Overview

This book presents the mathematical study of vortices of the two-dimensional Ginzburg-Landau model, an important phenomenological model used to describe superconductivity. The Ginzburg-Landau functionals considered include both the model cases with and without a magnetic field. The text introduces the reader to essential mathematical techniques and tools for analyzing the Ginzburg-Landau functional, such as the Ball-Method and the Jacobian estimate. These methods are used to determine vortex locations and densities, asymptotic expansions of energy in terms of the vortices, and rigorously derived values of the critical fields. The book concludes with a discussion of convergence and the results obtained through both minimizing and nonminimizing solutions. The book acts a guide to the various branches of Ginzburg-Landau studies, provides context for the study of vortices, and presents a list of open problems in the field. It provides an introduction to the Ginzburg-Landau model, and discusses current research and results.