Ginzburg-Landau equation on non-compact Riemann surfaces

In this talk I will consider Ginzburg-Landau equations (GLE) on non-compact Riemann surfaces, more precisely on line bundles over such surfaces modelling superconductivity in thin membranes. I will report on recent results on existence of energy minimizing solutions for such equations. I will introduce all necessary definitions and will structure the presentation so that no preliminary knowledge of line bundles over Riemann surfaces is required. The talk is based on the joint work with Nick Ercolani and Jingxuan Zhang.