Singularity formation in general relativity, and geometric inverse problems.

广义相对论中奇点的形成和几何逆问题。

• 批准号: RGPIN-2020-05108
• 负责人: Alexakis, Spyros
• 金额: $ 2.7万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=717390
• 关键词: Singularity; formation; general; relativity; geometric

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.

PI提出了几何分析和偏微分方程两个不同的中心领域的研究项目。第一个是关于爱因斯坦的广义相对论方程的研究，特别关注奇点和宇宙审查系统。第二个领域涉及几何逆问题;它主要是由医学成像中的实际应用激发的，并试图用可以转化为适用的重建算法的方法来解决这一领域的核心问题。 广义相对论的项目关注的是在光滑初始数据的发展过程中奇点的形成和未来可能的演化。PI提出了一个关于奇点形成（在黑洞内部）的项目，另一个关于检测奇点的存在。这两个问题最初都是在轴对称的情况下解决的，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。 第一个项目的目的是扩大许多强大的结果奇点形成（非常特殊）类的球面和其他2维对称群。从史瓦西奇点的稳定性开始，PI希望在球对称数十年的工作成果的基础上，揭示更一般的单端黑洞内部奇点形成的机制，这些黑洞出现在引力坍缩中，在轴对称中。第二个项目将讨论彭罗斯弱宇宙审查的另一种表述：如果能够从一个源接收信号的观察者从未沿着其轨迹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。 第三个项目将是建设高斯光束型geons，与能量集中沿着零测地线在这个对称类。 PI打算追求的最后一个项目是几何逆问题。他打算研究著名的透镜刚性问题与一个新的，更几何的方法：考虑一个紧凑的非捕获流形与凸边界，与边界几何已知和度量的内部未知。人们希望从散射映射的知识中重建内部的度量：给定任何进入流形的测地线，人们知道出口的点和方向沿着长度。Pestov-Uhlmann在2D中解决了这个问题，Stefanov-Vasy-Uhlmann在高维中解决了这个问题。我建议证明这个问题没有额外的条件，并在一个真正的新的纯几何方法，可以转化为一个适用的稳定重建算法。这个问题有具体的现实生活中的实现：在超声成像，这种散射数据可以通过产生特殊的声波，沿着沿着这样的测地线。 因此，这个项目涉及数学的重要应用。