Fibered optimal transport introduces heterogeneity to gradient flows

There is a well-known, deep relationship between continuity equation and gradient flows with respect to the 2-Wasserstein metric. This connection provides an alternative description of well-known models and PDEs: Fokker-Planck, Vlasov, Keller--Segel, Kuramoto and many models of first-order collective dynamics (interaction potentials). I am going to present a recent work (joint with David Poyato), wherein we introduce the so-called fibered 2-Wasserstein metric (which admits only transportation along fibers controlled by a prescribed marginal distribution) and explore its applicability in gradient flows. We end up with heterogeneous gradient flows, parameterized continuity equations, models of mixtures of substances, multispecies models and (our main motivation) the Kuramoto-Sakaguchi equation and models of second-order alignment dynamics.