CAREER: Geometric Aspects of Isoperimetric and Sobolev-type Inequalities

职业：等周和索博列夫型不等式的几何方面

• 批准号: 2340195
• 负责人: Robin Neumayer
• 金额: $ 50.7万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-09-01 至 2029-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2340195&HistoricalAwards=false
• 关键词: CAREER; Geometric; Aspects; Isoperimetric; Sobolev

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.

等周和索博列夫型不等式在分析和几何的数学领域中发挥着核心作用，并提供了描述各种工程问题和物理系统的最佳配置的数学框架。例如，一个等周型不等式给出了数学证明：如果金属杆的横截面是圆形的，那么它对扭转力的抵抗力最强。该项目研究了与等周和索博列夫型不等式相关的几个几何问题，包括以下内容：如果仅测量给定杆的抗扭转力，则可以恢复多少有关杆横截面形状的几何信息？这类问题在数学的其他分支中具有强大的、有时是意想不到的应用。该项目的一个基本部分是两个支柱的教育部分。首先，首席研究员 (PI) 将在卡内基梅隆大学为分析领域的女性组织一次研讨会，通过迷你课程、研究讲座和为初级研究人员提供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项目，提供垂直整合的数学和专业发展，并为大流行后重建两个院系之间的桥梁提供及时的机会。 该项目植根于变分法和偏微分方程，开发了等周和索博列夫型不等式的新颖应用，以解决分析和几何中的核心问题，并探索几何与最优常数和不等式的等式情况之间的相互作用。 PI将开发一个框架，用于在偏微分方程驱动的形函数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德域中等周问题的局部最小化的定量描述，扩展了该领域的工具箱；通过双约束Sobolev型最小化问题，建立具有规定区域的恒定平均曲率边界的恒定标量曲率共角度量；并证明具有标量曲率下界的黎曼流形的 epsilon 正则定理的局部版本，为奇点分析铺平了道路。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评估进行评估，被认为值得支持影响审查标准。