Mass Rigidity and Curvature Problems in Mathematical Relativity

数学相对论中的质量刚度和曲率问题

基本信息

批准号：
2005588
负责人：
Lan-Hsuan Huang
金额：
$ 25.03万
依托单位：
University of Connecticut
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005588&HistoricalAwards=false
关键词：
Mass Rigidity Curvature Problems Mathematical

项目摘要

Einstein’s theory of gravity has been a strong driving force for the modern development in several branches of mathematics. Among its profound implications and wide applications, the theory of gravity successfully describes the shape of our universe and predicts celestial objects that were not known to exist, such as black holes. Over the past few decades, remarkable progress using advanced techniques in geometry and analysis has been made to resolve fundamental questions in general relativity, which has also led to the astonishing realization that some celestial objects are governed by the same mathematical principles as daily life objects, such as soap films. This project employs frontier developments in mathematics to investigate those interconnections and to further advance our understanding on the geometric structures of mathematical models of our universe. The project also incorporates mentoring and educational activities to promote geometry, analysis, and their interrelations with other STEM disciplines, to broader communities and to the society. Because of rapid advancement in geometric analysis in recent years, several longstanding questions in general relativity have been largely resolved. A prominent example is the resolution to the positive mass conjecture, including recent work on the spacetime positive mass theorem. At the same time, those resolutions in general relativity have motivated the development of new and unexpected techniques in geometry and analysis. The goal of this project is to interconnect general relativity with neighboring areas in geometry and analysis where some of innovative techniques can be further developed and applied. The scope of the project is to analyze curvature and geometric structure of initial data sets and their spacetime development, arising from mass minimization problems related to quasi-local mass, to investigate scalar curvature problems for compact manifolds and classification of static manifolds, and to characterize Einstein manifolds from the aspect of hyperbolic and conformal geometry. The project develops new ideas and techniques from differential geometry, analysis, partial differential equations, functional analysis, and calculus of variations to tackle fundamental questions in mathematical relativity and in the neighboring areas of geometry and analysis and is anticipated to have impact in other areas of mathematics and in theoretical physics.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.

爱因斯坦的引力理论一直是数学多个分支现代发展的强大推动力，其深远的影响和广泛的应用，成功地描述了我们宇宙的形状并预测了未知的天体的存在。在过去的几十年里，利用先进的几何和分析技术在解决广义相对论的基本问题方面取得了显着的进展，这也导致了令人惊讶的认识，即一些天体受到相同的数学原理的支配。作为日常生活该项目利用数学的前沿发展来研究这些相互联系，并进一步增进我们对宇宙数学模型的几何结构的理解。该项目还纳入了指导和教育活动，以促进几何、分析和计算。由于近年来几何分析的快速发展，广义相对论中的几个长期存在的问题已基本得到解决，其中包括积极的大众猜想。最近的工作与此同时，广义相对论中的这些解决方案推动了几何和分析领域新的和意想不到的技术的发展，该项目的目标是将广义相对论与几何和分析中的一些创新领域联系起来。该项目的范围是分析由与准局部质量相关的质量最小化问题引起的初始数据集的曲率和几何结构及其时空发展，以研究紧凑的标量曲率问题。该项目从微分几何、分析、偏微分方程、泛函分析和变分计算中开发了新的思想和技术，以解决流形的基本问题。数学相对论以及几何和分析的邻近领域，预计会对数学和理论物理的其他领域产生影响。该奖项反映了 NSF 的法定使命，并通过使用基金会的评估进行评估，被认为值得支持智力价值和更广泛的影响审查标准。