Cantor and positive measure spectrum for magnetic Schrödinger operators

Schrödinger operators with periodic electric and magnetic fields play a central role in solid state physics for understanding the low temperature behavior of periodic crystals. Mathematically, periodic electric fields are best dealt with using Floquet/Bloch theory: The space of solutions of the periodic equation can be decomposed into subspaces consisting of quasiperiodic solutions, and the spectrum of the periodic Schrödinger operator is the union of the spectra in these subspaces.
I will briefly review this theory and its implications, and then proceed to describe the complications that arise under the presence of a magnetic field (such as gauge properties, magnetic translations). In some cases known as rational magnetic flux, we can deal with these complications by putting Bloch theory in a more geometric context, i.e. studying sections of vector bundles instead of functions. Unfortunately, the chances for a real number to be rational are pretty slim. I will describe two different approaches for the case of irrational flux: non-commutative Bloch theory, which leads to some results on weak genericity of Cantor spectra; and a "partial Bloch theory", which leads to results about the measure of the spectrum. I will try and keep at least the first part of the talk accessible to a general audience (including graduate students); just bring a basic understanding of "Laplace operator" and "Fourier transform", and you will be fine.