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

几何测度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，如肥皂泡簇中的角点和奇点的形成。术语“测量”是指长度、面积和体积的抽象概括，它为每个数学集合赋予了大小的概念。尽管现代数学中的措施普遍存在，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措施。拟议的调查旨在开发新的和强大的定量方法来研究几何的一般集合和措施，在没有传统的简化假设。该项目将探索这些工具和方法在理想环境和非均匀介质中分析具有粗糙边界的域中的调和函数和温度（热方程的解）的应用。该项目还将为康涅狄格大学的研究生研究助理和访问博士提供培训。本计画将发展四个相互关联的研究主题，分别是集合的精细几何、测度的结构与规则性，以及偏微分方程的解。调查的第一条线追求的问题，在无限维Banach空间的可求长曲线的子集，与具体的目标，解决分析师的旅行推销员问题在非希尔伯特设置。第二条调查线涉及Lipschitz图像的平面和分类的参数化问题的2-可求Radon措施。更多的工作将集中在结构上的措施支持的图表的Hölder连续功能。研究的第三个方向将涉及改进的上界一般域上的调和和热量措施的Hausdorff维数，沿着与静态和时间依赖的情况下之间的关系。该研究计划的最后一个环节将面对当代的挑战，并在两个或多个阶段的谐波和椭圆措施的非变分自由边界问题的理论中发起新的调查。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.