Mathematics

The theory of optimal transport aims at studying how to bring a distribution of mass from a place into another in the cheapest possible way with respect to a given cost functional. Starting from some basic results in Riemannian geometry, I will describe the so-called Benamou-Brenier formula, which is a link between the static formulation and the dynamical formulation of this transportation problem. Then, I will move on to describe optimal transport in the setting of general relativity, where the cost is given by the time separation function on a spacetime, that is to say the intrinsic proper time that passes between two events. In particular, I will describe a new Benamou-Brenier type formula in spacetimes, and I will discuss some applications of it. This talk is based on a joint work with N.Gigli and F.Rott.