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.

等级和Sobolev型不平等在分析和几何学的数学领域中起着核心作用，并提供了数学框架，以描述针对各种工程问题和物理系统的最佳配置。例如，一种等术类型的不平等，给出了数学上的理由，即如果金属杆的横截面是圆形的，则金属棒对扭曲力的阻力最强。该项目研究了几种几何问题与等术和Sobolev型不平等相关，包括以下几何问题：如果仅对给定杆对扭曲力的抗性进行测量，那么可以在杆的横截面的形状上恢复多少几何信息？这种类型的问题在数学的其他分支中具有强大的，有时是意外的应用程序。该项目的基本部分是两柱教育组成部分。首先，首席研究员（PI）将在卡内基·梅隆大学（Carnegie Mellon University）为妇女分析的妇女研讨会，通过迷你库，研究演讲和初级研究人员的机会整合研究和教育。其次，PI将在卡内基·梅隆大学（Carnegie Mellon University）和邻近的匹兹堡大学（University of Pittsburgh）之间启动一项联合定向的阅读计划，并提供垂直整合的数学和专业发展，并及时进行重建大桥梁之间的桥梁。 该项目植根于变化和部分微分方程的计算，开发了等等和Sobolev-Type的不平等的新颖应用，以攻击分析和几何学中的中心问题，并探索几何和最佳常数与不平等现象之间的相互作用。 PI将开发一个框架，以证明在由部分微分方程驱动的形状函数的相关性的背景下，在形状函数的背景下证明了一种新型的广泛的定量稳定性估计；证明了Riemannian歧管和欧几里得域中等次问题的局部最小化的定量描述，从而扩展了该区域的工具箱；通过双重约束的Sobolev型最小化问题，构建具有规定区域的恒定平均曲率边界的恒定标态曲率共形度量；并在标量曲率上具有下限的Riemannian流形的Epsilon-Regullity定理的局部版本，为分析奇异性铺平了道路。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和广泛的影响来评估Criteria的智力功能和广泛影响。