To participate in this seminar, please register here: https://www.msri.org/seminars/25205

This is one of the research seminars for the RAS program, that distinguishes itself from the postdocs and program associates seminars in that speakers are chosen among Research Members, Research Professors with occasional outside speakers.

Abstract: A natural dynamical system associated with any given countable group G is the space PD_1(G) consisting of (normalized) positive definite functions on G. More generally, if G acts on H by automorphisms then PD_1(H) is a G-dynamical system. The case H=Z^n and G=GL_n(Z) is a prominent example. Here, by duality, the extremal positive definite functions could be seen as points in the n-dimensional torus, providing one of the most ever studied dynamical systems. Here I will only mention Furstenberg's stiffness result for a fine ergodic theoretical perspective on this system. In contrast, it is interesting to note how little attention such dynamical systems were obtained in case the group H is assumed to be a nilpotent group, e.g the discrete Heisenberg group. On the other extreme, the action of G on PD_1(G) itself was recently obtained much attention by operator theorists, culminating into a recent breakthrough by Boutonnet-Houdayer. In my talk I will survey the subject, explain why the above described dynamical systems are crucial to our understanding of dynamical, ergodical and representation theoretical properties of groups, and I will explain the state of the art and our conjectural picture for arithmetic groups. The talk will be based on joint works with Boutonnet-Houdayer-Petreson and with Itamar Vigdorovich.