General Relativity and geometric hypersolic PDEs

这个项目的重点是研究在广义相对论和规范理论中使用的模型中产生的几何双曲偏微分方程解的局部和全局行为。在广义相对论中，爱因斯坦方程的完整非线性系统将在弱引力场和强引力场两种情况下被考虑。首席研究员将继续他的工作有界曲率猜想，声称爱因斯坦真空方程的度量解可以局部扩展，只要其曲率张量在平方可积范数有界。一个强引力场制度将研究的范围内的扩展和发展的短脉冲方法最近介绍了由Christodoulou在形成的问题被困的表面。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图和杨-米尔斯方程的奇点形成的稳定状态，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。对广义相对论中出现的数学和物理现象的研究具有根本的重要性。虽然对这一问题的物理理解取得了迅速的进展，并产生了许多杰出的成果，但严格的数学图景尚未出现。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性质和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.