Geometry and Analysis of Einstein Metrics

这个项目涉及对爱因斯坦方程几个方面的研究。这些方程源于爱因斯坦在广义相对论中对空间、时间和引力的理解上的根本性突破。在物理学中，这些方程支配着宇宙的大尺度结构，但在理解精细尺度结构方面也至关重要。例如，如果不完全理解爱因斯坦方程，GPS系统将无法工作。它们在重力量子化的主要努力中也扮演着重要的角色，即弦理论。在数学上，爱因斯坦方程是最有趣的关于几何和弯曲空间分析的方程。将要进行的调查涉及到来自几个领域的思想和方法的相互融合。主要焦点将是爱因斯坦度量的边值问题的全局问题（存在性和唯一性），包括在经典微分几何中的应用，如等距嵌入问题。广义相对论的研究将包括准局部质量和爱因斯坦方程的初边值问题的研究。研究弦理论的反德西特/共形场论（AdS/CFT）对应关系以及共形紧致爱因斯坦度量的存在性和结构。

This project concerns investigations on several aspects of the Einstein equations. These equations arose from Einstein's fundamental breakthroughs in understanding space, time, and gravity in his theory of general relativity. In physics, the equations govern the large scale structure of the universe, but are also crucial in understanding fine scale structures. For example, the GPS system would be unworkable without a full understanding of the Einstein equations. They also play an important role in the main efforts to quantize gravity, namely string theory. Mathematically, the Einstein equations are the most interesting equations relating geometry and analysis on curved spaces. The investigations to be undertaken involve a cross-fertilization of ideas and methods from several areas. A main focus will be global issues (existence and uniqueness) for boundary value problems for Einstein metrics, including applications to classical differential geometry such as the isometric embedding problem. Research in general relativity will include studies of quasi-local mass and the initial boundary value problem for the Einstein equations. Studies in the anti-de Sitter/conformal field theory (AdS/CFT) correspondence from string theory and the existence and structure of conformally compact Einstein metrics will also be undertaken.