**Courses**

### David Kerr

**Title: ** Sofic entropy and orbit equivalence

**Abstract:** The first part of this minicourse will be an introduction to amenable and sofic groups and the entropy theory of their actions, both on probability spaces by measure-preserving transformations and on compact metrizable spaces by homeomorphisms. The second part will examine the relation between dynamical entropy and orbit equivalence, with a focus on results of Austin and of Kerr and Li that hinge around the geometric notion of sparse connectivity.

### Florian Richter

**Title: ** Dynamical methods in additive combinatorics

**Abstract:** The purpose of this mini-course is to explore various applications of topological dynamics and ergodic theory to combinatorics and number theory. The mutually perpetuating relationship between these fields started with Furstenberg’s ergodic-theoretic proof of Szemeredi’s Theorem, and has since evolved into its own area of research called ergodic Ramsey theory, which brings together ideas from number theory, harmonic analysis, combinatorics, and dynamical systems. One of the basic general questions in this area asks what types of arithmetic patterns are unavoidable in large subsets of the integers. Problems of this kind become especially formidable if the patterns in question are infinite. Results dealing with infinite patterns, such as Ramsey’s Theorem, Jin’s Theorem, Hindman’s Theorem, and Erdos’ Sumset Conjectures, will be one of the focal points of this course. We will cover most of the dynamical background needed and the course should be accessible to a wide audience.

**Short Talks**

### Anush Tserunyan

**Title: ** Cost and treeability

**Abstract:** “Groups, as [people], will be known by their actions”–Guillermo Moreno. Measured group theory studies groups via their probability-measure-preserving (pmp) actions, often forgetting the action itself and only remembering its orbit equivalence relation. This study is partly fuelled by drawing parallels between properties of groups and their orbit equivalence relations. In this talk we discuss cost and treeability of pmp equivalence relations – the analogues of rank and freeness of groups. We illustrate how, just like rank, the cost of the orbit equivalence relations of free pmp actions of free groups determines the group (Gaboriau 1997) and discuss further related results and open questions.

### Ben Hayes

**Title: ** Sofic entropy of algebraic actions

**Abstract:** An algebraic action is an action of a countable, discrete, group G on a compact, metrizable group X by continuous automorphisms. The study of algebraic actions has exciting connections to noncommutative algebra, algebraic geometry, fourier analysis, and functional analysis. I will give a brief introduction to the entropy theory of algebraic actions and discuss how it is related to functional analysis.

### Donald Robertson

**Title: ** IP Sets in Amenable Groups

**Abstract:** Hindman proved that however the natural numbers are finitely partitioned there will be an IP set in one of the cells. Hindman’s result holds in every countable group. In this talk I will describe how dynamics can help to characterize those countable amenable groups in which a density analogue of Schur’s result does hold.

### Renling Jin

**Title: ** Nonstandard but elementary proof of Szemeredi’s Theorem

**Abstract:** We illustrate, with some help of nonstandard analysis, Szemeredi’s original proof of Szemeredi’s Theorem on arithmetic progressions. The goal of the talk is to make Szemeredi’s idea as transparent as possible.

### Joel Moreira

**Title: ** Multiplicative recurrence and partition regularity

**Abstract:** The theory of recurrence and multiple recurrence in ergodic theory is intimately connected with problems in Ramsey theory seeking to find linear configurations in every set with positive density, or to establish partition regularity of linear equations. In recent years on the Ramsey theory side, attention has turned to polynomial equations and configurations, and the classical theory of recurrence has to be adapted in order to deal with these problems. One natural direction is to look for sets of recurrence for multiplciative actions of the natural numbers. I will introduce these ideas, present some new results in this area and pose several natural related open questions.

### Nikos Frantzikinakis

**Title: ** Furstenberg systems of bounded sequences

**Abstract:** Furstenberg systems are measure preserving systems that are used to model statistical properties of bounded sequences of complex numbers. They offer a different viewpoint for a variety of problems for which progress can be made by a partial or complete description of Furstenberg systems of suitably chosen sequences. In this lecture I will give some examples of Furstenberg systems, and record a few applications and open problems.

### François Le Maître

**Title: ** Belinskaya’s theorem is optimal

**Abstract:** Dye’s theorem states that given any two ergodic measure-preserving transformations on two standard probability spaces, there is an identification of the spaces which also identifies their orbits: they are orbit equivalent. Various refinements of orbit equivalence have been proposed in order to get a nontrivial theory, notably by requiring that the identification of the orbits does not distort them too much. Belinskaya proved in 1968 that for the seemingly mild notion of \(L^1\) orbit equivalence, one gets a completely opposite phenomenon: any two ergodic \(L^1\) orbit equivalent measure-preserving transformations must actually be flip conjugate.

I will report on an ongoing joint work with Alessandro Carderi, Matthieu Joseph and Romain Tessera where we show that the same statement is false for any \(p\) in \((0,1)\) if you replace \(L^1\) by \(L^p\). Noting that \(L^p\) orbit equivalence for some \(p\) in \((0,1)\) implies Shannon orbit equivalence for \(\mathbb{Z}\)-actions, we answer by the negative a question of Kerr and Li who asked whether Shannon orbit equivalence implies flip conjugacy for \(\mathbb{Z}\)-actions.

### Lewis Bowen

**Title: ** Weak Pinsker Entropy

**Abstract:** The weak Pinsker entropy of a group action on a probability space is the supremum of entropies of direct Bernoulli factors. This notion is well-defined for arbitrary probability-measure-preserving group actions. In breakthrough work, Tim Austin proved that wP entropy is equal to classical entropy for ergodic actions of amenable groups. I proved that the analogous statement for free groups is false; with an explicit counterexample based on independent subsets of random regular graphs. With Robin Tucker-Drob, we proved that if a group \(G\) is Bernoulli cocycle-superrigid (e.g. \(\text{PSL}(3,{\mathbb{Z}}))\) then weak Pinsker entropy is an orbit-equivalence invariant of \(G\)-actions.