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 …

Optimal Transport (OT) has recently emerged as a central tool in data sciences to compare
in a geometrically faithful way point clouds and more generally probability distributions. The …

This textbook originated from the teaching experience of the first author at the Scuola
Normale Superiore, where a course on optimal transport and its applications has been given …

The purpose of this book is to describe the global properties of complete simply connected
spaces that are non-positively curved in the sense of AD Alexandrov and to examine the …

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 …

At the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and
John Mather launched a revolution in the venerable field of optimal transport founded by G …

The book is devoted to the theory of gradient flows in the general framework of metric
spaces, and in the more specific setting of the space of probability measures, which provide …

Geodesic convexity generalizes the notion of (vector space) convexity to nonlinear metric
spaces. But unlike convex optimization, geodesically convex (g-convex) optimization is …

Metric theory has undergone a dramatic phase transition in the last decades when its focus
moved from the foundations of real analysis to Riemannian geometry and algebraic …

Riemannian geometry is characterized, and research is oriented towards and shaped by
concepts (geodesics, connections, curvature,...) and objectives, in particular to understand …