The characteristic Cauchy problem for linear wave equations consists of imposing initial values for the solution on a characteristic hypersurface instead of initial values for the function and its normal derivative on a spacelike Cauchy hypersurface. After a general introduction to the relevant notions we show that this problem is well posed on globally hyperbolic Lorentzian manifolds under suitable assumptions. This is joint work with Roger Tagne Wafo and it generalizes classical results by Hörmander.[-]

We prove a local version of the index theorem for Dirac-type operators on globally hyperbolic Lorentzian manifolds with Cauchy boundary. In case the Cauchy hypersurface is compact, we do not assume self-adjointness of the Dirac operator on the spacetime or of the associated elliptic Dirac operator on the boundary.In this case, integration of our local index theorem results in a generalization of previously known index theorems for globally hyperbolic spacetimes that allows for twisting bundles associated with non-compact gauge groups. This is joint work with Alexander Strohmaier.[-]