This talk by Tim Austin, at the University of Warwick, will be an introduction to "almost periodic entropy".  This quantity is defined for positive definite functions on a countable group, or more generally for positive functionals on a separable C*-algebra.  It is an analog of Lewis Bowen's "sofic entropy" from ergodic theory.  This analogy extends to many of its properties, but some important differences also emerge.  Tim will not assume any prior knowledge about sofic entropy.

After setting up the basic definition, Tim will focus on the special case of finitely generated free groups, about which the most is known.  For free groups, results include a large deviations principle in a fairly strong topology for uniformly random representations.  This, in turn, offers a new proof of the Collins—Male theorem on strong convergence of independent tuples of random unitary matrices, and a large deviations principle for operator norms to accompany that theorem.

In this talk, Konstantin Recke, University of Oxford,  will report on some results pertaining to the interplay between geometric group theory, operator algebras and probability theory. Konstantin will introduce so-called invariant percolation models from probability theory and discuss their relation to geometric and analytic properties of groups such as amenability, the Haagerup property (a-T-menability), $L^p$-compression and Kazhdan's property (T). Based on joint work with Chiranjib Mukherjee (Münster).