Stability and Regularity: From Analysis to Geometry

稳定性和规律性：从分析到几何

• 批准号: 2155054
• 负责人: Robin Neumayer
• 金额: $ 18.8万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2155054&HistoricalAwards=false
• 关键词: Stability; Regularity; Analysis; Geometry

This project focuses on several problems at the crossroads of analysis and geometry that are united by a central theme: the use of functional and geometric inequalities as a tool to explore the geometry of a manifold or domain. These kinds of inequalities are used to describe ground states for physical systems, for instance, in acoustics and materials science; a classical example is the isoperimetric inequality, which mathematically describes why soap bubbles take a spherical shape. The project provides research training opportunities for graduate students. The results obtained as part of the project will be broadly disseminated to the scientific community.The project centers on stability and regularity estimates in different settings, including regularity of Riemannian manifolds with scalar curvature bounds and stability properties of their Laplace spectra; rectifiability results using the Alt-Caffarelli-Friedman monotonicity formula; and quantitative estimates for the Yamabe problem in conformal geometry. The research will bring together ideas from geometric analysis, partial differential equations and the calculus of variations, in terms of both techniques and applications.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.

该项目着重于分析和几何形状的十字路口的几个问题，这些问题由中心主题结合在一起：使用功能和几何不平等作为探索歧管或域的几何形状的工具。这些不平等用于描述物理系统的基态，例如在声学和材料科学中；一个经典的例子是等等的不平等，它在数学上描述了为什么肥皂泡具有球形形状。该项目为研究生提供了研究培训机会。作为项目一部分获得的结果将大致传播到科学界。该项目集中于不同环境中的稳定性和规律性估计，包括具有标态曲率界限的Riemannian歧管的规律性以及其拉普拉斯光谱的稳定性；使用Alt-caffarelli-Friedman单调性公式进行可重新讨论结果；以及保形几何形状中Yamabe问题的定量估计。这项研究将从几何分析，部分微分方程和变化的计算中汇集在一起​​，从技术和应用方面。该奖项反映了NSF的法定任务，并认为值得通过基金会的知识分子优点和更广泛的影响来通过评估来获得支持。