Global Studies of Einstein Spacetimes

这个项目的主要目标之一是继续发展爱因斯坦方程的光锥估计，这些估计与已经开发并成功用于证明杨-米尔斯-希格斯方程的整体存在定理的光锥估计类似。虽然严格的整体存在对于广义相对论是无效的，但彭罗斯宇宙审查猜想提供了它的自然的、引力的类比，这个猜想的解决被认为是经典广义相对论中主要的公开数学问题之一。最近的分析已经成功地导出了真空时空在任意点的曲率的显式、精确的积分公式。这个积分公式还应该适用于其他基本的、公开的数学问题，例如黑洞稳定性的严格证明。该项目的另一个主要目标是继续开发用于分析引力波的改进的数值方法。一个主要目标是确定如何最好地将这一发现与标准的、不受限制的数字进化技术相结合。这个项目的最终目标是将最近发展起来的欧几里德签名半经典近似方法应用于形式量子广义相对论的Wheeler-DeWitt方程的研究。这个项目涉及的许多研究都是几何和偏微分方程组的数学家感兴趣的。为了从数值上研究波的传播而开发的技术可以很好地被证明在科学和工程的其他领域是有用的。预计，当这些方法应用于引力辐射研究时，将对目前全世界正在进行的对这类波的有趣来源进行数值模拟的努力产生重大影响，从而最终影响实验性的搜索工作。进一步发展“欧几里得签名半经典”方法来研究量子场论，特别是规范理论，对于理解这类系统应该有深远的影响。预计在这些努力中争取学生和年轻研究人员的贡献将对数学和物理科学的人力资源开发产生重大影响，这是该项目的另一个主要目标。

One of the main aims of this project is to continue the development of light-cone estimates for the Einstein equations that parallel those that have already been developed and successfully used to prove a global existence theorem for the Yang-Mills-Higgs equations. Though strict global existence is not valid for general relativity the Penrose cosmic censorship conjecture provides its natural, gravitational analogue and the resolution of this conjecture is considered to be one of the main open mathematical problems in classical general relativity. Recent analysis has succeeded to derive an explicit, exact integral formula for the curvature of a vacuum spacetime, at an arbitrary point. This integral formula should also admit applications to other fundamental, open mathematical problems such as the rigorous proof of black hole stability. An additional, principal aim of this project is to continue work on the development of improved numerical methods for the analysis of gravitational waves. A primary goal is to determine how best to integrate this discovery with standard, unconstrained, numerical evolution techniques. A final goal of this project is to adapt recently developed "Euclidean signature semi-classical" approximation methods to the study of the Wheeler-DeWitt equation of formal, quantum general relativity.Much of the research involved in this project is of interest to mathematicians working in geometry and partial differential equations. Techniques developed for studying, numerically, the propagation of waves that literally extend all the way to infinity could well prove to be useful in other areas of science and engineering. It is anticipated that these same methods, when applied to the study of gravitational radiation, will have a significant influence on the ongoing worldwide efforts to model, numerically, interesting sources of such waves and thereby, ultimately, to impact the experimental search efforts. The proposal to further develop "Euclidean-signature semi-classical" methods for the study of quantum field theories, especially gauge theories, should have far-reaching implications for the understanding of such systems. It is expected that enlisting the contributions of students and younger researchers in these endeavors should have significant implications for the development of human resources in the mathematical and physical sciences and this is an additional, principal aim of the project.