Rigidity and quantitative rigidity in General Relativity

Rigidity theorems play an important role in Riemannian and Lorentzian geometry. Well-known examples are
given by the Splitting Theorems. A natural question is trying to prove quantitative versions of these
statements. On the Riemannian side, this problem was solved by Cheeger and Colding with their celebrated
Almost-splitting Theorem. In this talk, I will propose a quantitative version of the Lorentzian splitting Theorem
under timelike sectional curvature bounds. This talk is based on an ongoing joint work with N. Gigli, M. Graf,
R. McCann, A. Ohanyan, E. Woolgar.