Stability and Regularity: From Analysis to Geometry

这个项目的重点是几个问题的分析和几何的十字路口是统一的一个中心主题：使用功能和几何不等式作为工具，探索几何形状的流形或域。这类不等式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周不等式，它在数学上描述了为什么肥皂泡呈球形。该项目为研究生提供研究培训机会。作为该项目的一部分，所获得的结果将广泛传播给科学界。该项目集中于不同设置下的稳定性和正则性估计，包括具有标量曲率边界的黎曼流形的正则性及其拉普拉斯谱的稳定性;使用Alt-Caffarelli-Friedman单调性公式的可求正性结果;以及共形几何中Yamabe问题的定量估计。该研究将汇集来自几何分析、偏微分方程和变分法的技术和应用方面的想法。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.