Existence and Stability of Vortex Lattices

Vortex lattices are celebrated solutions of the Ginzburg-Landau equations, which describe macroscopic properties of superconductors and play a prominent role far beyond its original area of superconductivity. For instance, the Ginzburg-Landau equations and their non-Abelian generalizations - the Yang-Mills-Higgs equations - are a key part of the standard model in elementary particle physics.

 

I will review recent results involving the existence and stability of the vortex lattices -  and how they relate to the modified theta functions appearing in number theory and algebraic geometry. If time permits, I will also describe the existence results for the Ginzburg-Landau equations on Riemann surfaces.