Project: One of the tools often used in the setting of non-smooth differential geometry is the notion of Gromov-Hausdorff convergence. Amongst its uses this notion has proved to be powerful tool to produce examples of spaces that satisfy certain curvature bounds but that exhibit geometric/topological propoerties that can vastly differ from those of a smooth manifold.

In the Lorentzian setting one needs to handle things with extra care, since the signature of the metric (and its consequences) should be reflected on any sensible notion of convergence being defined. Several researchers have aimed to tackle this problem. For instance, Sakovich and Sormani have introduced various notions by considering compactifications of spacetimes with respect to the null distance, a notion previously defined by Sormani and Vega. Briefly, this distance is induced by a cosmologial time function, which in turn may be defined using the Lorentzian time separation function. Then, one can use the classical Gromov-Hausdorff convergence to study the convergence of spacetimes.

Mondino and Sämann defined a different notion for convergence of Lorentzian length spaces by using causal diamonds as an analog of epsilon-nets, which allows them to obtain precompactness results à la Gromov. Moreover, they obtain the stability of lower sectional curvature bounds for Lorentzian spaces under this notion of convergence.

Our goal is to study these recently introduce notions to obtain some results analogous to those established in the Riemannian setting. We also want to explore instnaces where there is compatibility between these notions.

Research Team: Jaime Santos Rodríguez (U Politécnica de Madrid), Mauricio Che Moguel (U of Vienna)

Dates of Stay: June 1 - June 30, 2026