Singularity formation in general relativity, and geometric inverse problems.

PI提出了几何分析和偏微分方程这两个不同的中心领域的研究项目。第一个是关于爱因斯坦广义相对论方程的研究，特别关注奇点和宇宙审查猜想。第二个领域涉及几何逆问题；它主要是由医学成像中的真实应用程序驱动的，并试图通过可以转换为适用的重建算法的方法来解决这一领域的中心问题。广义相对论中的项目关注在平滑初始数据的发展中奇点的形成和未来可能的演变。PI提出了一个关于奇点形成(在黑洞内部)的项目，以及另一个关于检测奇点缺失的项目。这两个问题最初都是在轴对称的背景下解决的，PI的最新工作为研究爱因斯坦方程提供了一种强大的新方法。第一个项目的目标是在(非常特殊的)类球面和其他二维对称群中推广许多关于奇点形成的强大结果。从Schwarzschild奇点的稳定性开始，PI希望在球对称的几十年工作成果的基础上，揭示在更一般的一端黑洞内形成奇点的机制，这些黑洞是在引力坍缩中出现的轴对称黑洞。第二个项目将解决彭罗斯提出的弱宇宙审查的另一种表述：证明如果可以从源接收信号的观察者从未在其轨迹上检测到时空中的奇点，那么光信号的发射者也没有经历奇点。第三个项目将是构建高斯光束类型的几何粒子，能量集中在这个对称类中的零测地线上。PI打算追求的最后一个项目是几何逆问题。他打算用一种新的、更几何的方法来研究著名的透镜刚性问题：考虑一个紧凑的非陷阱流形，具有凸边界，边界几何已知，内部度量未知。人们希望根据散射图的知识重建内部的度规：给定进入流形的任何测地线，就可以知道出射的点和方向，以及长度。Pestov-Uhlmann在二维简单流形上解决了这个问题，而在高维凸叶化条件下，Stefan ov-Vasy-Uhlmann解决了这个问题。我建议在没有额外条件的情况下证明这个问题，并用一种真正新的纯几何方法来证明这个问题，这种方法可以转换为适用的稳定重建算法。这个问题有具体的现实实现：在超声成像中，可以通过产生沿这样的测地线传播的特殊声波来获得这样的散射数据。因此，这个项目涉及到数学的重要应用。

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.