Geometrical description of vortices in Ginzburg-Landau billiards

In these notes we discuss the topological nature of some problems in condensed matter physics. We adopt the language of differential geometry to present this subject and our aim is to develop some intuition towards concepts like curvature, fiber bundles, connexions, characteristic classes and topological invariants. The lectures end with a theoretical analysis of recent experimental results obtained on mesoscopic superconducting films.