Georgia Institute of Technology College of Sciences

We will give a brief introduction to the spectral theory of ergodic operators. Then we discuss several remarkable spectral phenomena present in the class of quasiperiodic operators, as well as the nonperturbative approach to small denominator problems that has been behind much of the related progress.  In particular, we will talk about the almost Mathieu (aka Harper's) operator - a model heavily studied in physics literature and linked to several Nobel prizes (in addition to one Fields medal). We will describe several results on this model that resolve some long-standing conjectures.