Evolution of Ginzburg Landau vortices for vector fields on surfaces

In this talk I will report on a joint work with Giacomo Canevari. I will discuss a parabolic Ginzburg-Landau equation for vector fields on a 2 dimensional closed and oriented Riemannian manifold. I will show that in a suitable asymptotic regime the energy of the solutions concentrates on a finite number of points. These points are called vortices and their evolution is governed by gradient flow of the so-called renormalized energy.