### Speakers

Miklós Abért (MTA Alfréd Rényi Institute of Mathematics): **A measured approach to sofic entropy.**

Vadim Alekseev (Technische Universität Dresden): **Coarse geometry of sofic approximations.**

Andrei Alpeev (St. Petersburg State University): **Maximal sofic approximations.**

Lewis Bowen (University of Texas at Austin): **A counterexample to the Weak Pinsker conjecture for free group actions.**

Felipe García-Ramos (CONACyT & Universidad Autonoma de San Luis Potosi): **Entropy pairs for pseudo orbit tracing amenable actions.**

Anna Giordano Bruno (Università di Udine):** Algebraic entropy for amenable semigroup actions on abelian groups.**

Benjamin Hayes (University of Virginia): **Algebraic Actions: A Max-Min Principle for Local Wk*-Convergence.**

David Kerr (Texas A&M University): **Almost finiteness and the small boundary property.**

Hanfeng Li (University at Buffalo): **Garden of Eden and Specification.**

Douglas Lind (University of Washington): **Sub-entropies for algebraic Z**^{d}-actions.

Alejandro Maass (DIM-UChile): **Local entropy theory, a “kind” of review.**

Klaus Schmidt (University of Vienna): **Entropy and periodic points of algebraic actions of discrete groups.**

Brandon Seward (NYU Courant Institute): **Bernoulli shifts with bases of equal entropy are isomorphic.**

Balázs Szegedy (MTA Alfréd Rényi Institute of Mathematics): **Spectral information via entropy methods.**

Andreas Thom (Technische Universität Dresden): **Stability of asymptotic representations, cohomology vanishing, and non-approximable groups.**

Benjamin Weiss (Hebrew University of Jerusalem): **Some open problems in entropy theory for groups.**

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### Abstracts

Miklós Abért (MTA Alfréd Rényi Institute of Mathematics): **A measured approach to sofic entropy.**

**Abstract: **Classical sofic entropy theory, a la Lewis Bowen, counts the number of good colorings (models) on a sofic approximation to define the basic invariant of interest. In our approach (also independently investigated by Tim Austin) one models the infinite process by a random coloring and uses classical Shannon's entropy to define sofic entropy. This approach suggests a natural Yuzvinsky addition formula for factor maps. We prove a version of that, but the strongest possible result in this direction is still open. A joint work with Benjy Weiss.

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Vadim Alekseev (Technische Universität Dresden): **Coarse geometry of sofic approximations**

**Abstract: **In joint work with Martin Finn-Sell, we prove that coarse geometric properties of the box space of graphs (property A, asymptotic coarse embeddability into Hilbert space, geometric property (T)) attached to a sofic approximation of a finitely generated group imply corresponding analytic properties of the group (amenability, a-T-menability and property (T)), thus extending ideas and results present in the literature for residually finite groups and their box spaces to sofic groups. Moreover, we generalise coarse rigidity results for box spaces due to Kajal Das, proving that coarsely equivalent sofic approximations of two groups give rise to a uniform measure equivalence between the groups.

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Andrei Alpeev (St. Petersburg State University): **Maximal sofic approximations.**

**Abstract: **We prove the existence of (non-unique) maximal sofic action with respect to the weak containment order. For any sofic group we define a non-empty class of sofic approximations such tha all the sofic actions could be approximated by any of them in LDE way. As a corollary we obtain that for these approximations measure sofic entropy of algebraic actions is either minus infinity or equals to the topological one.

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Lewis Bowen (University of Texas at Austin): **A counterexample to the Weak Pinsker conjecture for free group actions.**

**Abstract: **The Weak Pinsker conjecture posits that any ergodic measure-preserving transformation decomposes as a direct product of a Bernoulli shift and a low-entropy transformation. There is a natural generalization to ergodic actions of groups. I’ll explain a counterexample, which is an action of a nonabelian free group, that arises from the hardcore model on the d-regular tree. The proof is based on a new measure-conjugacy invariant that counts the number of clusters of pseudo-periodic orbits.

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Felipe García-Ramos (CONACyT & Universidad Autonoma de San Luis Potosi): **Entropy pairs for pseudo orbit tracing amenable actions.**

**Abstract: **Joint work with Sebastián Barbieri. Using asymptotic pairs we will characterize topological entropy pairs for pseudo orbit tracing amenable actions. We will construct different shift of finite type examples with completely positive entropy answering a question by Ronnie Pavlov.

Anna Giordano Bruno (Università di Udine):** Algebraic entropy for amenable semigroup actions on abelian groups.**

**Abstract: **[pdf]

Benjamin Hayes (University of Virginia): **Algebraic Actions: A Max-Min Principle for Local Wk*-Convergence.**

**Abstract: **Given an algebraic action of G on X, I will discuss results investigating when there is a sequence of measures supported on topological microstates which locally wk* converge to the Haar measure. In particular, I show that the largest “generalized subgroup” which is a “generalized local wk*-limit” is equal to the smallest “generalized subgroup” which “absorbs all topological microstates.” The goal of the talk will be to define these scare quotes and make them less scary. This involves use of the Loeb measure space.

