Mass/Momentum beyond Classical Gravity and Submanifolds of Higher Codimensions

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.