Geometry and Topology Seminar: An introduction to the theory of Optimal Transport
22 June 2017 13:00 in CM221
This will be a wrap-up of the study group on optimal transport led by Norbert Peyerimhoff with participants from the probability, applied, and pure groups. We will give an accessible introduction to the question in the plane : move a pile of sand into a hole with the same volume by minimizing the transportation cost. First we introduce the measure-theoretic formulation of Leonid Kantorovich. This allows for duality of the variational problem and the Kantorovich potential and, by the Direct Method of the Calculus of Variations, results in existence of a weak solution in the space of measures. Secondly we proceed with Yann Brenier's representation of the minimizer for quadratic cost. This is based on the Legendre transform used in the passage from Lagrangian mechanics to Hamiltonian mechanics. Thirdly, time allowing, we describe Alessio Figalli's C(1,\alpha) regularity of the Brenier potential via the fully nonlinear elliptic Monge-Ampere equation.