Problems in General Relativity and Geometric Flows

广义相对论和几何流问题

基本信息

批准号：
1810856
负责人：
Mu-Tao Wang
金额：
$ 23.53万
依托单位：
Columbia University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1810856&HistoricalAwards=false
关键词：
Problems General Relativity Geometric Flows

项目摘要

This project concerns investigations of fundamental problems at the interface of general relativity, geometry, and differential equations. Einstein's theory of general relativity describes how the spacetime is curved by gravitation. The language of his theory is geometry and the phenomenon is governed by his eponymous equation. There have been great advances in both theoretical and experimental aspects of general relativity, for example the recent detection of gravitational waves by LIGO. However, due to the complexity of Einstein's equation, most of our knowledge of the universe are global and large scale, such as the observation of an astrophysical event from a very faraway distance. Recently, the PI and his collaborators applied the tools of geometry and differential equations to give the most precise measurement of gravitational energy and mass on any finitely extended region of the universe. This is essential in understanding the fine and local structure of our universe, with applications in, for example, GPS technology and space exploration. It is also crucial in studying non-isolated large-scale phenomena such as black hole mergers and collisions. The success of this project will deepen our understanding of gravitational energy/mass and the nonlinear local/global nature of the spacetime. The PI also plans to study geometric flows, which are differential equations that model how a geometric shape deforms and evolves to an optimal form in the most efficient way. The PI has been engaging himself in educating a diversified body of graduate students and young researchers, and the project will be instrumental for his continued efforts along this direction.The PI plans to resolve several outstanding problems in general relativity and geometric flows by the method of geometric analysis. In particular, the quasilocal mass definition the PI discovered with Yau will be a key ingredient in this research. Immediate goals include proving the general invariant mass conjecture, establishing a positive mass theorem for the Robinson-Trautman spacetime and the linear stability of higher dimensional Schwarzschild spacetimes, and the solution of a conjecture of Arnol'd by the Lagrangian mean curvature flow. This research project will also advance our understanding of nonlinear partial differential systems such as the optimal isometric embedding equation, the mean curvature flow, and the Einstein equation.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.

该项目涉及广义相对论、几何和微分方程接口的基本问题的研究。爱因斯坦的广义相对论描述了时空如何因引力而弯曲。他的理论语言是几何学，现象是由他的同名方程控制的。广义相对论在理论和实验方面都取得了巨大进展，例如最近 LIGO 探测到了引力波。然而，由于爱因斯坦方程的复杂性，我们对宇宙的大部分认识都是全局性的、大尺度的，比如从很远的距离观测天体物理事件。最近，PI 和他的合作者应用几何和微分方程工具，对宇宙任何有限延伸区域的引力能和质量进行了最精确的测量。这对于理解宇宙的精细局部结构及其在 GPS 技术和太空探索等领域的应用至关重要。它对于研究黑洞合并和碰撞等非孤立的大规模现象也至关重要。该项目的成功将加深我们对引力能量/质量以及时空的非线性局部/全局性质的理解。 PI 还计划研究几何流，这是一种微分方程，用于模拟几何形状如何变形并以最有效的方式演化为最佳形式。 PI一直致力于培养多元化的研究生和年轻研究人员，该项目将有助于他沿着这个方向继续努力。PI计划通过以下方法解决广义相对论和几何流中的几个突出问题：几何分析。特别是，PI 与 Yau 一起发现的准局域质量定义将成为这项研究的关键要素。近期目标包括证明一般不变质量猜想，建立罗宾逊-特劳特曼时空的正质量定理和高维史瓦西时空的线性稳定性，以及通过拉格朗日平均曲率流解决阿诺德猜想。该研究项目还将增进我们对非线性偏微分系统的理解，例如最优等距嵌入方程、平均曲率流和爱因斯坦方程。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力评估进行评估，认为值得支持。优点和更广泛的影响审查标准。