The Ginzburg-Landau model is a phenomenological description of superconductivity. An essential feature of type-II superconductors is the presence of quantized vortices, which appear above a certain value of the external magnetic field called the first critical field. This talk will review some known mathematical tools developed to analyze the vortices, and will provide a new polyhedral approximation of the vorticity measure in three dimensions, which allows one to obtain a new lower bound for the energy and a new vorticity estimate. This construction is at the $\varepsilon$-level and yields optimal estimates. As an application, we will describe the behavior of global minimizers for the three-dimensional Ginzburg-Landau functional below and near the first critical field.
