Singular Ginzburg-Landau Vortices

Sammanfattning: In this thesis we study the critical Ginzburg-Landau action, defined on fields in the plane which are allowed to have a finite number of singularities. We show that a topological invariant, the degree, can be defined under the assumption of finite action only. The action is bounded below by a constant times the degree, and the fields which realize this lower bound satisfy a first order differential equation. The critical points of the action satisfy a second order differential equation. Using methods of C. Taubes, we give a classification of all finite action solutions to the first order equation, and we show that if there is at most one singularity, then the first and second order equations are equivalent. By different methods we then construct solutions to the second order equation with more than one singularity which does not solve the first order equation.