AMMCS-2013 Plenary Talk

Superconductivity and automorphic functions

Israel Michael Sigal, Univ. of Toronto

Abstract: Macroscopic theory of superconductivity is based on the celebrated Ginzburg - Landau equations. First developed to explain and predict properties of superconductors, these equations had a profound influence on physics well beyond their original designation area. These are a pair of coupled nonlinear equations for a complex function (called order parameter or Higgs field) and a vector field (magnetic potential or gauge field). They are the simplest representatives of a large family of equations appearing in physics and mathematics. (The latest variant of these equations is the Seiberg - Witten equations.) Besides of importance in physics, they contain beautiful mathematics (some of the mathematics was discovered independently by A. Turing in his explanation of patterns of animal coats). In this talk I will review recent results involving key solutions of these equations - the magnetic vortices and vortex lattices, their existence, stability and dynamics, and how they relate to the modified theta functions appearing in number theory. Some automorphic functions play a key role in this theory.

Israel Michael Sigal is the Norman Stuart Robertson Chair in Applied Mathematics and University Professor at the University of Toronto. He works in several areas of mathematical physics. Among his results are the proof (jointly with Avy Soffer) of asymptotic completeness of the quantum many-body scattering for short-range potentials and the development of a mathematical framework (jointly with Volker Bach and Jurg Frohlich) of the theory of emission and absorption of quantum radiation by non-relativistic quantum systems such as atoms and molecules, as well as several important results on the nonlinear Schrodinger, Ginzburg-Landau, mean-curvature and wave equations. Professor Sigal was an invited speaker at several International Congresses of Mathematical Physics and at an International Congress of Mathematicians.