Svetlana Jitomirskaya, Irvine – University of California, Irvine
Wednesday, October 26, 10:45AM (Zoom meeting starts at 10:30)
"Fractal properties of the Hofstadter butterfly, eigenvalues of the almost Mathieu operator, and topological phase transitions"
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Harper's operator - the 2D discrete magnetic Laplacian - is the model behind the Hofstadter's butterfly and Thouless theory of the Quantum Hall Effect. It reduces to the critical almost Mathieu family, indexed by the phase. We will discuss the Aubry-Andre metal-insulator conjecture, as well as the state-of-the-art on the arithmetic transition conjecture, including the discovery of the universal (reflective-)hierarchical structure of the eigenfunctions throughout the localization regime. We will then present a complete proof of singular continuous spectrum for the critical family, for all phases, finishing a program with a long history. The proof is based on a simple Fourier analysis and a new Aubry duality-type transform. We will also explain how these ideas provide for a very simple proof of zero measure of the spectrum of Harper's operator, a problem previosly solved by sophisticated dynamical systems techniques, as well as progress on some other outstanding conjectures.