CAREER: Geometric Aspects of Isoperimetric and Sobolev-type Inequalities

等周不等式和Sobolev型不等式在分析和几何的数学领域中起着核心作用，并提供了一个数学框架来描述各种工程问题和物理系统的最佳配置。例如，一个等周型不等式给出了数学证明，如果金属棒的横截面是圆形的，则它对扭转力的抵抗力最强。这个项目调查几个几何问题有关的等周和Sobolev型不等式，包括以下内容：如果一个只有测量一个给定的杆的阻力扭转力，有多少几何信息可以恢复有关形状的杆的横截面？这种类型的问题在数学的其他分支中有着强大的，有时是意想不到的应用。该项目的一个基本部分是一个双支柱教育组成部分。首先，首席研究员（PI）将在卡内基梅隆大学组织一个妇女分析讲习班，通过小型课程，研究讲座和初级研究人员的机会，将研究和教育结合起来。其次，PI将在卡内基梅隆大学和邻近的匹兹堡大学之间发起一个联合定向阅读计划，提供垂直整合的数学和专业发展，并及时提供机会在疫情后重建两个部门之间的桥梁。 这个项目，植根于变分和偏微分方程的微积分，开发等周和Sobolev型不等式的新应用，以攻击分析和几何中的中心问题，并探讨几何和最佳常数之间的相互作用和平等的情况下的不平等。PI将开发一个框架，用于在偏微分方程驱动的形状泛函的等周问题的背景下证明一种新型的广泛适用的定量稳定性估计;证明黎曼流形和欧几里得域中等周问题的局部极小值的定量描述，扩展该领域的工具箱;通过双约束Sobolev型极小化问题，建立了具有给定区域的常平均曲率边界的常数量曲率共形度量;证明了黎曼流形的ε正则性定理的局部化形式，该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的学术价值和更广泛的影响审查标准。

Isoperimetric and Sobolev-type inequalities play a central role in the mathematical fields of analysis and geometry and provide a mathematical framework to describe optimal configurations for various engineering problems and physical systems. For example, one isoperimetric-type inequality gives the mathematical justification that a metal rod will have the strongest resistance to twisting forces if its cross-sections are circular. This project investigates several geometric questions related to isoperimetric and Sobolev-type inequalities, including the following: if one only has measurements of a given rod's resistance to twisting forces, how much geometric information can be recovered about the shape of the rod's cross-sections? Questions of this type have powerful and sometimes unexpected applications in other branches of mathematics. A fundamental part of the project is a two-pillared educational component. First, the Principal Investigator (PI) will organize a workshop for women in analysis at Carnegie Mellon University, integrating research and education through mini-courses, research talks, and opportunities for junior researchers. Second, the PI will initiate a joint Directed Reading Program between Carnegie Mellon University and the neighboring University of Pittsburgh, delivering vertically integrated mathematical and professional development and a timely opportunity to rebuild bridges between the two departments post-pandemic. This project, rooted in the calculus of variations and partial differential equations, develops novel applications of isoperimetric and Sobolev-type inequalities to attack central questions in analysis and geometry, and explores the interplay between geometry and optimal constants and equality cases for the inequalities. The PI will develop a framework for proving a new type of broadly applicable quantitative stability estimate in the context of isoperimetric problems for shape functionals driven by partial differential equations; prove quantitative descriptions of local minimizers of isoperimetric problems in Riemannian manifolds and Euclidean domains, expanding the toolbox in this area; via doubly-constrained Sobolev-type minimization problems, build constant scalar curvature conformal metrics with constant mean curvature boundary of prescribed area; and prove localized versions of epsilon-regularity theorems for Riemannian manifolds with lower bounds on scalar curvature, paving the way for the analysis of singularities.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.