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The Role of Gromov Hyperbolicity and Besov Spaces in Quasiconformal Analysis

格罗莫夫双曲性和贝索夫空间在拟共形分析中的作用

基本信息

批准号：
2054960
负责人：
Nageswari Shanmugalingam
金额：
$ 30万
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：
2021
资助国家：
美国
起止时间：
2021-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054960&HistoricalAwards=false
关键词：
Role Gromov Hyperbolicity Besov Spaces

项目摘要

This research project concerns the development of mathematical tools for analysis on metric measure spaces (spaces with well-defined notions of distance and volume), inspired by basic tools used in ordinary Euclidean space. The work has implications for many fields of mathematics, including dynamical systems, harmonic analysis, and partial differential equations. One useful measurement in geometric analysis is the so-called Besov energy. This project will explore links between the nonlocal Besov-type energy in a compact metric measure space and local energy encoded in the space, as measured through a well-connected super-space that sees the compact space as its boundary. The work aims to use these links to explore how non-local energies are transformed by well-regulated changes in the metric, or distance. Parts of this project will be done in collaboration with graduate students and other early career mathematicians, thus also providing training of the future STEM workforce. In this project, the principal investigator will develop the potential theoretic tools related to Besov energies by using Gromov hyperbolic fillings of the compact metric measure space so that the compact space is realized as the boundary of the Gromov hyperbolic space. Such a Gromov hyperbolic space is of controlled geometry, and by exploiting this control, the principal investigator will seek to gain control of the nonlocal energies and to obtain information about how these energies are transformed by quasisymmetric maps. The project will also augment the potential theoretic properties of Besov energy by developing the perspective of fractional p-Laplacian (nonlinear versions of the fractional Laplacian).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.

这项研究项目涉及在普通欧氏空间中使用的基本工具的启发下，开发用于分析度量度量空间(具有明确定义的距离和体积概念的空间)的数学工具。这项工作涉及到许多数学领域，包括动力系统、调和分析和偏微分方程式。几何分析中一种有用的测量方法是所谓的贝索夫能量。这个项目将探索紧致度量空间中的非局域Besov型能量与该空间中编码的局域能量之间的联系，通过一个以紧致空间为边界的连接良好的超空间来测量。这项工作旨在利用这些联系来探索非局部能量是如何通过规范的度量或距离变化来转换的。该项目的一部分将与研究生和其他职业生涯早期的数学家合作完成，从而也为未来的STEM劳动力提供培训。在这个项目中，主要研究人员将利用紧度量度量空间的Gromov双曲填充来开发与Besov能量相关的潜在理论工具，从而将紧致空间实现为Gromov双曲空间的边界。这样的格罗莫夫双曲空间具有受控几何，通过利用这种控制，主要研究者将寻求获得对非局域能量的控制，并获得关于这些能量如何通过准对称映射进行转换的信息。该项目还将通过发展分数p-拉普拉斯(分数拉普拉斯的非线性版本)的观点来增强贝索夫能量的潜在理论属性。该奖项反映了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。