Past Meetings

Speaker: Petra Staynova

Title: A cornucopia of bounds

Abstract: In this talk we consider right-infinite words over a finite alphabet, which are generated via substitution rules. A well-known way of studying complexity of these words is via the question 'how many finite subwords of a given length does this infinite word have?', which gives rise to the notion of subword complexity. If instead one considers what types of (finite) subwords occur as arithmetic subsequences, one can obtain a different and very interesting measure of complexity. In this talk, we consider the occurrence of monochromatic (i.e. same letter) arithmetic progressions within right-infinite words, and provide asymptotic growth rates in some (general) cases. No previous knowledge is assumed, and the talk will start from the basics of substitution systems. 

Speaker: Dan Rust

Title: Random substitutions, entropy and fractals

Abstract: Random substitutions are a relatively new generalisation of the classical notion of a substitution and give rise to tilings that possess long-range order but local disorder, and these two properties 'fight' with each other at all levels of the hierarchy of the tiling. In particular, they have non-trivial long-range correlations but also positive entropy, which is typically rare for tilings. They can be defined and studied in any dimension but for the purposes of this talk, I'll give an introduction to random substitutions in the 1-dimensional (symbolic) setting. After giving a basic overview of their construction, I'll focus on two aspects of their study. First I'll give a brief tour of their word complexity and hence how to calculate their entropy. Then, I'll introduce the Pisot property for random substitutions and discuss how this allows us to construct Rauzy fractals for random substitutions, as well as a closely related family of measures that arises from the probabilistic nature of their construction.

Speaker: Felipe García Ramos

Title: From chaos to order

Abstract: This talk is an overview of the different behaviour we see in dynamical systems. 

Speaker: Reem Yassawi

Title: Mahler equations for Zeckendorf numeration

Abstract: Fixed points of Pisot substitutions can be visualised as projections, via a precompact acceptance window, of cut and project schemes. Fixed points of constant length substitutions are projections, along a diagonal, of a two dimensional array which is the sequence of coefficients of the expansion of a rational function of two variables. In an attempt to understand the relationship between these two results, we define generalised versions of equations of q-Mahler type, which fixed points of constant length-q substitutions satisfy. For simplicity I will focus on substitutions whose characteristic polynomial has the golden mean as leading root. We show that  fixed points of these substitutions satisfy a Zeckendorf-Mahler equation, and conversely, that isolating Zeckendorf-Mahler equations generate such fixed points. This is joint work with Olivier Carton.

Speaker: Daniel Clarke

Title: Investigations into the mechanical properties of aperiodic honeycombs

Abstract: Honeycombs are a class of cellular solids that are widely used in engineering due to desirable properties such as high specific stiffness and energy absorption. Traditionally comprised of periodic arrays of hexagonal, triangular or square cells, these materials have attracted much attention due to the development of additive manufacturing. This removal of manufacturing constraints has led to the study of many novel honeycombs based on periodic honeycombs using a number of strategies. Recent works have focussed on honeycombs and composites based on aperiodic tilings.  Unlike periodic lattices aperiodic lattices lack translational symmetry but can have rotational symmetries other than those permitted by conventional crystallographic arrangements. This promising new area of study has attracted more attention recently due to the discovery of the aperiodic “hat” monotile. Aperiodic lattices provide opportunities to increase the range of available mechanical properties of honeycombs, however they also pose challenges in their generation, testing and simulation due to the lack of periodicity. This presentation aims to disseminate methods and results relating to the study of honeycombs based on aperiodic tilings.

Speaker: Iestyn Jowers

Title: Mechanical properties of the ‘hat’ and ‘hat’ family 

Abstract: Cellular structures are synthetic materials, and can be engineered to have desirable mechanical properties. Our research is concerned with a specific class of cellular structures, honeycombs based on tilings with aperiodic order. Through testing and simulation, we have discovered that these aperiodic honeycombs can give rise to a useful range of isotropic mechanical properties. The recently discovered ‘hat’ monotile introduced a new aperiodic pattern to investigate as the basis of honeycomb structures, and in this presentation the remarkable mechanical properties of these structures will be discussed. Also, unlike most other aperiodic tilings, the ‘hat’ is part of a continuous family of aperiodic tilings, and the mechanical properties of the resulting family of honeycomb structures is also reviewed. We envisage that our findings will benefit the design of components, for example in aerospace, automotive or medical engineering, where cellular structures are used due to their improved performance in energy absorption, heat transfer and weight-to-strength ratio.

Speaker: Jamie Walton

Title: When is a polytopal cut and project set a substitution pattern?

