University of Seville

Topological data analysis (TDA) comprises a set of techniques of computational topology that has had enormous growth in the last decade, with applications to a wide variety of fields, such as images,  biological data, meteorology, materials science, time-dependent data, economics, etc. In this talk, we will first have a walk through a typical pipeline in TDA, to move later to its adaptation to a specific context: the topological characterization of the spatial distribution of regions in a segmented image

In this talk we consider several scenarios involving the interaction among incompressible fluids of different nature. The main concern is the dynamics of the free boundary separating the fluids, which evolves with the velocity flow. The important question to address is whether the regularity is preserved in time or, on the other hand, the system develops singularities. We focus on Navier-Stokes models, where the viscosity of the fluids play a crucial role. At first showing results of finite time blow-up for the case of vacuum-fluid interaction. Later discussing new recent results on global existence for 1996 P.L. Lions' conjecture for density patches evolving by inhomogeneous Navier-Stokes equations.