Mathematical Problems in General Relativity

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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.