Abstract: The two main methods of constructing aperiodically ordered patterns are by substitution and the cut and project method. Many interesting examples in Aperiodic Order, such as the Penrose tilings and Ammann–Beenker tilings, are both substitution patterns and cut and project sets with polytopal windows. In this talk I will discuss recent work with Edmund Harriss and Henna Koivusalo that gives a simple set of necessary and sufficient conditions for a cut and project scheme with polytopal window to define patterns that are also generated by substitution. This builds off previous results of Harriss and Lamb in this area. A key new feature is a simplification of some core definitions, in particular what is meant by a ‘substitution pattern’. We use a definition similar to that of a LIDS (local inflation deflation symmetry), although one that applies to all elements of the hull. This definition covers all examples of interest (whether they are stone or non-stone tile substitutions, or Delone set substitutions), still implies the main useful properties of being generated by substitution (such as having linear repetitivity in the self-similar case) and allows for simpler proofs of our main results.

Speaker: Ian Short

Title: Frieze patterns and Farey complexes

Abstract: Frieze patterns are periodic arrays of integers introduced by Coxeter in the 1970s. Conway and Coxeter discovered an elegant way of classifying friezes of positive integers using triangulated polygons. Recently there has been a resurgence of interest in friezes because of connections with cluster algebras, continued fractions, and integer tilings. An open problem was to provide a combinatorial model to classify friezes over the ring of integers modulo n. We describe such a model using Farey complexes, which are 2-complexes akin to the Farey tessellation of the hyperbolic plane. The model offers insight into friezes, allowing us to determine the width of a frieze, enumerate friezes over finite fields, and specify when a frieze over the ring of integers modulo n lifts to a frieze over the integers.

Speaker: Maciej Koch-Janusz

Title: Aperiodic systems through the lens of information theory

Abstract: Identifying relevant degrees of freedom is key to developing an effective theory of a complex system. Relevance is well defined within the renormalisation group (RG) program, whose practical execution is, however, often difficult. This is certainly true in aperiodic systems, where analytical and numerical methods cannot benefit from translation symmetry. 

I will discuss how the above problem can be addressed from the point of view of information theory. Field-theoretic relevance can be shown to be equivalent to the notion of “relevant information” in the Information Bottleneck (IB) formalism of compression theory. This gives rise both to theoretical insights, as well as numerical methods employing machine learning (ML), applicable also to aperiodic systems. I will showcase this on the example of the dimer model on the Amman-Beenker tiling, where our approach explicitly reveals self-similar degrees of freedom and discrete scale invariance of the interacting theory. I will also mention applications do dynamical systems.

Speaker: Shobhna Singh

Title: The O(n) Loop Model on the Ammann-Beenker tiling

Abstract: In statistical physics, universality is the observation that there are properties for a large class of systems that are independent of the details of the system at a critical point. A universality class is a collection of mathematical models which may differ dramatically at finite scales, but their behavior will become increasingly similar as the continuum limit is approached. In particular, asymptotic phenomena such as critical exponents will be the same for all models in a class. 

In this talk I will consider how universality survives in an aperiodic setting: the Ammann-Beenker tiling which has quasicrystalline symmetries. Considering the O(n) loop model on the AB tiling, I consider in detail two special limits - the Hamiltonian cycle and the fully packed loop (FPL) model. I compare the critical behavior of the latter with its equivalent in the periodic square lattice. I discuss applications, including polymer physics and other 'hard' problems on AB. I also discuss insights about studying quantum dimer and loop models on AB using variational matrix product states algorithm DMRG,  a future direction of our research.

Speaker: Franz Gähler

Title: The hat tiling is topologically conjugate to a model set

Abstract: The by now well-known hat tiling is related by a shape change to a self-similar (meta-tile) tiling with geometric φ^2 inflation. We show that this shape change is asymptotically negligible in the sense of Clark/Sadun, meaning that it does not mess up the long-range aperiodic order. As a result, these tilings form conjugate dynamical systems under the translation action, and their dynamical spectra must coincide. By the overlap algorithm, we can show that the spectra of both tilings are pure-point.

Inflation tilings with pure-point spectrum are expected to be cut-and-project sets (model sets), which is also the case here. We construct an equivalent cut-and-project set obtained from a 4d lattice with triangular symmetry and φ^2 scaling, and a reasonably simple window having a regular hexagon as outer shape. Only the internal boundaries between different tile types are fractal.

Speaker: Michael Baake

Title: A cocycle approach to the Fourier transform of Rauzy fractals and the point spectrum of Pisot inflation tilings

Abstract: PV inflations give rise to a covering model set with pure point diffraction, which typically has Rauzy fractals as windows. Here, we show how to use the model set theorem and how to calculate the Fourier transform of Rauzy fractals. This is joint work with Uwe Grimm.