Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications

The goal of the present work is three-fold. The first goal is to set foundational results on optimal transport in Lorentzian synthetic spaces, including cyclical monotonicity, stability of optimal couplings and Kantorovich duality (several results are new even for smooth Lorentzian manifolds). The second one is to give a synthetic notion of "timelike Ricci curvature bounded below and dimension bounded above" for a Lorentzian space using optimal transport. The key idea being to analyse convexity properties of Entropy functionals along future directed timelike geodesics of probability measures. Such a notion is proved to be stable under a suitable weak convergence of Lorentzian synthetic spaces. The third goal is to draw applications, most notably extending volume comparisons and Hawking singularity Theorem (in sharp form) to the synthetic setting. The framework of Lorentzian synthetic spaces includes as remarkable classes of examples: space-times endowed with a causally plain (or, more strongly, locally Lipschitz) continuous Lorentzian metric, closed cone structures, some approaches to quantum gravity (e.g. causal Fermion systems).