Singularity formation in general relativity, and geometric inverse problems.

PI在几何分析和偏微分方程两个不同的中心领域提出研究项目。第一部分涉及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；它主要是由现实生活中的医学成像应用所驱动的，并试图用可以转化为适用重建算法的方法来解决这一领域的核心问题。广义相对论中的项目涉及光滑初始数据发展过程中奇点的形成和可能的未来演变。PI提出了一个关于奇点形成（黑洞内部）的项目，另一个关于探测奇点缺失的项目。这两个问题最初都是在轴对称的背景下解决的，PI最近的工作为研究爱因斯坦方程提供了一种强大的新方法。第一个项目旨在扩展（非常特殊的）球形和其他二维对称群中奇点形成的许多强有力的结果。从史瓦西奇点的稳定性开始，PI希望揭示奇点形成的机制，在引力坍缩和轴对称中产生的更一般的单端黑洞，建立在几十年的球对称结果的基础上。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：表明如果能够从一个源接收信号的观测者在其轨迹上从未探测到时空中的奇点，那么光信号的发射者也没有经历过奇点。第三个项目将是建造高斯光束型直线，能量集中在这个对称类的零测地线上。PI打算进行的最后一个项目是几何逆问题。他打算用一种新的、更几何化的方法来研究著名的透镜刚性问题：考虑一个紧致的非俘获流形，它具有凸边界，边界几何已知，内部度规未知。我们希望从散射图的知识中重建内部的度量：给定任何进入流形的测地线，我们知道出口的点和方向，以及长度。Pestov-Uhlmann在二维中对简单流形解决了这个问题，Stefanov-Vasy-Uhlmann在高维凸叶化条件下解决了这个问题。我建议在没有额外条件的情况下，用一种真正新的纯几何方法来证明这个问题，这种方法可以转化为一种适用的稳定重构算法。这个问题在现实生活中有具体的实现：在超声成像中，这种散射数据可以通过产生沿着这种测地线传播的特殊声波来获得。因此，这个项目涉及数学的重要应用。

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.