Stability of Functional and Geometric Inequalities and Applications

泛函和几何不等式的研究在现代数学中起着举足轻重的作用。几何不平等的一个例子可以追溯到古希腊人，等周不等论是指在所有固定体积的形状中，球的周长最小。这类不等式的稳定性是描述几乎平衡状态的行为的一个性质。近年来，稳定性的话题引起了人们的极大兴趣，这在很大程度上是因为它在物理模型中的令人兴奋的应用。这个项目分析了某些泛函和几何不等式的稳定性性质，并将它们作为工具来解决跨越不同数学领域的问题，并在物理和材料科学中有不同的应用。给定一个可以刻画所有等式情形的不等式，稳定性问题如下：假设一个函数(或集合)几乎达到该不等式中的相等。那么，在适当的意义上，它是否接近于平等的情况？在这个项目中，稳定性估计将在不同的环境中使用，包括研究具有标量曲率下界的黎曼流形的极限空间的正则性，谱形状优化问题的正则性，以及原子核的液滴模型中的能量最小化的性质。该项目汇集了几何分析、偏微分方程式和变分法的思想，在技术和应用方面都是如此。这个奖项反映了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。

The study of functional and geometric inequalities plays a pivotal role in modern mathematics. An example of a geometric inequality, with a history dating back to the ancient Greeks, is the isoperimetric inequality, which states that a ball has the least perimeter among all shapes of a fixed volume. The stability of such inequalities is a property that describes behavior of almost-equilibrium states. The topic of stability has generated substantial interest in recent years, due in large part to its exciting applications to physical models. This project analyzes the stability properties of certain functional and geometric inequalities, and applies them as a tool to address problems across various mathematical fields and with diverse applications in physics and materials science.Given an inequality for which all equality cases are characterized, the question of stability is the following: suppose a function (or set) almost achieves equality in the inequality. Then is it close, in a suitable sense, to an equality case? In this project, stability estimates will be employed in varied contexts, including the study of regularity properties for limit spaces of Riemannian manifolds with lower bounds on scalar curvature, regularity in spectral shape optimization problems, and properties of energy minimizers in the liquid drop model for atomic nuclei. The project brings together ideas across 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.