We shall review basic properties of cocycles over a minimal dynamical system, taking values in the special linear group of two by two matrices over the real numbers. It turns out that dynamical properties of such cocycles play a central role in the spectral theory of quasiperiodic one-dimensional Hamiltonians. We shall review those dynamical properties and connections with spectral theory. This talk will be of expository nature, and technical details will be kept to a minimum (respectively, we shall assume no prior background in the subject). 

Given a regular cardinal $\kappa$, an uncountably cofinal ordinal $\nu<\kappa$ is a reflection point of the stationry set $S\subseteq\kappa$ just in the case where $S\cap\alpha$ is stationary in $\alpha$. Starting from ininitely many supercompact cardinals, Magidor constructed a model of set theory where every stationary $S\subseteq\aleph_{\omega+1}$ has a reflection point. In this series of talks we present a construction of a model of set theory where we obtain a large amount of stationary reflection (although not full) using a significantly weaker large cardinal hypothesis. We start from a quasicompact (quasicompactness is a large cardinal hypothesis significantly weaker than any nontrivial variant of supercompactness) cardinal $\kappa$ and use modified Prikry forcing to turn $\kappa$ into $\aleph_{\omega+1}$. We then show that in the resulting model every stationray $S\subeteq\aleph_{\omega+1}$ not concentrating on ordinals of ground model cofinality $\kappa$ has a reflection point.