Mass Rigidity and Curvature Problems in Mathematical Relativity

爱因斯坦的引力理论是现代数学几个分支发展的强大动力。在其深刻的影响和广泛的应用中，引力理论成功地描述了我们宇宙的形状，并预测了未知存在的天体，如黑洞。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，即一些天体与日常生活中的物体（如肥皂膜）受到相同的数学原理的支配。该项目利用数学的前沿发展来研究这些相互联系，并进一步促进我们对宇宙数学模型几何结构的理解。该项目还包括指导和教育活动，以促进几何，分析及其与其他STEM学科的相互关系，更广泛的社区和社会。由于近年来几何分析的快速发展，广义相对论中几个长期存在的问题已经在很大程度上得到解决。一个突出的例子是解决了正质量猜想，包括最近关于时空正质量定理的工作。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中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相邻领域联系起来，以便进一步开发和应用一些创新技术。该项目的范围是分析曲率和几何结构的初始数据集和他们的时空发展，所产生的质量最小化问题有关的准本地质量，研究标量曲率问题的紧凑流形和分类的静态流形，并从双曲和共形几何方面的特点爱因斯坦流形。该项目从微分几何，分析，偏微分方程，泛函分析，和变分法，以解决数学相对论和几何和分析的邻近领域的基本问题，预计将在数学和理论物理的其他领域产生影响。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估支持使用基金会的知识价值和更广泛的影响审查标准。

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.