### Session 1: Vitaly Bergelson – Daniel Glasscock

**Title: ** The Density Polynomial Hales-Jewett (DPHJ) Problem.

**Abstract:** The main topic of the presentation is the Density version of the Polynomial Hales Jewett Theorem, DPHJ. Many known and conjectured results in Ramsey Theory are special cases of DPHJ, but even special cases of the quadratic version of DPHJ remain open. We will discuss the background, some special cases, and, if time permits, some equivalent forms..

### Session 2: Valerie Berthé – Jorg Thuswaldner

**Title: ** Balanced words with factor frequencies.

**Abstract:** The question is to find infinite words with prescribed letter frequencies being balanced on factors and having at most linear factor complexity. An infinite word is balanced on letters if the number of occurrences of this letter in any two factors (i.e., subwords) of this word of the same length differs by a fixed constant. This notion extends to factors by considering occurrences of a given factor. Words that are balanced on factors have bounded symbolic discrepancy.

### Session 3: Jarkko Kari – Siamak Taati

**Title: ** Do low-complexity aperiodic SFTs exist?.

**Abstract:** Given a finite shape \(D\subseteq\mathbb{Z}^d\) and a set of allowed symbolic patterns on \(D\), a configuration of the entire lattice $\mathbb{Z}^d$ in which all the \(D\)-shaped patterns are allowed may or may not exist. In dimensions \(d\geq 2\), there are famous examples in which valid configurations exist but are all non-periodic. In other words, aperiodic SFTs exist. The proposed open problem asks about the low-complexity variant of this result, that is, when the number of allowed patterns on \(D\) is at most \(|D|\). Do low-complexity aperiodic SFTs exist? Some variants of this question as well as partial results will be discussed.

### Session 4: Benjy Weiss – Nishant Chandgotia

**Title: ** Recognising processes from partial data.

**Abstract:** How much of a sample do we need to recognise an ergodic process? While for ergodic processes defined on a probability space there is a very simple answer: one needs to know the process only on a thick set of times, the situation is a little more complicated for ergodic, (Radon, conservative) processes arising from spaces with infinite measure. While we have some partial results, the general question is wide open and shall be the topic of discussion during the talk.

### Session 5: Tomasz Downarowicz – Adam Abrams

**Title: ** Explicit construction of a multiplicatively circle-normal number.

**Abstract:** Classically, a number is “normal in base \(m\)” if it is generic for the Lebesgue measure under the action \((n,x) \to m^n x \mod 1\) of the semigroup \((\mathbb{N},+)\) acting on \([0,1]\). If we instead consider the \(({\mathbb{N}},\cdot)\)-action, \((n,\cdot) \to nx \mod 1\), we get a system that is in some ways familiar and in some ways very different. Unlike the \((\mathbb{N},+)\)-action, it has entropy zero. Defining normality in this multiplicative setting requires Følner sets (a concept from the theory of amenable semigroups). A point is multiplicatively \((F_n)\)-circle normal if it is \((F_n)\)-generic for the Lebesgue measure under the above \((\mathbb{N},\cdot)\)-action. By ergodicity, we know that almost every number is multiplicatively \((F_n)\)-circle-normal, but explicit examples of such numbers are still unknown.

### Session 6: Anthony Quas – Mark Piraino

**Title: ** Phase Transition(s) for matrix equilibrium states.

**Abstract:** Given a family of matrices, a matrix equilibrium state maximizes entropy plus logarithmic growth rate of the norm scaled by a parameter beta. These measures arise in some questions of geometric measure theory. The question that we pose concerns phase transition(s) that occur as beta is varied. There seems to be an analogy with a well-known phase transition in an example of a scalar equilibrium state due to Hofbauer.

### Session 7: Jon Chaika – Samantha Fairchild

**Title: ** Classifying horocycle ergodic measures on translation surfaces.

**Abstract:** Ratner’s measure classification theorem for SL(2,R) acting on a homogeneous space says that any ergodic measure under the action of the unipotent upper triangular matrices (horocycles) are “algebraic.” In other words the horocycle ergodic measures cannot be too exotic. The same result is not true in general when acting on non-homogenous spaces.

Specifically families of translation surfaces are non-homogenous. There are examples of translation surface spaces where an analog of Ratner’s classification does not hold. However in one of the most simple cases where there is only one singularity on the translation surface, the classification is still an open problem.

Slides (PDF)### Session 8: Andrés Navas – Kim Sang-Hyun

**Title: ** Distortion elements in diffeomorphism groups.

**Abstract:** A group element is distorted inside a group \(G\) if the world length of its powers grows sublinearly (with respect to the exponent) inside a finitely generated subgroup containing it. This notion was introduced by Gromov and studied in geometric and dynamical contexts. We will introduce the following question: given \(r > s\), does every manifold support a diffeomorphism that is distorted in the group of \(C^s\) diffeomorphisms yet undistorted in the group of \(C^r\) diffeomorphisms ? Motivation and a very particular (though illustrative) example will be presented for this question.