The geometry of space-time used for the theory of relativity is Lorentzian geometry. Gravity is expressed through the curvature of space-time. I will introduce Lorentzian geometry and describe some of its key features. I will explain some physical motivation for generalizing this geometry and demonstrate some of the potentially undesirable phenomena that can arise as we do this.

Recently Kunzinger and Sämann proposed Lorentzian length spaces as a synthetic generalization of Lorentzian geometry. Generally, a synthetic geometry is a theory which takes certain theorems from a well-established smooth geometric theory and uses them instead as axioms. I will discuss these spaces and, in particular, ways of defining curvature on them.

This will include recent work I have done with Tobias Beran, Lewis Napper and Felix Rott showing that curvature bounds, which are essentially local, can have implications on the global geometry of the space.