Title: Universal minimal dynamical systems

Abstract: Each topological group G admits a unique universal minimal dynamical system (M(G), G). For a locally compact non-compact group this is a nonmetrizable dynamical system with rich algebraic structure, on which G acts effectively. However there are topological groups for which M(G) is the trivial one point system (extremely amenable groups), as well as topological groups G for which M(G) is a metrizable space and for which one has an explicit description. It turns out that these questions are connected with the phenomenon of concentration of mass in high dimensional structures. I will review some results obtained recently in this area by Pestov, Uspenskji and Weiss and myself. In particular I will identify (M(S), S), for the topological group S = S(ℤ) of all the permutations of the integers ℤ with the topology of pointwise convergence, as a symbolic dynamical system.