The Mathematics of Gravitational Collapse

This project is concerned with some of the most fundamental open questions in the physics of gravitational collapse, including the formation and stability of so-called extreme black holes and the structure of the singularities these black holes hide deep in their interior. These questions can be completely expressed in the precise language of mathematics as conjectures about the behavior of solutions to a set of differential equations known as the Einstein equations, equations first written down in 1915. The project will thus further contribute to the venerable tradition of rigorous mathematics making fundamental statements about the nature of our physical world, a tradition which over the centuries has proven central to advances in science and technology. The subject of black holes and spacetime singularities continues to inspire the popular imagination well beyond the confines of the scientific community, as was apparent most recently in the public reception of the first stunning images of the immediate vicinity of a supermassive black hole. The project provides research training opportunities for graduate students and postdoctoral researchers. The project seeks to answer fundamental mathematical questions about gravitational collapse in general relativity, including the dynamic formation and stability of rapidly rotating and extremal black holes and the structure and stability of spacetime singularities. The first major objective of the project is to probe the extremal black hole formation threshold, also addressing the full nonlinear stability problem for the subextremal case. The project's second major objective concerns probing the presence and stability of strong, spacelike portions of the singular boundary of black holes arising in gravitational collapse without symmetry. Finally, the project's third major objective aims to prove stability and instability results of vacuum naked singularities, exploring further the problem of critical collapse. 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.