Mar 24

Entropy, ultralimits and Poisson boundariesOnline

Elad Sayag (TelAviv University)

In many important actions of groups there are no invariant measures. For example: the action of a free group on its boundary and the action of any discrete infinite group on itself. The problem we will discuss in this talk is ‘On a given action, how invariant measure can be?’. Our measuring of noninvariance will be based on entropy (fdivergence).
In the talk I will describe the solution of this problem for the free group acting on its boundary and on itself. For doing so we will introduce the notion of ultralimit of Gspaces, and give a new description of the PoissonFurstenberg boundary of (G,k) as an ultralimit of G action on itself, with ‘Abel sum’ measures. Another application will be that amenable groups possess KLalmostinvariant measures (KL stands for the KullbackLeibler divergence).
All relevant notions, including the notion of PoissonFurstenberg boundary and the notion of ultrafilters will be explained during the talk.
This is a master thesis work under the supervision of Yehuda Shalom.

Mar 31

The rigidity of lattices in products of treesOnline

Annette Karrer (Technion)

Each complete CAT(0) space has an associated topological space, called visual boundary, that coincides with the Gromov boundary in case that the space is hyperbolic. A CAT(0) group G is called boundary rigid if the visual boundaries of all CAT(0) spaces admitting a geometric action by G are homeomorphic. If G is hyperbolic, G is boundary rigid. If G is not hyperbolic, G is not always boundary rigid. The first such example was found by CrokeKleiner.
In this talk we will see that every group acting freely and cocompactly on a product of two regular trees of finite valence is boundary rigid. That means that every CAT(0) space that admits a geometric action of any such group has the boundary homeomorphic to a join of two copies of the Cantor set. The proof of this result uses a generalization of classical dynamics on boundaries introduced by Guralnik and Swenson. I will explain the idea of this generalization by explaining a higherdimensional version of classical Northsouthdynamics obtained this way.
This is a joint work with Kasia Jankiewicz, Kim Ruane and Bakul Sathaye.

Apr 7

TBAOnline



Apr 14

Passover break

Holiday


Apr 21

Passover break

Holiday


Apr 28

Mean dimension of an action and the radius of comparison of its C*algebraOnline

Chris Phillips (University of Oregon)

For an action of a countable amenable group $G$ on a compact metric
space $X$, the mean dimension $mdim (G, X)$ was introduced by
Lindenstrauss and Weiss, for reasons unrelated to $C^*$algebras. The
radius of comparison $rc (A)$ of a $C^*$algebra $A$ was introduced by
Toms, for use on $C^*$algebras having nothing to do with dynamics.
A construction called the crossed product $C^* (G, X)$ associates a
$C^*$algebra to a dynamical system. There is significant evidence for
the conjecture that $rc ( C^* (G, X) ) = (1/2) mdim (G, X)$ when the
action is free and minimal. We give the first general partial results
towards the direction $rc ( C^* (G, X) ) \geq (1/2) mdim (G, X)$.
We don’t get the exact conjectured bound, but we get nontrivial
results for many of the known examples of free minimal systems with
$mdim (G, X) > 0$. The proof depends, among other things, on Cech
cohomology, and uses something we call the mean cohomological
independence dimension. Unlike the currently known results in the
other direction, it works for all choices of $G$.
The talk will include something about the crossed product
construction; no previous knowledge of it will be assumed.
This is joint work with Ilan Hirshberg.

May 5

Yom Ha’Atzmaut

Holiday


May 12

Density of oscillating sequences in the real lineOnline

Ioannis Tsokanos (The University of Manchester)

In this talk, we study the density properties in the real line of oscillating sequences of the form
$( g(k) \cdot F(kα) )_{k \in \mathbb{N}}$,
where $g$ is a positive increasing function and $F$ a real continuous $1$periodic function.
This extends work by Berend, Boshernitzan and Kolesnik who established differential properties on the function F ensuring that the oscillating sequence is dense modulo 1.
More precisely, when $F$ has finitely many roots in $[0,1)$, we provide necessary and sufficient conditions for the oscillating sequence under consideration to be dense in $\mathbb{R}$. All the related results are stated in terms of the Diophantine properties of $α$, with the help of the theory of continued fractions.

May 19

Characters of groups, stability and sofic dynamical systemsOnline

Arie Levit (TelAviv University)

We study the character theory of infinite solvable groups, focusing on the metabelian and polycyclic cases. This theory has applications towards the HilbertSchmidt stability of such groups  whether almosthomomorphisms into the unitary groups U(n) are nearby honest homomorphisms? We explore an interesting link between stability and topological dynamics via a notion of “sofic dynamical systems”. I will introduce all relevant notions.
The talk is based on a joint work with Itamar Vigdorovich.
