Non-regular spacetime geometry, curvature and analysis

I present an approach to Lorentzian geometry and General Relativity that does neither rely on smoothness nor
on manifolds, thereby leaving the framework of classical differential geometry. This opens up the possibility to study
curvature (bounds) for spacetimes of low regularity or even more general spaces. An analogous shift in perspective
proved extremely fruitful in the Riemannian case (Alexandrov- and CAT(k)-spaces). After introducing the basics of our
approach, we report on recent progress in developing a Sobolev calculus for time functions on such non-smooth
Lorentzian spaces. This seminar talk can also be viewed as a primer and advertisement for my mini course in
May: Current topics in Lorentzian geometric analysis: Non-regular spacetimes