## Departmental PhD Thesis Exam – Tim Tzaneteas

Monday, July 19, 2010, 2:00 p.m., in BA 6183,
40 St. George Street

PhD Candidate:  Tim Tzaneteas

Thesis Title:  Abrikosov Lattice Solutions of the Ginzburg-Landau
Equations of Superconductivity
(http://www.math.toronto.edu/~ttzanete/thesis/)

Thesis Abstract:

In this thesis we study the Ginzburg-Landau equations of superconductivity,
which are among the basic nonlinear partial differential equations of
Theoretical and Mathematical Physics. These equations also have geometric
interest as equations for the section and connection of certain principal
bundles and are related to Seiberg-Witten equations used extensively in
Differential Geometry. In 1957, Abrokosov suggested that for sufficiently
high magnetic fields there exist solutions for which all physical quantities
have the periodicity of a lattice, with the magnetic field penetrating the
superconductor at the vertices of the lattice (Abrikosov lattice solutions).
The corresponding phenomenon was confirmed experimentally and is among the
most interesting aspects of  superconductivity and is discussed in every
book on the subject. In 2003, Abrikosov was awarded  the Nobel Prize in
Physics for this discovery.

Building on the previous results in the subject we prove the existence of
such lattices in the case where each lattice cell contains a single quantum
of magnetic flux, and in the general case reduce the problem to an
$n$-dimensional problem, where $n$ is the number of quanta of flux. We prove
that for Type II superconductors, these solutions are stable, and in the
case $n = 1$, we show that as the external magnetic field approaches the
critical value at which superconductivity first appears, the lattice which
minimizes the average free energy per lattice cell is the triangular
lattice.