Mathematical problems in gravitational collapse

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The thesis project involves understanding basic science questions concerning gravitational collapse of isolated systems in general relativity. Problems to be addressed include the issue of critical collapse, the cosmic censorship conjectures and the instability of naked singularities, the relevance of shock formation in self-gravitating fluids, and the nature of singularities inside black holes. These questions have precise mathematical formulations and can be addressed using the qualitative theory of partial differential equations. The thesis will build on many recent mathematical advances in the field over the past few years, including advances in the mathematical technology for handling the Einstein equations of general relativity and the Euler equations of fluid mechanics in non-trivial geometric settings