Ebrahim Samei, Saskatoon: Let $G$ be a countable discrete group, and let $\mu$ be a probability measure on $G$ with finite (Shannon) entropy. We initiate the study of several related concepts associate to a probability measure $\mu$ and exploit their relations. First, we look at the concept of {\it Lyapunov exponent} of $\mu$ with respect to weights on $G$ and build a framework that connects it to the entropy of $\mu$ in $G$. This is done by introducing a generalization of Avez entropy, taking into account the given weight, and investigating in details their relations together as well as to the actions of $G$ on measurable stationary spaces. As a byproduct of our techniques, we show that if $G$ has rapid decay w.r.t. a length function $\fL$ and $\mu$ has a finite logarithm moment (w.r.t. $\fL$), the weak containment of the representation $\pi_X$ of $G$ on a $\mu$-stationary space $(X,\xi)$ implies that $$h(G,\mu)=h_\mu(X,\xi),$$ where $h_\mu(X,\xi)$ is the Furstenberg entropy of $(X,\xi)$. This allows us to characterize amenable action of $(G,\mu)$ on stationary spaces: $(X,\xi)$ is an amenable $(G,\mu)$-space if and only if it is a measure-preserving extension of the Poisson boundary of $(G,\mu)$. In particular, if $(X,\xi)$ is a boundary, then $\pi_X$ is weakly contained in $\lambda_G$ if and only if $(X,\xi)$ coincides with the Furstenberg-Poisson boundary of $(G,\mu)$. Hence the action of $G$ on a proper $\mu$-boundary of $G$ is not amenable. This extends the results of Nevo, Zimmer, and others on many hyperbolic like groups. This is a join work with Benjamin Anderson-Sackaney, Tim de Laat, and Matthew Wiersma.

Dietmar Bisch, Vanderbilt University: Since Vaughan Jones introduced the theory of subfactors in 1983, it has been an open problem to determine the set of Jones indices of irreducible, hyperfinite subfactors. Not much is known about this set. Julio Caceres and I have recently shown that all indices of finite depth subfactors between 4 and 5 are realized by new hyperfinite subfactors with Temperley-Lieb-Jones standard invariant as well. They are non-amenable, but have certain nice asymptotic commutativity properties. Our work leads to a conjecture and some results regarding Jones' problem. The construction involves new families of commuting squares, a graph planar algebra embedding theorem, and a few tricks that allow us to avoid solving large systems of linear equations to compute invariants of our subfactors. If there is time, I will mention connections to quantum Fourier analysis and QIT.