**Title:** Renormalization and deformation methods in parabolic dynamics

**Abstract:** Parabolic dynamics is characterized by slow, at most polynomial divergence of nearby orbits with time. Since most techniques of dynamical systems have been developed for systems with some hyperbolicity, that is, with exponential divergence of orbits, parabolic systems are often hard to study.

Important examples include homogeneous unipotent flows, for which Ratner proved fundamental results on the classification of invariant measures, and piecewise linear systems such as billiards in polygons. In this talk we will focus on the problem of establishing speed of convergence of ergodic averages for a class of nillflows closely related to Weyl sums of analytic number theory and on the problem of establishing ergodicity for billiards in polygons. The approach we will emphasize is derived from the idea of renormalization that applies to systems which are approximately self-similar at different scales. However, renormalization seems to fail in many cases (nilflows on higher step nilmanifolds, billiards in non-rational polygons) leading to extremely challenging problems. We will argue that renormalization can be generalized in an effective way drawing from ideas coming from the degeneration of Riemannian manifolds under deformations of the metric.