Singularity formation in general relativity, and geometric inverse problems.

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The PI proposes research projects in two distinct central areas of Geometric Analysis and PDEs. The first concerns the study of Einstein's equations of general relativity, with special focus on singularities and the cosmic censorship conjectures. The second area concerns geometric inverse problems; it is very much motivated by real-life applications in medical imaging primarily, and seeks to address central questions in this area with methods that can be transformed into applicable reconstruction algorithms. The projects in general relativity are concerned with the formation and possible future evolution of singularities in the development of smooth initial data. The PI proposes one project on singularity formation (in black hole interiors), and another on detecting the absence of singularities. Both are to be initially addressed in the setting of axial symmetry, where very recent work of the PI has provided a powerful new method to study Einstein's equations. The first project aims to extend the many powerful results of singularity formation in the (very special) class of spherical and other 2-dimensional symmetry groups. Having commenced with the stability of the Schwarzschild singularity, the PI hopes to uncover the mechanism of singularity formation inside more general, one-ended black holes that arise in gravitational collapse, in axial symmetry, building on decades of work of results in spherical symmetry. The second project would address an alternative formulation of weak cosmic censorship due to Penrose: To show that if observers that can receive signals from a source never detect a singularity in space-time along their trajectory, then the emiter of light signals did not experience a singularity either. A third project would be the construction of Gaussian-beam type geons, with energy focused along null geodesics in this symmetry class. The final project the PI intends to pursue is in geometric inverse problems. He intends to study the celebrated lens rigidity question with a new, more geometric approach: Consider a compact non-trapping manifold with convex boundary, with boundary geometry known and metric in the interior unknown. One wishes to reconstruct the metric in the interior from knowledge of the scattering map: Given any geodesic that enters the manifold, one knows the point and direction of exit, along with the length. This problem was solved for simple manifolds by Pestov-Uhlmann in 2D, and under a convex foliation condition in higher dimensions by Stefanov-Vasy-Uhlmann. I propose to prove this question without extra conditions, and in a genuinely new purely geometric approach that can be transformed to an applicable stable reconstruction algorithm. This problem has concrete real-life realizations: In ultrasound imaging, such scattering data can be derived by generating special acoustic waves that travel along such geodesics. Thus this projects touches on important applications of mathematics.