Inverse problems for Einstein equations and related topics of Lorentzian geometry

尽管双曲系统的逆问题理论发展迅速，但现有的结果仍然远远不足以处理具有时间相关系数的方程（解析相关的情况除外）或非线性方程。特别是，目前还没有研究广义相对论基本方程的IP的方法。在这个项目中，我们打算开始解决知识产权理论中的这些缺陷。虽然，很明显，研究具有时间相关系数的非线性双曲方程的IP将主要需要一些新的想法和方法，但我们相信我们有一些重要的想法来开始解决这些问题。它们涉及洛伦兹流形上破碎的类光测地线的刚性分析。它们还涉及使用非线性来产生具有所需的法向奇点的二次波，以及通过入射奇异波的相互作用产生的二次波的微局部分析。我们的目标是稍后将这两个想法结合起来，以便从逆数据的观测中提取有关破碎的类光测地线的行为的信息。因此，对这个项目的研究将包括两个主要部分；i)洛伦兹流形上破碎的类光测地线流的刚性研究。在这里，我们相信第一个可发布的结果将在12个月的项目期限结束时出现。ii)奇异入射波相互作用的分析和通过这种相互作用产生的法向奇点的传播研究。在这里，我们期望第一个初步结果（如预印本格式）将在项目结束时出现。

In spite of the rapid developments into the theory of inverse problems (IP) for the hyperbolic systems, existing results stay well short of dealing with either equations with time-dependent coefficients (except for the case of analytic dependence), or non-linear equations. In particular, there exist no approaches to study IP for the fundamental equations of general relativity. In this project we intend to start addressing these deficiencies in the theory of IP. Although, clearly, the study of IP for the non-linear hyperbolic equation with time-dependent coefficients would require some principally new ideas and methods, we believe we have some important ideas to start tackling these problems. They involve analysis of rigidity of the broken light-like geodesics on Lorentzian manifold. They also involve the use of non-linearity to generate secondary waves with desired conormal singularities and microlocal analysis of the secondary waves which are generated through the interaction of the incoming singular waves. We aim later to combine these two ideas in order to extract information about the behaviour of the broken light-like geodesics from observations of inverse data . Therefore, research into this project would consist of two principal parts; i) Study into rigidity of the broken light-like geodesics flow on Lorentzian manifolds. Here we believe that first publishable results will appear by the end of the 12-months duration of the project. ii) Analysis of the interaction of singular incoming waves and study of the propagation of conormal singularities generated through this interaction. Here we expect that the first preliminary results, eg in the preprint format, would appear by the end of the project.