QLunch: Jingxuan Zhang

Speaker: Jingxuan Zhang

Title: Ginzburg-Landau Equations on Hyperbolic Surfaces

Abstract: We construct equivariant solutions to the magnetic Ginzburg-Landau equations on line bundles over hyperbolic surfaces. These solutions are the non-commutative generalizations of the Abrikosov vortex lattice of superconductivity. We obtain precise asymptotic expansions of these solutions and their energies in terms of the curvature of the underlying surface. Among other things, our result shows the spontaneous breaking of the gauge-translational symmetry of the Ginzburg-Landau equations on hyperbolic surfaces. This talk is based on recent joint work with Nicolas M. Ercolani and Israel Michael Sigal, see arXiv:2203.14179 [math.AP].