Mass/Momentum beyond Classical Gravity and Submanifolds of Higher Codimensions

这个项目研究广义相对论、几何和微分方程交叉的基本问题。爱因斯坦的广义相对论描述了时空是如何被引力弯曲的。他的理论语言是几何学，这种现象是由他的同名方程控制的。最近的进展，如LIGO探测到引力波和视界望远镜观测到黑洞图像，证实了爱因斯坦理论的预言，并增强了我们对天体物理事件和物体的全局和大尺度结构的理解。PI的研究将应用时空几何和爱因斯坦方程的最新数学突破，以获得对宇宙中任何有限扩展区域的能量和角动量等基本概念的最精确描述和测量。这对于理解我们宇宙的局部和精细结构至关重要，例如GPS技术和空间探索，以及引力系统（如黑洞合并）的相互作用。这些概念的一个新应用是四维以上的时空，这是在统一广义相对论和量子物理学的最可行方法中出现的。PI还将研究存在于更大维度的环境空间中的多种维度的几何物体。这样的例子包括依赖于受多种约束的多个变量的庞大数据集。PI将应用微分方程的方法来研究这些物体的最佳形状/相位。该项目的研究将用于促进本科生对数学的兴趣，并为研究项目提供动力。PI一直致力于培养多样化的本科生/研究生和年轻研究人员，该项目将有助于他继续朝着这个方向努力。此外，本提案所研究的几个研究问题是数学之外的兴趣，具有相当大的跨学科合作潜力。PI计划用几何分析的方法解决与高维重力和高维子流形有关的几个突出问题。特别是，PI将定义高维时空零无穷附近的准局部质量和线性/角动量。直接目标包括证明准局部质量的正性/单调性定理和高协维一般子流形的刚性/正则性定理，以及建立零无穷处线性/角动量的守恒定律和超平移不变性。该研究将促进我们对非线性偏微分系统的理解，如高维空间中的爱因斯坦方程和平均曲率方程，并为高维时空中的引力能和角动量等重要物理量提供新的视角。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project investigates fundamental problems at the intersection of general relativity, geometry, and differential equations. Einstein’s theory of general relativity describes how spacetime is curved by gravitation. The language of his theory is geometry and the phenomenon is governed by his eponymous equation. Recent advances such as the detection of gravitational waves by LIGO and the observation of black hole images by the Event Horizon Telescope confirmed predictions made by Einstein’s theory, and enhanced our understanding of the global and large scale structure of astrophysical events and objects. The PI's research will apply the latest mathematical breakthroughs in spacetime geometry and Einstein’s equation to obtain the most precise descriptions and measurements of fundamental concepts such as energy and angular momentum on any finitely extended region of the universe. This is essential in understanding the local and fine structure of our universe, with applications in, for example, GPS technology and space exploration, as well as the interaction of gravitating systems such as black hole coalescence. A novel application of the concepts is to space-times beyond four dimensions, which arise in the most viable approach in unifying general relativity and quantum physics. The PI will also study geometric objects of manifold dimensions that live in ambient spaces of even greater dimensions. Examples of such include gigantic data sets that rely on multiple variables subject to multiple constraints. The PI will apply the method of differential equations to investigate the optimal shapes/phases of these objects. The research in the project will be used to promote interest in mathematics among undergraduate students and to provide motivations for research projects. The PI has been engaging himself in educating a diversified body of undergraduate/graduate students and young researchers, and the project will be instrumental for his continued efforts along this direction. In addition, several research problems studied in this proposal are of interest beyond mathematics and there is considerable potential for interdisciplinary cooperations. The PI plans to resolve several outstanding problems related to higher dimensional gravity and submanifolds of higher codimensions by the method of geometric analysis. In particular, the PI will define quasilocal mass and linear/angular momentum near null infinity of higher dimensional spacetimes. Immediate goals include proving positivity/monotonicity theorems for quasilocal mass and rigidity/regularity theorems for general submanifolds of higher codimensions, in addition to establishing conservation laws and supertranslation invariance for linear/angular momentum at null infinity. The proposed research will advance our understanding of nonlinear partial differential systems, such as the Einstein equation and mean curvature equations in higher codimensions, and cast new light on important physical quantities such as gravitational energy and angular momentum in higher dimensional spacetimes.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.