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David Kerr (Texas A&M University): **Almost finiteness and the small boundary property.**

**Abstract: **The notion of almost finiteness for group actions on compact spaces is an analogue of both hyperfiniteness in the setting of measure-preserving actions and of Z-stability in the setting of C*-algebras and is related to dynamical comparison in a way that is reminiscent of the link between Z-stability and strict comparison in the Toms-Winter conjecture.

I will explain how this relationship with dynamical comparison can be illuminated by means of the small boundary property and the concept of almost finiteness in measure, leading to new classification results for crossed product C*-algebras.

This is joint work with Gabor Szabo.

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Hanfeng Li (University at Buffalo): **Garden of Eden and Specification.**

**Abstract: **A garden of Eden theorem for a topological dynamical system means that for every continuous equivariant map T from the underlying space to itself, T is surjective if and only if it is injective on each homoclinic class. I will discuss a garden of Eden theorem for expansive algebraic actions of amenable groups with the weak specification property. The main tool is entropy.

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Douglas Lind (University of Washington): **Sub-entropies for algebraic Z**^{d}-actions.

**Abstract: **An action of Z^{d} by commuting automorphisms of a compact abelian group may have zero entropy, and yet finite positive entropies along lower dimensional subspaces. Ledrappier’s classic example illustrates this phenomenon very well. For prime actions, commutative algebra predicts the correct dimension k for such behavior, and so for every k-tuple of linearly independent vectors in R^{d} there is an associated “directional” entropy. For k-tuples spanning the same subspace, this is actually a k-form. A consequence of the expansive subdynamics machinery is that within an expansive component of subspaces this form has constant coefficients. When k = 1 there is an explicit description of this form, but for larger k the problem of computing this form is a fascinating open problem. I will discuss aspects of this question, as well as the more general problem of computing Milnor’s “entropy geometry” for algebraic actions, which is open even for Ledrappier’s example.

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Alejandro Maass (DIM-UChile): **Local entropy theory, a “kind” of review.**

**Abstract: **In this talk I will review the first steps together with some new advances and application of the local entropy theory developed from the 90’s by Francois Blanchard and collaborators.

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Klaus Schmidt (University of Vienna): **Entropy and periodic points of algebraic actions of discrete groups.**

**Abstract: **Let G be a countable group of automorphisms of a compact abelian group X. A point x in X is G-periodic if its G-orbit is finite. This talk addresses the following problems:

1. When does G have nontrivial periodic points?

2. When are the G-periodic points dense in X?

3. When is the logarithmic growth rate of the number of G-periodic points equal to the topological entropy of the G-action on X?

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Brandon Seward (NYU Courant Institute): **Bernoulli shifts with bases of equal entropy are isomorphic.**

**Abstract:** The well known Ornstein isomorphism theorem states that if two alphabets have the same Shannon entropy, then the corresponding Bernoulli shifts over the group integers are isomorphic. Stepin proved that the class of groups satisfying this theorem is closed under extensions, and Ornstein and Weiss proved that all amenable groups satisfy this theorem. A few years ago Lewis Bowen made significant progress by proving that the isomorphism theorem holds for all countably infinite groups provided that both alphabets have at least 3 letters (ignoring sets of measure zero). In this talk I will show that the isomorphism theorem holds for all countably infinite groups without any additional assumptions.

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Balázs Szegedy (MTA Alfréd Rényi Institute of Mathematics): **Spectral information via entropy methods.**

**Abstract: **In this talk, we describe a set of tools based on graph limits and entropy inequalities that can be used to characterise almost eigenvectors in various models of random matrices. One particular model of interest is random regular graphs. Another one is a random matrix with +-1 entries. Joint work with Agnes Backhausz.

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Andreas Thom (Technische Universität Dresden): **Stability of asymptotic representations, cohomology vanishing, and non-approximable groups.**

**Abstract: **First of all, I plan to survey various approximation properties of discrete groups and mention a few applications to the theory of groups and group rings. We provide the first examples of countable groups which are not Frobenius-approximated. Our strategy is to use higher-dimensional cohomology vanishing phenomena to prove that any Frobenius-almost homomorphism into finite-dimensional unitary groups is close to an actual homomorphism and combine this with existence results of certain non-residually finite central extensions of lattices of higher rank. We ultimately rely on work of Garland, Ballmann-Świątkowski, Deligne, Rapinchuk and others.* *

Benjamin Weiss (Hebrew University of Jerusalem): **Some open problems in entropy theory for groups.**

**Abstract: **The last decade has seen many dramatic new developments in the entropy theory for non-amenable groups. I will briefly mention some of these and describe a few of the open problems that remain.

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