On the Mathematical Theory of Black Holes

这个项目处理与黑洞的数学理论有关的问题，如果这些问题得到解决，将极大地有助于我们对物理宇宙中这些最迷人的物理物体的理解。该项目不仅将推动黑洞科学的发展，而且通过开发与数学物理其他领域相关的新数学技术，它将在最广泛的意义上促进数学和科学的进步。首席研究员（PI）在针对偏微分方程（PDE）各个领域具有广泛吸引力的特定问题方面有着良好的记录。从这个意义上说，作为这个项目的一部分的问题，虽然特定于手头的主题，也可以在数学物理的其他重要问题中重新表述。该项目也非常符合PI的长期目标，即帮助创建一个充满活力的科学社区，研究与这些领域相关的数学问题。特别是，该项目为研究生提供了研究训练的机会。该项目侧重于一些关于黑洞的主要开放问题，如刚性、稳定性和坍缩，特别强调克尔家族稳定性的基本问题。这是爱因斯坦真空方程的一个精确、明确的解族，它取决于两个参数（质量和角动量），这是我们对黑洞的理论理解的基础。因此，上面提到的问题不仅从数学的角度来看是深刻的，而且在天体物理学中也有严重的影响。在稳定性问题上尤其如此，因为如果克尔家族被发现是不稳定的，黑洞就只不过是数学上的人工制品。PI将这三个相关问题解释为黑洞的数学“现实测试”。这三个问题都需要对强引力场下爱因斯坦场方程的动力学有深刻的理解。这是一项艰巨的任务，只能通过使用新的几何偏微分方程技术对问题的底层结构进行系统的数学分析来实现。这些问题的解决促进了数学和科学的进步，超出了这个特定项目的主题。事实上，正如过去经常发生的那样，人们期望与这些问题相关的一些技术将与数学物理中其他重要的偏微分方程相关。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project deals with problems connected to the mathematical theory of black holes which, if solved, would contribute greatly to our understanding of these most fascinating physical objects of the physical Universe. The project will not only advance the science of black holes, but, by developing new mathematical techniques of relevance to other fields of mathematical physics, it promotes the progress of Mathematics and Science in the broadest possible sense. The Principal Investigator (PI) has a strong track record in targeting specific problems with broad appeal in various areas of Partial Differential Equations (PDE). In that sense, the problems considered as part of this project, though specific to the subject at hand, can also be reformulated in other important problems of mathematical physics. The project also fits well with respect to the long term goal of the PI to help create a vibrant scientific community working on mathematical problems connected to these areas. In particular, the project provides research training opportunities for graduate students. The project focuses on some of the main open problems concerning black holes such as rigidity, stability, and collapse, with special emphasis being given to the fundamental problem of the stability of the Kerr family. This is a precise, explicit, family of solutions to the Einstein vacuum equations, depending on two parameters (the mass and the angular momentum), on which our theoretical understanding of black holes is based. Thus, the problems mentioned above are not only deep from a mathematical point of view, they also have serious implications in Astrophysics. This is particularly true about the problem of stability, since if the Kerr family were to be found unstable, black holes would be nothing more than mathematical artifacts. The PI interprets these three related problems as mathematical "tests of reality" for black holes. All three problems require a deep understanding of the dynamics of the Einstein field equations in a strong gravitational field regime. This is a tall task which can only be achieved by a systematic mathematical analysis of the underlining structure of the problems, using new geometric PDE techniques. The resolution of these problems promotes the progress of Mathematics and Science beyond the subject matter of this particular project. Indeed, it is expected, as it often happened in the past, that some of the techniques developed in connection to these problems will be relevant to other important partial differential equations of mathematical physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.