From the Ginzburg-Landau energy to vortex lattice problems

In a joint work with Etienne Sandier, we study minimizers the
two-dimensional Ginzburg-Landau energy of superconductivity.  The main
question is to understand the vortex-locations of minimizers according to
the applied magnetic field.
By using Gamma-convergence techniques i.e. precise lower bounds
  and upper bounds for the minimal energy, we prove that at leading order
the vortices arrange themselves according to a uniform density. We then
show that the next order of the energy governs the vortex interaction
after blow-up at a suitable scale, we derive a limiting "renormalized"
energy that governs this interaction of points in the plane. A striking
result is that this interaction energy can be computed explicitely in the
case of lattice configurations and that it is uniquely minimized by the
triangular lattice, thus providing a rigorous explanation of the Abrikosov
triangular lattice known in physics.