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Monday, July 19, 2010, 2:00 p.m., in BA 6183, 40 St. George Street PhD Candidate: Tim Tzaneteas PhD Advisor: I.M. Sigal 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.
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