Optimal Transport theory emerged more than two centuries ago as an engineering problem posed by Gaspard Monge just before the French revolution. Its rich mathematical structure was first revealed by the Russian Nobel-prize winner Kantorovich during world War II, and then about 30 years ago by Yann Brenier in his work on fluid dynamics, Robert McCann in his forays in mathematical physics, and by Wilfrid Gangbo and Craig Evans who, using PDE methods, eventually solved the original problem of Monge. Many other breakthroughs followed, led by Luigi Ambrosio, Felix Otto and their schools leading to the recent Fields medals for Cedric Villani and Alessio Figalli.
The richness of the theory of optimal transport stems from its central role in many branches of mathematics, be they theoretical, applied or computational. Indeed, the basic problem of transporting a probability measure onto another probability measure, while minimizing a given cost of the transport, is now at the core of a wide range of problems in mathematics, physics, economics, statistics, computer science, biology and neuroscience. Recent theoretical and computational advances have paved the way for major breakthroughs in all these areas.
The theory of optimal mass transport has had an impact on various classical branches of mathematics: geometry, analysis, dynamics, partial differential equations, and fluid mechanics. Since it defines a distance between very general distributions and entities of various nature, essential for object recognition and classification, it is now widely investigated in signal processing, machine learning, weather prediction, neuroscience, computer vision, and astrophysics.