Exploring Large-Scale Geometry via Local and Nonlocal Potential Theory

通过局部和非局部势理论探索大尺度几何

基本信息

批准号：
2348748
负责人：
Nageswari Shanmugalingam
金额：
$ 34.39万
依托单位：
University of Cincinnati Main Campus
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2024
资助国家：
美国
起止时间：
2024-09-01 至 2027-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348748&HistoricalAwards=false
关键词：
Exploring Large Scale Geometry via

项目摘要

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.

该项目将开发用于分析措施空间的新数学工具，即装备的距离（例如欧几里得空间）的距离和体积 - 重点是（与欧几里得空间不同）缺乏平滑结构的环形空间。平滑空间是一个重要的研究领域，在数学和物理科学机能，神经生理学和分形几何形状中，PI将使用局部和非局部能量的数学工具在这些空间中研究这些空间小规模的振荡，而非局部能量在长距离上进行了变化。地下学生。称为Sobolev的功能。通过在调节本身上与局部能量的小规模联系，在该地区边界上的非局部能量的大规模几何形状。使用基金会的智力优点和更广泛的影响审查标准进行评估。