Ressource pédagogique : A. Mondino - Time-like Ricci curvature bounds via optimal transport

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The goal of the talk is to present a recent work in collaboration with Cavalletti (SISSA) on optimal transport in Lorentzian synthetic spaces. The aim is to set up a “Lorentzian analog” of the celebrated Lott-Sturm-Villani theory of CD(K,N) metric measure spaces. The key idea being to analyse convexity properties of Entropy functionals along future directed timelike geodesics (with respect to a suitable Lorentzian Wasserstein distance) of probability measures. The smooth Lorentzian setting was previously investigated by McCann and Mondino-Suhr. After recalling the general setting of Lorentzian synthetic spaces (including remarkable examples fitting the framework), I will discuss some basics of optimal transport theory thereof in order to define "timelike Ricci curvature bounded below and dimension bounded above''  for a (possibly non-smooth)  Lorentzian space. The notion of  "timelike Ricci curvature bounded below and dimension bounded above''  for a (possibly non-smooth)  Lorentzian space is stable under a suitable weak convergence of Lorentzian synthetic spaces, giving a glimpse on the strength of the proposed approach. As an application of the optimal transport approach to timelike Ricci curvature lower bounds, I will discuss an extension of the  Hawking's Singularity Theorem (in sharp form) to the synthetic setting.

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