CAREER: Geometric Aspects of Isoperimetric and Sobolev-type Inequalities

等周和Sobolev型不等式在数学分析和几何领域中发挥着核心作用，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优配置。例如，一个等周型不等式给出了一个数学证明，即如果金属棒的横截面是圆形的，那么它将具有最强的抗扭力。这个项目研究了几个与等周型和索博列夫型不等式有关的几何问题，包括以下问题：如果只有给定杆件的扭力阻力的测量，可以恢复多少关于杆件横截面形状的几何信息？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的教育部分。首先，首席调查员(PI)将在卡内基梅隆大学为妇女举办一次分析讲习班，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将启动卡内基梅隆大学和邻近的匹兹堡大学之间的联合定向阅读计划，提供垂直整合的数学和专业发展，并提供一个及时的机会，在大流行后重建两个系之间的桥梁。这个项目植根于变分和偏微分方程组，开发了等周和Sobolev型不等式的新应用，以解决分析和几何中的中心问题，并探索几何和最优常数之间的相互作用以及不等式的等式情况。PI将在偏微分方程驱动的形状泛函的等周问题背景下发展一种新的广泛适用的定量稳定性估计的框架；证明黎曼流形和欧几里德区域中等周问题的局部极小点的定量描述，扩展该领域的工具箱；通过双约束Sobolev型极小化问题，建立具有指定面积的常平均曲率边界的常标量曲率共形度量；并证明了具有标量曲率下界的黎曼流形的局部版本的epsilon正则性定理，为奇点分析铺平了道路。这一奖项反映了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。

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.