Mathematical Problems in General Relativity

这个项目研究爱因斯坦的广义相对论，用爱因斯坦方程来描述。爱因斯坦理论的一个引人入胜的预言是奇点无处不在。这项研究的目的是对这些奇点给出精确的数学描述。例如，我们研究黑洞内部的奇点结构，以及高度奇异和高度振荡的引力波的影响。此外，PI还将承担学生的培训（他目前有一名博士生），并继续在数学界和物理界传播他的工作。其中一些作品将被传播给更广泛的公众；PI的工作已经在一本科普杂志上发表过。我们描述了一个我们试图回答的问题样本。一般黑洞的奇异边界是什么样的？在极端黑洞的特殊情况下情况是如何变化的？还有其他类型的奇点吗？多重脉冲引力波的相互作用是什么样的？能否描述高振荡解的适当极限？最后，我们还提出了一些问题来理解爱因斯坦方程在物质场存在下的动力学。虽然我们考虑的问题范围很广，但它们是由潜在的解析和几何技术统一的。通过解决这些问题，我们也将开发出对广义相对论以外的双曲型偏微分方程有用的数学工具。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 studies the Einstein's remarkable theory of general relativity, described by the Einstein equations. One fascinating prediction of Einstein's theory is the ubiquity of singularities. The research aims at giving a precise mathematical description of these singularities. For instance, we study the structure of singularities inside black holes, and also the effects of highly singular and highly oscillatory gravitational waves. In addition, the PI will also undertake training of students (he currently has a PhD student) and continue disseminating his work in both the mathematics and physics communities. Some of the work will be communicated to a wider public audience; the PI's work has already been featured in a popular science magazine.We describe a sample of questions that we seek to answer. What does the singular boundary of generic black holes look like? How does the situation change in the special case of extremal black holes? Are there other types of singularities? What does the interaction of multiple impulsive gravitational waves look like? Can one describe appropriate limits of highly oscillatory solutions?Finally, we also propose some questions to understand the dynamics of the Einstein equations in the presence of matter fields. While we consider a wide range of questions, they are unified by the underlying analytic and geometric techniques. Through solving these problems we will also develop mathematical tools useful for hyperbolic partial differential equations beyond the application in general relativity.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.