EVENT DETAILS AND ABSTRACT

Title: Critical phenomena, arithmetic phase transitions, and universality: some recent results on the Harper's/almost Mathieu operator
Speaker: Svetlana Jitomirskaya
Speaker Info: UC Irvine
Brief Description:
Special Note:
Abstract:

Harper's operator - the 2D discrete magnetic Laplacian - is the model behind the Hofstadter's butterfly and Thouless theory of the Quantum Hall Effect. The model reduces to the almost Mathieu family, indexed by phase, and we will present some recent results on both models some finishing a program with a long history and some opening new directions. Time permitting, we will include a complete proof of absence of point spectrum of critical almost Mathieu operators, based on a simple Fourier analysis and a new 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 previously solved by sophisticated dynamical systems techniques.