Mathematical Problems in General Relativity

广义相对论有非常重要的长期目标，其中最突出的是克尔猜想的弱宇宙审查性、强宇宙审查性、末态和稳定性。要实现这一目标，需要解决许多基本困难，其中PI的提出集中在Kerr族的唯一性和稳定性问题以及有界曲率猜想上。PI打算继续他与A.Ionescu以及最近S.Alexakis的工作，以消除S.霍金和卡特-罗宾逊著名结果的限制性真实分析性假设。PI还计划继续与I.Rodnianski和J.Szeftel合作，解决重要的有界曲率猜想。他们和I.Rodnianski一起计划改编Christodoulou最近的工作，基于他所说的短脉冲方法，他能够用这种方法推导出关于爱因斯坦积分方程解的陷阱表面形成的第一个结果。PI和他的合作者认为，新方法可以显著改进并应用于各种情况。如果成功，该提议将使人们更好地理解广义相对论的一些主要理论开放问题。长期的希望是，在这个过程中获得的一些见解将影响到相关领域，如理论物理和数值相对论。引力波实验最感兴趣的是设计能够真实地模拟黑洞相互作用的数值算法。PI还希望正在开发的几何方法和分析方法之间的相互作用将影响到非线性偏微分方程组、微分几何和分析的其他领域。PI计划吸引和培训更多的博士生和博士后，比他目前为止在广义相对论方面所做的要多。国际和平研究所认为，这一课题现在正在经历一个激动人心的时期，重要问题得到解决，许多新的研究途径正在开辟。

General Relativity has very important long term goals among whichthe weak and strong cosmic censorship, final state and stability of Kerrconjectures are the most outstanding. To complete such goals oneneeds to solve many fundamental difficulties among which the PI proposesto concentrate on the problems of uniqueness and stability of the Kerrfamily as well as the bounded curvature conjecture. The PI intends tocontinue his work with A. Ionescu, and more recently S. Alexakis, toremove the restrictive real analyticity assumptions of the well knownresults of S. Hawking and Carter-Robinson. The PI also plans to continueto work with I. Rodnianski and J. Szeftel and settle the importantbounded curvature conjecture. Together with I. Rodnianski they planto adapt the recent work of Christodoulou, based on what he callsthe short pulse method, with which he was able to derive the firstresult on the formation of trapped surfaces for solutions of the Einsteinvacuum equations. The PI and his collaborator believe that the new method can be signifficantly improved and applied to various situations.If successful the proposal will give a better understanding of someof the main theoretical open problems of General Relativity. The longterm hope is that some of the insight gained in the process will influence related areas such Theoretical Physics and Numerical Relativity.Of paramount interest to the gravitational wave experiments is to de-sign numerical algorithms which can simulate, realistically, black holeinteractions. The PI also hopes that the interplay between geometric and an-alytic methods, which are being developed, will influence other areas ofnonlinear PDE, Differential Geometry and Analysis. The PI plans to attractand train a larger number of PhD students and postdocs, than he hasdone so far, in General Relativity. The PI believes that the subject is goingnow through an exciting period when important problems are solved and many new avenues of research are being open.