Mathematical problems in general relativity

点击翻译按钮获取中文摘要

General relativity is a theory proposed by Einstein in 1915 as a unified theory of space, time and gravitation and is fundamentally rooted in classical mechanics. Its main objective is the understanding of the evolution of gravitational physical systems such as planetary systems, multiple stars, black holes, (a cluster of) galaxies and ultimately the universe as a whole. What distinguishes general relativity from other physical theories is that "general relativity describes the evolution of systems in terms of differential equations (i.e.~physical laws) imposed directly on the space-time metric'', connecting thus geometry, analysis and physics. These differential equations, known as the Einstein equations, give rise to a theory of great mathematical subtlety and beauty. ******The research program will strive to resolve certain fundamental mathematical questions pertaining to the main conjectures in general relativity. Specifically, the project will investigate the stability problem for the so-called extremal and sub-extremal Kerr black holes aiming at establishing the relevance of black holes from a theoretical point of view. It will also develop a scattering theory for the Einstein equations which is expected to provide new insights into the study of the propagation of gravitational waves. Finally, the program will investigate the gluing problem for characteristic initial data of hyperbolic equations. The latter is an extension of the Riemannian gluing problem for elliptic equations and is intensively studied in geometric analysis.******The proposal, if successful, will lead to the discovery of genuinely new analytical techniques applicable to a wider spectrum of problems in geometry, analysis, partial differential equations and mathematical physics. We have previously used modern techniques of mathematical analysis to obtain new and mathematically rigorous results that led to very fruitful and productive contacts with members of the high-energy physics, numerical relativity and astrophysics communities. The program will contribute to further strengthen the ties between these communities and pure mathematics.******As part of this research program we will train undergraduate and graduate students, and attract and supervise highly qualified doctoral students and post-docs from Canada and worldwide aiming to establish the University of Toronto as a central place for research in the rapidly expanding collaborative area of mathematical general relativity.