Mathematical problems in gravitational collapse

该论文项目涉及理解关于广义相对论中孤立系统引力坍缩的基础科学问题。需要解决的问题包括临界坍缩问题、宇宙审查猜想和裸奇点的不稳定性、自引力流体中激波形成的相关性以及黑洞内奇点的性质。这些问题有精确的数学公式，可以用偏微分方程组的定性理论来解决。这篇论文将建立在过去几年中该领域的许多最新数学进展的基础上，包括在非平凡几何环境中处理爱因斯坦广义相对论方程和流体力学欧拉方程的数学技术的进步。

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