Periodic surfaces and deterministic random walks: dynamics between geometry and probability

Many deterministic systems display chaotic features: the stronger the chaotic features, the better the system can be by approximated by a probabilistic model, an idea that can be traced back to Boltzmann and explains the success of the branch of dynamical systems known as ergodic theory. In this talk we will discuss systems which display only 'mild’ chaotic features, such as the geodesic flow on surfaces which a flat geometry or the Ehrenfest model in mathematical physics.
Recent breakthroughs on our understanding of the latter model, introduced more than a century ago, were made possible by the powerful tools exploiting moduli spaces of surfaces and Teichmueller dynamics, an area which has attracted the work of several Fields medallists. We will in particular highlight some results of probabilistic flavor that can still be proven for these deterministic systems, hidden in the fractal structure of trajectories and the 'deterministic random walks' that describe them.