While the roots of ergodic theory are in the kinetic theory of gases and celestial mechanics, modern ergodic theory stands at the junction of many areas. The goal of these two talks is to acquaint the students with some of the promising directions of research by focusing the discussion on three central topics: recurrence, equidistribution and independence.

We shall discuss the notion of a dynamical system and consider some instructive examples: billiards, unimodal maps, continued-fraction transformations, automorphisms of compact groups and others. Connections with and applications to statistical mechanics, number theory, and combinatorics will be pointed out.

Throughout these lectures we will constantly emphasize the ubiquity of dynamical ideas in diverse, seemingly static situations.