Some Mathematical Problems in the Ginzburg-Landau Theory of Superconductivity

In the Ginzburg-Landau theory (1950), the state of a superconductor is described by solutions of a pair of nonlinear PDEs called the Ginzburg-Landau equations. Abrikosov (1957) observed that a fundamental role is played by a family of symmetric solutions known as vortices, which are classified by a topological degree, and which represent localized defects in the superconductor. Recently, the Ginzburg-Landau theory has received a lot of attention from mathematicians.

I will describe some of the interesting issues, focusing on vortices and their dynamics. In particular, I will present results which settle an old conjecture about how vortex stability depends on the topology, and on the nature of the superconducting material.