Hydrodynamics and optimal transport: from Euler to Monge and vice versa
The foundation of Hydrodynamics was done in 1755 by Leonhard Euler in his memoir for the Berlin Academy "Principes generaux du mouvement des fuides". Few years later, Gaspard Monge wrote in 1780 his "memoire sur les deblais et les remblais", about a civil engineering problem. This was the starting point of what is now called "optimal transport theory"and turned out, after its revival by Kantorovich in 1942, to be incredibly rich from the mathematical point of view, connecting calculus of variations, probability theory, dierential geometry, partial di erential equations, functional analysis, statistics and computer sciences. In these lectures, I will emphasize the interplay between Euler and Monge theories. I will also discuss their link with the nonlinear theory of Electromagnetism, developped by Max Born and Leopold Infeld in 1934, which has known a revival in high energy physics in the last 20 years. This theory can be interpreted as well as as a theory of optimal transport of currents or as an unusual model of Magnetohydrodynamics, renewing in some sense the link between Euler and Monge.
Yann Brenier, CNRS, Centre de Mathematiques Laurent Schwartz, Ecole Polytechnique, Palaiseau, France.