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.

May 26

Universality for R^dflowsOnline

Shrey Sanadhya (BenGurion University)

A dynamical system is called universal if any system with lower entropy can be embedded into it. In this talk, we will discuss universality for $R^d$ flows $(d>1)$ both in ergodic and Borel contexts. We will discuss a specification property that implies universality for $R^d$ flows and provide an example of a tiling dynamical system with this specification property. This is ongoing work with Tom Meyerovitch. This talk is a preliminary report.

Jun 2, In room 106, building 28

Probabilistic Takens timedelay embeddingsOnline

Adam Śpiewak (BarIlan University)

Consider a dynamical system (X,T) consisting of a compact set X in the Euclidean space and a transformation T on X. Takenstype timedelay embedding theorems state that for a generic realvalued observable h on X, one can reconstruct uniquely the initial state x of the system from a sequence of values of h(x), h(Tx), …, h(T^{k1} x), provided that k is large enough. In the deterministic setting, the number of measurements k has to be at least twice the dimension of the state space X. This was proved in several categories and can be seen as dynamical versions of the classical (nondynamical) embedding theorems. We provide a probabilistic counterpart of this theory, in which one is interested in reconstructing almost every state x, subject to a given probability measure. We prove that in this setting it suffices to take k greater than the Hausdorff dimension of the considered measure, hence reducing the number of measurements at least twice. Using this, we prove a related conjecture of Shroer, Sauer, Ott and Yorke in the ergodic case. We also construct an example showing that the conjecture does not hold in its original formulation. This is based on joint works with Krzysztof Barański and Yonatan Gutman.

Jun 9, In room 106, building 28

Topological models of abstract commensuratorsOnline

Edgar Bering (Technion)

Given a group G, an EilenbergMacLane space X = K(G,1) provides a topological model of both G and Aut(G). The latter is understood via Whitehead’s theorem as the group of pointed homotopy equivalences of X up to homotopy. When X has rich structure, such as the case of a closed surface group, this point of view leads to a rich understanding of Aut(G). Motivated by dynamics and mathematical physics, Biswas, Nag, and Sullivan initiated the study of the universal hyperbolic solenoid, the inverse limit of all finite covers of a closed surface of genus at least two. Following their work, Odden proved that the mapping class group of the universal hyperbolic solenoid is isomorphic to the abstract commensurator of a closed surface group. In this talk I will present a general topological analog of Odden’s theorem, realising Comm(G) as a group of homotopy equivalences of a space for any group of type F. I will then use this realisation to classify the locally finite subgroups of the abstract commensurator of a finiterank free group. This is joint work with Daniel Studenmund.

Jun 16, In room 106, building 28

(Non)Integrability of quaternionKähler symmetric spacesOnline

Anton Hase (BenGurion University)

It is a famous result of HarishChandra that every noncompact Hermitian symmetric space can be realized as a bounded domain in a complex vector spaces. If we replace the complex numbers by the division algebra of quaternions in the definition of Hermitian symmetric spaces, we obtain the class of quaternionKähler symmetric spaces. While these spaces emerge in an analogous way, we show that there is no quaternionic analogue of HarishChandra’s embedding theorem: A quaternionKähler symmetric space is integrable if and only if it is a quaternionic vector space, quaternionic hyperbolic space or quaternionic projective space. In the talk I will explain some of the background and some of the tools used in the proof.

Jun 23, In room 106, building 28

Hausdorff and packing measure of some decimal and Luroth expansionsOnline

Daniel Ingebretson (BenGurion University)

A common method for quantifying the size of sets of Lebesgue measure zero is via Hausdorff or packing dimension. A more delicate question is to determine the value of the corresponding Hausdorff or packing measure at dimension. In this talk we will show a way to approach this question for some simple fractal sets arising from numeration systems.

Jun 30, In room 106, building 28

Extremal independence in discrete random systemsOnline

Maksim Zhukovskii (Weizmann Institute)

Let G be a graph with several vertices v_1,..,v_r being roots. A Gextension of u_1,..,u_r in a graph H is a subgraph G* of H such that there exists a bijection from V(G) to V(G*) that maps v_i to u_i and preserves edges of G with at least one nonroot vertex. It is well known that, in dense binomial random graphs, the maximum number of Gextensions obeys the law of large numbers. The talk is devoted to new results describing the limit distribution of the maximum number of Gextensions. To prove these results, we develop new bounds on the probability that none of a given finite set of events occur. Our inequalities allow us to distinguish between weakly and strongly dependent events in contrast to wellknown Janson’s and Suen’s inequalities as well as Lovasz Local Lemma. These bounds imply a general result on the convergence of maxima of dependent random variables.
