Stability and Regularity: From Analysis to Geometry

该项目专注于分析和几何交叉的几个问题，这些问题由一个中心主题联合起来：使用函数和几何不等式作为探索流形或域几何的工具。这些不等式被用来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周不等式，它从数学上描述了为什么肥皂泡呈球形。本项目为研究生提供研究训练机会。作为该项目的一部分获得的结果将广泛传播给科学界。本项目主要研究不同情况下的稳定性和正则性估计，包括具有标量曲率界的黎曼流形的正则性及其拉普拉斯谱的稳定性；用Alt-Caffarelli-Friedman单调性公式计算可整流性结果；以及保形几何中Yamabe问题的定量估计。这项研究将在技术和应用方面汇集几何分析、偏微分方程和变分学的思想。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.