Mass/Momentum beyond Classical Gravity and Submanifolds of Higher Codimensions

超越经典引力的质量/动量和更高维数的子流形

基本信息

批准号：
2104212
负责人：
Mu-Tao Wang
金额：
$ 33.71万
依托单位：
Columbia University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104212&HistoricalAwards=false
关键词：
Mass Momentum beyond Classical Gravity

项目摘要

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.

该项目研究了广义相对论、几何学和微分方程的交叉点，爱因斯坦的广义相对论描述了时空是如何被引力弯曲的，而这种现象是由他的同名方程控制的。 LIGO对引力波的探测、视界望远镜对黑洞图像的观测等证实了爱因斯坦理论的预测，增强了我们对全球大尺度结构的认识。 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，以获得对宇宙任何有限扩展区域的能量和角动量等基本概念的最精确的描述和测量。了解宇宙的局部和精细结构，并应用于 GPS 技术和太空探索，以及黑洞合并等引力系统的相互作用。这些概念的一个新颖应用是时空。超越四个维度，这是统一广义相对论和量子物理学的最可行的方法，PI还将研究存在于更大维度的周围空间中的多维几何对象，此类例子包括依赖于多个维度的巨大数据集。 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提高本科生对数学的兴趣并为研究项目提供动力。 PI 一直在参与他本人致力于教育多元化的本科生/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。此外，本提案中研究的几个研究问题超出了数学范围，并且具有相当大的潜力。 PI计划通过几何分析的方法解决与高维引力和高维子流形相关的几个突出问题，特别是，PI将定义高维的准局部质量和接近零无穷大的线性/角动量。近期目标包括证明准局部质量的正性/单调性定理和高维一般子流形的刚性/正则性定理，以及建立零无穷大线性/角动量的守恒定律和超平移不变性。拟议的研究将增进我们的理解。非线性偏微分系统的研究，例如爱因斯坦方程和高维平均曲率方程，并为重要的物理量提供了新的视角例如高维时空中的引力能和角动量。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。