Exploring Large-Scale Geometry via Local and Nonlocal Potential Theory

这个项目将开发新的数学工具来分析度量空间--即配备了距离和体积概念的空间(如欧几里得空间)--重点关注(不同于欧几里得空间)缺乏光滑结构的度量空间。非光滑空间的分析是一个重要的研究领域，在数学和物理科学中有着广泛的应用，包括流体力学、神经生理学和分形几何。PI将使用局域和非局域能量的数学工具来研究这些空间中对象的大规模几何行为。给定一个测量物理现象的函数，例如温度或动量，局部能量测量该函数的附近或小范围的振荡，而非局部能量测量其在长距离上的变化。这项工作的一个主要目标是开发急需的数学工具来分析非局域能量。该项目还将通过合作研究项目、有效数学交流的指导以及与本科生进行研究互动的机会，加强对研究生和博士后学者的专业培训。本课题的主要研究对象是缺乏光滑结构的度量度量空间。环境空间的有限维用支持加倍Radon测度的性质来表示。在这种情况下，函数的邻近或渐近振荡是使用上梯度来测量的，上梯度是函数导数的可行替代品，并且局部能量与对象上的函数集合相关联，称为索博列夫函数。大尺度的变化能量与被称为函数的贝索夫空间的集合有关。最近的研究揭示了度量度量空间中区域上的局部能量与区域边界上的非局部能量之间的联系。该项目将利用这种联系，通过将非局部能量与区域本身的局部能量的小尺度行为联系起来，来探索区域边界上的非局部能量的大规模几何形状。特别是Dirichlet类型的边值问题和Neumann类型的边值问题之间的联系将被调查。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project will develop new mathematical tools for the analysis of metric measure spaces – that is, spaces equipped (like Euclidean space) with notions of distance and volume – with a focus on metric measure spaces that (unlike Euclidean space) lack smooth structure. The analysis of non-smooth spaces is a vital area of research with diverse applications across the mathematical and physical sciences, including fluid mechanics, neurophysiology, and fractal geometry. The PI will investigate the large-scale geometric behavior of objects in these spaces using the mathematical tools of local and nonlocal energies. Given a function measuring a physical phenomenon, such as temperature or momentum, local energies measure the function’s nearby or small-scale oscillations, while nonlocal energies measure its variations over long distances. A primary goal of this work is to develop much-needed mathematical tools for analyzing nonlocal energies. The project will also enhance the professional training of graduate students and postdoctoral scholars, through collaborative research projects, instruction in effective mathematical communication, and opportunities for research interactions with undergraduate students. The primary objects of study in this project are represented as metric measure spaces that lack smooth structure. The finite dimensionality of the ambient space is represented by the property of supporting a doubling Radon measure. In such a setting, nearby or asymptotic oscillation of a function is measured using upper gradients, which are viable substitutes for the derivative of a function, and the local energy is associated with the collection of functions on the object, called Sobolev functions. The large-scale variation energy is associated with the collection called Besov space of functions. Recent research has uncovered a connection between local energies on a region in a metric measure space and nonlocal energies on the boundary of the region. The project will leverage this connection to explore the large-scale geometry of nonlocal energies on the boundary of the region by linking them with small-scale behavior of local energies on the region itself. In particular, connections between Dirichlet-type boundary value problems and Neumann-type boundary value problems will be investigated.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.