Title: Global existence versus singularities for incompressible fluid interfaceserfaces.
Abstract: The evolution of an interface between two immiscible incompressible fluids can develop singularities in finite time. In particular those contour dynamics that are given by basic fluid mechanics systems: Euler´s equations, Darcy´s law and the Quasi-geostrophic equation. These give rise to problems such as water wave, Muskat, and the evolution of sharp fronts of temperature. In this lecture we will present recent developments on the subject.