Glassy dynamics: mean-field theory and beyond

The slowing down of a liquid's dynamics upon approaching the glass transition is accompanied by a number of novel dynamical phenomena: transient localization of particles' positions, strong spatial and temporal fluctuations of the dynamics of individual particles, etc. The description of these dynamics, even at the level of a mean-field theory, is still being developed. I will present a simple theory for the dynamics of mean-field-like models of glass forming liquids. Next, I will turn to the glassy dynamics of one particle moving in a frozen matrix and, in this simpler case, show how one can incorporate dynamic events neglected in the mean-field approach.