In this work a reduced order method is proposed that exploit optimal transport theory to build advection modes. The classical methods of model reduction (Proper Orthogonal Decomposition among
them) are based on a principal component analysis of the L2 projections of some snapshots of the evolution, and tend to perform poorly when systems featured by transport of concentrated structures are investigated. The main goal is to set up a principal component analysis framework that allows to use the Wasserstein distance instead of the Lp one. The structure of the seminar is as follows: in the first part some key properties of the Monge-Kantorovich problem are recalled and a Lagrangian numerical solver is introduced. Then, the model reduction technique is detailed, followed by some applications: 2D ideal vortex scattering, Korteweg de Vries equation, vortex shadding, real data Dean hurricane evolution.