This is an expository paper on the theory of gradient flows, and in particular of those PDEs
which can be interpreted as gradient flows for the Wasserstein metric on the space of …

In this introductory chapter we first give a brief historical review of optimal transport, then we
recall some basic definitions and facts from measure theory and Riemannian geometry, and …

We develop a full theory for the new class of Optimal Entropy-Transport problems between
nonnegative and finite Radon measures in general topological spaces. These problems …

The goal of this survey is to give a self-contained introduction to synthetic timelike Ricci
curvature bounds for (possibly non-smooth) Lorentzian spaces via optimal transport and …

We prove the equivalence of the curvature-dimension bounds of Lott–Sturm–Villani (via
entropy and optimal transport) and of Bakry–Émery (via energy and Γ _2 Γ 2-calculus) in …

This paper is devoted to a deeper understanding of the heat flow and to the refinement of
calculus tools on metric measure spaces (X,d,m). Our main results are: A general study of …

The main goals of this paper are:(i) To develop an abstract differential calculus on metric
measure spaces by investigating the duality relations between differentials and gradients of …

The theory of curvature-dimension bounds for nonsmooth spaces has several motivations:
the study of functional and geometric inequalities in structures which arc very far from being …

The aim of this paper is to discuss convergence of pointed metric measure spaces in the
absence of any compactness condition. We propose various definitions, and show that all of …

The aim of the present paper is to bridge the gap between the Bakry–Émery and the Lott–
Sturm–Villani approaches to provide synthetic and abstract notions of lower Ricci curvature …