The seminar meets on Thursdays, 11:10-12:00, in -101

2021–22–A meetings

Oct 21, In Building 34, room 14 Intermediate subalgebras of commutative crossed products of discrete group actions.Online Tattwamasi Amrutam (Ben-Gurion University)

In this talk, we shall focus our attention on intermediate subalgebras of $C(X)\rtimes_r\Gamma$ (and $L^{\infty}(X,\nu)\ltimes\Gamma$). We begin by describing the construction of the commutative crossed product $C(X)\rtimes_r\Gamma$ and how the group contributes to its structure. We shall talk about various (generalized) averaging properties in this context. As a first application, we will show that every intermediate $C^*$-subalgebra $\mathcal{A}$ of the form $C(Y)\rtimes_r\Gamma\subseteq\mathcal{A}\subseteq C(X)\rtimes_r\Gamma$ is simple for an inclusion $C(Y)\subset C(X)$ of minimal $\Gamma$-spaces whenever $C(Y)\rtimes_r\Gamma$ is simple. We shall also show that, for a large class of actions of $C^*$-simple groups $\Gamma\curvearrowright X$, including non-faithful action of any hyperbolic group with trivial amenable radical, every intermediate $C^*$-algebra $\mathcal{A}$, $C_r^*(\Gamma)\subset \mathcal{A}\subset C(X)\rtimes_r\Gamma$ is a crossed product of the form $C(Y)\rtimes_r\Gamma$, $C(Y)\subset C(X)$ is an inclusion of $\Gamma$-$C^*$-algebras.

Nov 4, In Building 34, room 14 Bohr Chaos and Invariant MeasuresOnline Matan Tal (The Hebrew University)

A topological dynamical system is said to be Bohr chaotic if for any bounded sequence it possesses a continuous function that correlates with the sequence when evaluated along some orbit. The theme of the lecture will be the relation of this property to an abundance of invariant measures of the system.

Nov 11 Allosteric actions of surface groupsOnline Matthieu Joseph (École normale supérieure de Lyon)

In a recent work, I introduced the notion of allosteric actions: a minimal action of a countable group on a compact space, with an ergodic invariant measure, is allosteric if it is topologically free but not essentially free. In the first part of my talk I will explain some properties of allosteric actions, and their links with Invariant Random Subgroups (IRS) and Uniformly Recurrent Subgroups (URS). In the second part, I will explain a recent result of mine: the fundamental group of a closed hyperbolic surface admits allosteric actions.

Nov 18, In Building 34, room 14 Introduction to bounded cohomologyOnline Anton Hase (Ben-Gurion University)

While there are earlier works on bounded cohomology, the topic was popularized by Gromov in 1982. In this introductory talk, we will give definitions of bounded cohomology of discrete groups with trivial coefficients. We will interpret bounded cohomology in low degrees in terms of quasimorphisms and central extensions. Then we will mention a few examples of how bounded cohomology has proved useful in applications, before concentrating on the classification of circle actions

Nov 25 TBAOnline Trip to the desert of the PET seminar group!!
Dec 2 Stabilizers in group Cantor actions and measuresOnline Olga Lukina (University of Vienna)

Given a countable group G acting on a Cantor set X by transformations preserving a probability measure, the action is essentially free if the set of points with trivial stabilizers has a full measure. In this talk, we consider actions where no point has a trivial stabilizer, and investigate the properties of the points with non-trivial holonomy. We introduce the notion of a locally non-degenerate action, and show that if an action is locally non-degenerate, then the set of points with trivial holonomy has full measure in X. We discuss applications of this work to the study of invariant random subgroups, induced by actions of countable groups. This is joint work with Maik Gröger.

Dec 9 Symbolic discrepancy and Pisot dynamicsOnline Valérie Berthé (Université de Paris)

Discrepancy is a measure of equidistribution for sequences of points. A bounded remainder set is a set with bounded discrepancy, that is, the number of times it is visited differs by the expected time only by a constant. We discuss dynamical, symbolic, and spectral approaches to the study of bounded remainder sets for Kronecker sequences. We consider in particular discrepancy in the setting of symbolic dynamics and we discuss the existence of bounded remainder sets for some families of zero entropy subshifts. Note that bounded discrepancy has also to do with the notion of bounded displacement to a lattice in the context of Delone sets. We focus on the case of Pisot parameters for toral translations and then show how to construct symbolic codings in terms of multidimensional continued fraction algorithms.
This is joint work with W. Steiner and J. Thuswaldner.

Dec 16 Linnik’s basic lemma with uniformity over the base fieldOnline Andreas Wieser (The Hebrew University)

Long periodic geodesics on the unit tangent bundle of the modular surface are not necessarily equidistributed. However, there is a natural way to group finitely many geodesics together so that the so-obtained unions do equidistribute. This theorem (in this instance going back to Duke ‘88) is very well studied nowadays. In the talk, we discuss a dynamical approach due to Linnik through what is nowadays called Linnik’s basic lemma (providing in particular an entropy lower bound). We present here a new proof of Linnik’s basic lemma based on geometric invariant theory. This is joint work with Pengyu Yang.

Dec 23 TBAOnline
Dec 30 Non-classifiability of ergodic flows up to time changeOnline Philipp Kunde (Universität Hamburg )
Jan 6, In Building 34, room 14 The Ramanujan Machine: Polynomial Continued Fraction and Irrationality MeasureOnline Nadav Ben-David (Ben-Gurion University)

Apéry’s proof of the irrationality of ζ(3) used a specific linear recursion that formed a Polynomial Continued Fraction (PCF). Similar PCFs can prove the irrationality of other fundamental constants such as 𝜋 and e. However, in general, it is not known which ones create useful Diophantine approximations and under what conditions they can be used to prove irrationality. Here, we will present theorems and general conclusions about Diophantine approximations created from polynomial recursions. Specifically, we generalize Apéry’s work from his particular choice of PCF to any general PCF, finding the conditions under which a PCF can be used to prove irrationality or to provide an efficient Diophantine approximation. We further propose new conjectures about Diophantine approximations based on PCFs. Our study may contribute to ongoing efforts to answer open questions such as the proof of the irrationality of the Catalan constant or of values of the Riemann zeta function (e.g., ζ(5)).

Jan 13 Substitutions on compact alphabetsOnline Dan Rust (The Open University (UK))

Substitutions and their subshifts are classical objects in symbolic dynamics representing some of the most well-studied and ‘simple’ aperiodic systems. Classically they are defined on finite alphabets, but it has recently become clear that a systematic study of substitutions on infinite alphabets is needed. I’ll introduce natural generalizations of classical concepts like legal words, repetitivity, primitivity, etc. in the compact Hausdorff setting, and report on new progress towards characterising unique ergodicity of these systems, where surprisingly, primitivity is not sufficient. As Perron-Frobenius theory fails in infinite dimensions, more sophisticated technology from the theory of positive (quasicompact) operators is employed. There are still lots of open questions, and so a ground-level introduction to these systems will hopefully be approachable and stimulating.

This is joint work with Neil Mañibo and Jamie Walton.