The Ginzburg - Landau theory was first developed to explain magnetic and other properties of superconductors, but had a profound influence on physics well beyond its original area. The theory provided the first demonstration of the Higgs mechanism and it became a fundamental part of the standard model in elementary particle physics. The theory is based on 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. Geometrically, these are equations for a section of the principal bundle and a connection on this bundle. (The latest variant of these equations is the Seiberg – Witten equations.) In addition to their great importance in physics, they contain beautiful mathematics (see e.g. a review of E. Witten in Bulletin AMS, 2007; 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. http://math.as.uky.edu/van-winter

Refreshments will be served at 3:30 p.m. in 179 Chemistry-Physics Building