Geometrical Description of Vortices in Ginzburg-Landau Billiards

In these notes we discuss the topological nature of some problems in condensed matter physics. This topic has been widely studied in various contexts. In statistical mechanics, the possible stable defects in an ordered system have been classified according to the nature of the order parameter (e.g. scalar, vector, matrix) and the space dimensionality of the system using homotopy groups [1]. Then, the discovery of the quantum Hall effects and the role played by stable integers or rational numbers for systems with few or no conserved quantum symmetries have motivated several topological models of quantum condensed matter systems [2,4]. A combination of these two ideas of defects classification and microscopic quantum models has been used in the description of superfluid 3He [5].