Superconductivity, Hyperbolic Geometry and the Spectrum of Magnetic Laplacians
In this talk we will describe the existence of non-trivial solutions to the magnetic Ginzburg-Landau equations on non-compact (cuspidal) Riemann surfaces (joint work with I.M. Sigal and J. Zhang) and the Yang-Mills-Higgs (YMH) equations on singular (orbifold) spheres. These solutions are non-commutative generalizations of the Abrikosov vortex lattice in superconductivity. The bifurcation parameter underlying this defect formation is in terms of the constant negative curvature of the underlying surface with critical transition directly related to the spectral analysis of a magnetic Laplacian. In the case of YMH, this analysis is made constructively through a matrix Riemann-Hilbert problem.