We will review several open problems in dynamical systems (mostly related to hyperbolic and partially hyperbolic dynamics) that can be used as a starting point for independent graduate student's research.

The one dimensional quantum Ising model is used in quantum statistical physics to model interracting particles on a discrete lattice. While the classical model (in one and two dimensions) has long been solved (its origin dates back to 1930's), its quasiperiodic analog (dating back about 25 years) is still a source of interesting problems. We shall discuss our solution to one such problem: we'll rigorously prove that the energy spectrum of the one dimensional quantum quasiperiodic Ising model is a Cantor set, as has been long believed, and discuss some of its properties.
This is the first in a series of two seminars dedicated to this topic. In this seminar we'll present the problem and set up the main ideas.

We consider the one-dimensional Schrodinger operator with potential given by Period Doubling and Thue-Morse sequences. We describe the results of Bellisard et al. in which the spectrum of these operators are obtained through the trace map associated to this operator. We see that the spectrum for non-zero values of the potential is a Cantor set of zero Lebesque measure, and give explicit description of the spectral gaps.

We consider the classical 1D Ising model, where the coupling constants and the external magnetic field take one of two possible values at each site, according to a substitution rule. We shall introduce (briefly) the idea of a trace map corresponding to the given substitution rule and how its dynamical properties can be used to investigate the partition function and, consequently, the free energy function of the given Ising model.