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This book contains the proceedings of the summer school “Optimal transportation: Theory and Applications” held at the Fourier Institute (University of Grenoble I, France). The first 2 weeks were devoted to courses that described the main properties of optimal transportation and discussed its applications to analysis, differential geometry, dynamical systems, partial differential equations and probability theory. Courses were addressed both to students and researchers. A workshop took place during the last week. The aim of this conference was to present very recent developments of optimal transportation and also its applications in biology, mathematical physics, game theory and financial mathematics.
The first part of the book contains (expanded) versions of the courses. There are two sets of notes by F. Santambrogio. The first one gives a short introduction to optimal transport theory. In particular, the Kantorovich duality, the structure of Wasserstein spaces and the Monge–Ampère equations related to optimal transport are presented to the readers. These notes could be seen as an introduction for the other papers of the book. The second one describes applications to economics, game theory and urban planning.
The notes of I. Gentil, P. Topping and S.-I. Ohta describe (with different flavours) the connections between optimal transport and the notion of Ricci curvature, which is a very important tool in classical Riemannian geometry. A notion of curvature-dimension condition was defined by D. Bakry and M. Émery to study geometric properties of diffusions and to get functional inequalities.
The theory of optimal transportation has its origins in the eighteenth century when the problem of transporting resources at a minimal cost was first formalised. Through subsequent developments, particularly in recent decades, it has become a powerful modern theory. This book contains the proceedings of the summer school 'Optimal Transportation: Theory and Applications' held at the Fourier Institute in Grenoble. The event brought together mathematicians from pure and applied mathematics, astrophysics, economics and computer science. Part I of this book is devoted to introductory lecture notes accessible to graduate students, while Part II contains research papers. Together, they represent a valuable resource on both fundamental and advanced aspects of optimal transportation, its applications, and its interactions with analysis, geometry, PDE and probability, urban planning and economics. Topics covered include Ricci flow, the Euler equations, functional inequalities, curvature-dimension conditions, and traffic congestion.
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