Alessio Figalli ∗ Review of the book:

The optimal transport problem has received the attention of many researchers in the last two decades, and its popularity is still increasing. This is mainly motivated by the discovery of unexpected connections between optimal transport and problems in physics, geometry, partial differential equations, etc. To give an example, consider the following geometric statement: Let (Mk, gk, volk) be a sequence of smooth compact Riemannian manifolds with nonnegative Ricci curvature, converging in the measured GromovHausdorff sense to a smooth compact Riemannian manifold (M∞, g∞, vol∞). Then (M∞, g∞) has nonnegative Ricci curvature. At first sight, this statement may look surprising. Indeed the GromovHausdorff convergence is a very weak notion, so it may seem strange that it can control lower bounds on the Ricci curvature, which a priori should depend on second derivatives of the metric. However, optimal transport allows recasting lower Ricci bounds in terms of much more robust inequalities (see Theorem 4.1 below), and this fact is at the core of the proof of the above result [8, 14, 15]. The present article briefly surveys the exciting and extremely active field of optimal transport, with emphasis on the content and features of the book under review. ∗Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin Texas 78712, USA. E-mail address: [email protected] 2010 Mathematics Subject Classification: 49-02 (28Axx 37J50 49Q20 53Cxx 58Cxx 82C70)