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The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80ies and was pushed by Cheeger-Colding in the ‘90ies, who investigated the structure of spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach based on Optimal Transport was proposed by Lott-Villani and Sturm around ten years ago; via this approach, one can give a precise sense of what means for a non-smooth space (more precisely for a metric measure space) to satisfy a Ricci curvature lower bound and a dimensional upper bound. This approach has been refined in the last years by a number of authors (most notably Ambrosio-Gigli- Savarè) and a number of fundamental tools have now been established, permitting to give further insights in the theory and applications which are new even for smooth Riemannian manifolds. The goal of the lectures is to give an introduction to the theory and discuss some of the applications.

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80ies and was pushed by Cheeger-Colding in the ‘90ies, who investigated the structure of spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach based on Optimal Transport was proposed by Lott-Villani and Sturm around ten years ago; via this approach, one can give a precise sense of what means for a non-smooth space (more precisely for a metric measure space) to satisfy a Ricci curvature lower bound and a dimensional upper bound. This approach has been refined in the last years by a number of authors (most notably Ambrosio-Gigli- Savarè) and a number of fundamental tools have now been established, permitting to give further insights in the theory and applications which are new even for smooth Riemannian manifolds. The goal of the lectures is to give an introduction to the theory and discuss some of the applications.

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.

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80ies and was pushed by Cheeger-Colding in the ‘90ies, who investigated the structure of spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach based on Optimal Transport was proposed by Lott-Villani and Sturm around ten years ago; via this approach, one can give a precise sense of what means for a non-smooth space (more precisely for a metric measure space) to satisfy a Ricci curvature lower bound and a dimensional upper bound. This approach has been refined in the last years by a number of authors (most notably Ambrosio-Gigli- Savarè) and a number of fundamental tools have now been established, permitting to give further insights in the theory and applications which are new even for smooth Riemannian manifolds. The goal of the lectures is to give an introduction to the theory and discuss some of the applications.

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80ies and was pushed by Cheeger-Colding in the ‘90ies, who investigated the structure of spaces arising as Gromov-Hausdorff limits of smooth Riemannian manifolds satisfying Ricci curvature lower bounds. A completely new approach based on Optimal Transport was proposed by Lott-Villani and Sturm around ten years ago; via this approach, one can give a precise sense of what means for a non-smooth space (more precisely for a metric measure space) to satisfy a Ricci curvature lower bound and a dimensional upper bound. This approach has been refined in the last years by a number of authors (most notably Ambrosio-Gigli- Savarè) and a number of fundamental tools have now been established, permitting to give further insights in the theory and applications which are new even for smooth Riemannian manifolds. The goal of the lectures is to give an introduction to the theory and discuss some of the applications.