  In these four lectures I describe basic ideas and methods applicable to both
classical and quantum systems displaying slow relaxation and non-equilibrium
dynamics. The first half of these notes considers classical systems, and the
second half, quantum systems. In Lecture 1, I briefly review the glass
transition problem as a paradigm of slow relaxation and dynamical arrest in
classical many-body systems. I discuss theoretical perspectives on how to think
about glasses, and in particular how to model them in terms of kinetically
constrained dynamics. In Lecture 2, I describe how via large deviation methods
it is possible to define a statistical mechanics of trajectories which reveals
the dynamical phase structure of systems with complex relaxation such as
glasses. Lecture 3 is about closed (i.e. isolated) many-body quantum systems. I
review thermalisation and many-body localisation, and consider the possibility
of slow thermalisation and quantum non-ergodicity in the absence of disorder,
thus connecting with some of the ideas of the first lecture. Lecture 4 is about
open quantum systems, that is, quantum systems interacting with an environment.
I review the description of open quantum dynamics within the Markovian
approximation in terms of quantum master equations and stochastic quantum
trajectories, and explain how to extend the dynamical large deviation method to
study the statistical properties of ensembles of quantum jump trajectories. My
overall aim is to draw analogies between classical and quantum non-equilibrium
and find connections in the way we think about problems in these areas.
