General Relativity and geometric hypersolic PDEs

这个项目的重点是研究广义相对论中出现的几何双曲偏微分方程解的局部和全局行为以及规范理论中使用的模型。在广义相对论中，爱因斯坦方程的完全非线性系统将在弱引力场和强引力场两种情况下被考虑。主要研究人员将继续他关于有界曲率猜想的工作，断言爱因斯坦真空方程的度规解可以局部扩张，只要它的曲率张量在平方可积范数中有界。在Christodoulou最近引入的短脉冲方法的扩展和发展的背景下，将研究强引力场区域。该项目将继续对黑洞背景下的线性和非线性波进行严格的数学研究，其中克尔解的稳定性仍然是一个悬而未决的问题。它还将进一步研究波动映射和杨-米尔斯方程奇点形成的稳定机制。广义相对论的爱因斯坦方程提供了物理时空连续体演化的主要经典描述。对广义相对论中数学和物理现象的研究具有根本性的重要性。虽然对这一主题的物理理解已经取得了快速的进步，并产生了一些杰出的猜想，但严格的数学图景还没有出现。后一个事实在很大程度上是由于爱因斯坦方程的高度非线性，以及缺乏处理它的数学工具。一个令人满意的数学方法的发展取决于分析、几何和偏微分方程式之间的接口。

The focus of this project is the study of the local and global behavior of solutions of geometric hyperbolic PDE arising in general relativity and models used in gauge theories. In general relativity the full nonlinear system of the Einstein equations will be considered in both weak and strong gravitational field regimes. The principal investigator will continue his work on the bounded curvature conjecture, asserting that a metric solution of the Einstein vacuum equations can be locally extended as long as its curvature tensor is bounded in the square integrable norm. A strong gravitational field regime will be studied in the context of extensions and developments of the short pulse method recently introduced by Christodoulou in the problem of formation of trapped surfaces. The project will continue a rigorous mathematical study of linear and nonlinear waves on black hole backgrounds, where the stability of the Kerr solution remains an outstanding open problem. It will also pursue further study of the stable regime of singularity formation for the Wave Map and Yang-Mills equations.The Einstein equations of general relativity provide the main classical description of evolution of the physical space-time continuum. The study of mathematical and physical phenomena arising in general relativity is of the fundamental importance. While the physical understanding of the subject has made rapid advancement and generated a number of outstanding conjectures, the rigorous mathematical picture has yet to emerge. The latter fact is, to a large extent, due to the highly nonlinear nature of the Einstein equations and a lack of the mathematical tools to deal with it. The development of a satisfactory mathematical approach lies on the interface between analysis, geometry and partial differential equations.