新的和稳健的定量方法来研究一般集合和测度的几何在缺乏传统的简化假设。该项目将探索这些工具和方法在具有粗糙边界域的调和函数和温度（热方程的解）分析中的应用，无论是在理想环境中还是在非均匀介质中。该项目还将为康涅狄格大学的研究生研究助理以及在几何测量理论、谐波分析和偏微分方程方面工作的访问博士生提供培训。该项目将发展四个相互关联的研究线索：集合的精细几何、测度的结构和规则性，以及偏微分方程的解。第一行探究的是关于无限维Banach空间中可整流曲线子集的问题，其具体目标是求解非hilbert环境下的分析师旅行商问题。第二个问题涉及平面Lipschitz图像的参数化问题和2-可校正Radon测度的分类。额外的工作将集中在Hölder连续函数图所支持的测度的结构上。第三个研究方向将涉及改进一般域上谐波和热量测度的Hausdorff维的上界，以及静态和时间相关情况之间的关系。研究计划的最后一部分将面对当代的挑战，并在具有两个或多个相位的谐波和椭圆测度的非变分自由边界问题理论中发起新的探索。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.