This talk is about the entropy of algebraic actions of the Heisenberg groups.

Abstract: TBA

If $(g_1,...,g_k)$ is a $k$-tuple of elements of $R/Z$, then the inequality $|n_1g_1+...+n_kg_k| < (\max n_i)^{-a}$ always has infinitely many integer solutions $(n_i)$ as soon as $a < k$. Conversely, if $a>k$, then for almost every $(g_1,...,g_k)$, there is only finitely many solutions. Starting with the Heisenberg group of dimension $3$, I will explain how to generalize the above results to any connected nilpotent Lie group. One of the key tools in the proofs is the correspondence between homogeneous dynamics and Diophantine approximation, as developed by Dani, Kleinbock and Margulis. (Joint work with Menny Aka, Emmanuel Breuillard and Lior Rosenzweig.)

On the torus $R/Z$, there exists a continuous function $f$ such that all convolution powers $f^{*k}$, $k\geq 1$, are nowhere differentiable. One aim of the talk will be to explain why such a function cannot exist on a connected perfect Lie group (e.g. $SL(d,R)$ or the group of affine transformations of $R^3$).

The proof is based on a bound on coefficients of the regular representation of a perfect Lie group, due to Bourgain in the case of $SU(d)$ and to Boutonnet, Ioana and Salehi Golsefidi in the case of simple Lie groups. As another application of that bound, a perfect group does not admit proper Borel measurable subgroups of arbitrarily large dimension. (Joint work with Yves Benoist.)

The uniform spanning forest (USF) of an infinite connected graph $G$ is the weak limit of the uniform spanning tree measure taken on exhausting finite subgraphs of $G$. It is easy to see that it is supported on spanning graphs of $G$ with no cycles, but it need not be connected. Indeed, a classical result of Pemantle ('91) asserts that when $G=Z^d$, the USF is almost surely a connected tree if and only if $d=1,2,3,4$.

We prove that when $G$ is a Cayley graph (or more generally, a unimodular random network) one cannot distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely. This confirms a conjecture of Benjamini, Lyons, Peres and Schramm 2001.

Joint work with Tom Hutchcroft.

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA

Abstract: TBA