Inverse problems for nonlinear hyperbolic equations in space-time

We consider inverse problems consisting in determining space-time structures using nonlinear waves. The problem was introduced by Kurylev, Lassas and Uhlmann (2013) for the Einstein-scalar field equations in general relativity. In this talk, we start with the inverse problem for semilinear wave equations on a 4-dimensional Lorentzian manifold. We discuss the results of determining the conformal and isometry class of the Lorentzian metric, and some information about the nonlinear term. Then we report the results for Einstein-Maxwell equations i.e. determining (vacuum) space-time structures using electromagnetic sources. The talk is based on some joint work with M. Lassas and G. Uhlmann.