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 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 …

In this paper, we introduce a synthetic notion of Riemannian Ricci bounds from below for
metric measure spaces (X, d, m) which is stable under measured Gromov–Hausdorff …

In the recent paper [24] the Cheeger-Colding-Gromoll splitting theorem has been
generalized to the abstract class of metric measure spaces with Riemannian Ricci curvature …

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 provide new characterizations of the curvature dimension
condition in the context of metric measure spaces $(X,\mathsf {d},\mathfrak {m}) $. On the …

We prove that a metric measure space (X, d, m) satisfying finite-dimensional lower Ricci
curvature bounds and whose Sobolev space W1, 2 is Hilbert is rectifiable. That is, an …

We prove that if (X, d, m)(X, d, m) is a metric measure space with m (X)= 1 m (X)= 1 having
(in a synthetic sense) Ricci curvature bounded from below by K> 0 K> 0 and dimension …

These are the lecture notes of the Ph. D. level course 'Nonsmooth Differential
Geometry'given by the first author at SISSA (Trieste, Italy) from October 2017 to March 2018 …

We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD (K, N)
metric measure spaces; regularity is understood with respect to a newly defined quasi …