Inverse problems for Einstein equations and related topics of Lorentzian geometry

尽管双曲组的反问题理论得到了迅速的发展，但现有的结果仍然远远不能处理含时变系数的方程(解析相关的情况除外)，也不能处理非线性方程。特别是，对于广义相对论的基本方程，还没有研究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不足。虽然，显然，研究含时系数的非线性双曲型方程的激元问题需要一些主要的新想法和方法，但我们相信我们有一些重要的想法来开始解决这些问题。它们涉及到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奇异性的次级波，并对通过输入的奇异波相互作用而产生的次级波进行微观局部分析。我们的目标是稍后将这两个想法结合起来，以便从反向数据的观测中提取有关破碎的类光测地线行为的信息。因此，本课题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。在此，我们相信，第一批可公布的成果将在项目为期12个月结束时公布。Ii)分析奇异入射波的相互作用，并研究通过这种相互作用产生的奇异点的传播。在这里，我们预计第一批初步结果，如预印本，将在项目结束时公布。

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.