Abstract: I will explain the basics of sub-Lorentzian geometry, a little-studied theory, through one of the simplest examples: the three-dimensional Heisenberg group. Roughly speaking, the geometry of this group is controlled by curves that are allowed to travel only in two out of three directions, and a Lorentzian metric defined on these preferred directions allows us to compute the time-separation between events. We will particularly focus on placing the Heisenberg group within the broader context of non-smooth Lorentzian length spaces. Finally, I will formulate the Lorentzian optimal transport problem and present a version of Brenier’s theorem. This talk will be based on a joint work with Wilhelm Klingenberg and Patrick Wood.