Past Meetings

Speaker: Pierre Arnoux (Aix-Marseille Université, CNRS, Marseille, France)
Title: Extensions of substitutions
Abstract: Substitutions are best defined as morphisms of the free monoid, or as positive automorphisms of the free group with a fixed basis. In that setting, primitive substitutions define a finite number of infinite fixed words, and an associated dynamical system. But these fixed words can also be seen as a self-similar tiling of the line, or as a broken line in a d-dimensional space (where d is the number of letters, whose projection on a well-chosen line gives this self similar tiling. We will show how this construction can be formalised in a very natural way, so that the substitutions acts on the set Z^d of integer points, on the set of broken lines, and more generally, on the set of k-dimensional faces for any 0≤k≤d. If the substitution is unimodular, the dual maps are also defined, and act as a kind on inverse to the substitution. In this way, we can extend a number of theorems known in the Pisot case to the general hyperbolic case. We will show some examples.

Speaker: Ion Wood-Thanan (Cardiff University, UK)
Title: A quantum mechanical road-coloring theorem
Abstract: The road-coloring theorem is a statement that, for certain type of directed graphs, it is always possible to color the graph as to permit a synchronizing instruction. A synchronizing instruction allows one to start at any vertex on the graph and travel to the same final vertex. Inspired by this theorem, we attempted to create quantum tight-binding models – models which are naturally represented on a graph – endowed with a synchronizing instruction. In the context of mechanics, a synchronizing instruction allows one to evolve a set of initial states to the same final states using the same time-dependent potential. The required irreversible time-evolution necessitates that the quantum system has a non-Hermitian Hamiltonian; the system is open and can have gain and loss from the environment. The talk will be centered around two attempts at creating these quantum road-colored systems: one attempt based on ancilla qubits used in quantum computing and one attempt based on the Hatano-Nelson model, a foundational non-Hermitian tight-binding model. 

Speaker: Gabriel Fuhrmann (Durham University, UK)
Title: On the lack of equidistribution on fat Cantor sets
Abstract: Given an irrational rotation, it is straightforward to see that for every Cantor set C, there is a dense (in fact, residual) set of points whose orbit does not intersect C. On the other hand, if C is a fat Cantor set (that is, of positive Lebesgue measure), almost every point visits C with a frequency equal to the measure of C. But what other frequencies of visits to C may occur? In the words of a recent MathOverflow post [1], what is the Birkhoff spectrum of fat Cantor sets? We give a first answer to this question by showing that every irrational rotation allows for certain fat Cantor sets C whose Birkhoff spectrum is maximal, that is, equal to the interval [0,Leb(C)]. In this talk, we will focus on discussing some of the basic tools behind this result and extensions of it.

[1] D. Kwietniak, Possible Birkhoff spectra for irrational rotations, MathOverflow (2020), https://mathoverflow.net/q/355860 (version: 2020-03-27).

Speaker: Felix Flicker (University of Bristol):
Title: Constraints and Aperiodicity
Abstract: Some of the most important phenomena in physics arise when correlations emerge from local constraints. I will outline results for a range of constrained models in the new setting of aperiodic tilings. On the Ammann Beenker tiling we prove the existence of Hamiltonian cycles (visiting each vertex precisely once), and thereby solve a range of related problems including the three-colouring problem and the travelling salesperson problem [1]. On the recently discovered 'Spectre' aperiodic monotiling we provide an exact analytic solution to the interacting quantum dimer model [2]. The result features deconfined particle-like excitations at all interaction strengths, which is impossible in the periodic square and hexagonal tilings. I will highlight recent and ongoing work in which we generalise these results to random graphs [3,4].

References:

[1] Shobhna Singh, Jerome Lloyd, and Felix Flicker, Hamiltonian cycles on Ammann-Beenker Tilings, Physical Review X 14, 031005 (2024)

[2] Shobhna Singh and Felix Flicker, Exact Solution to the Quantum and Classical Dimer Models on the Spectre Aperiodic Monotiling, Physical Review B 109, L220303 (2024)

[3] Doruk Efe Gökmen, Sounak Biswas, Sebastian D. Huber, Zohar Ringel, Felix Flicker, and Maciej Koch-Janusz, Compression theory for inhomogeneous systems, Nature Communications 15, 10214 (2024)

[4] Shobhna Singh, Lizzy Rieth, Peru d'Ornellas, and Felix Flicker, Classical Dimer Models on Bicubic Planar Graphs, Upcoming

Speaker: John Hunton (University of Durham)
Title: A complete invariant for 1d tiling spaces, and a lot of other things
Abstract: In this talk we shall consider one dimensional matchbox manifolds that admit a fixed point free, minimal action by the real numbers, objects we shall call `flow spaces'. These objects include, among many other things, the tiling spaces of 1d repetitive tilings, which in turn include the spaces of 1d primitive substitution tilings. This latter class was provided by Marcy Barge and Bev Diamond in 2001 with a complete invariant up to topological equivalence (i.e., a homeomorphism between their tiling spaces that preserves the direction of the translation flows along the path components). 

      Unfortunately, the Barge-Diamond construction relies heavily on the substitution nature of the tilings they consider, and does not generalise. In this talk, using a different perspective, we give a complete invariant up to an analogous notion of equivalence for all flow spaces, whether they come from substitutions, represent other types of tilings, or no tilings at all. Moreover, we will also see that all such flow spaces are in fact just limits of tiling spaces.

      There are links with the work of Giordano, Putnam and Skau on orbit equivalence of Z-Cantor dynamics, which I hope to say something about too.

      This is joint work with Alex Clark.

Speaker: Jianlong Liu (University of Texas at Austin)
Title: Étale topology for substitution tiling spaces
Abstract: We introduce an étale topology for tiling spaces, and use it produce an isomorphism between K-theory and cohomology in low dimensions. We then discuss its applications to gap-labelling and deformations.

Speaker: Sam Coates

Title: Hexagonal quasiperiodic tilings: decorations and fractals

Abstract: I will introduce a family of Fibonacci or golden-mean hexagonal and trigonal aperiodic tilings, produced using the dual-grid method, whose structures are determined by a single parameter . I will then move on to single edge length variants of these tilings, discuss their properties in terms of the periodic triangular lattice, and briefly describe the results of some applied research in simple fluid dynamics. Then, I will introduce a related tiling, produced by rotating a subset of vertices on the dice lattice and expanding certain tile edge lengths. If time permits, I will discuss a related set of metallic-mean fractals.

Speaker: Neil Manibo

Title: Continuous diffraction in aperiodic structures

Abstract: Aperiodic structures with long-range order are those which exhibit pure point diffraction. Recent results provide an equivalent characterisation at the level of the atomic configuration, complementing the well-known requirement at the level of the autocorrelation coming from the Eberlein decomposition. On the other hand, the continuous component of the diffraction remains relatively less explored, despite recent progress. This talk will centre on criteria related to the nature of continuous diffraction, in particular, for structures arising from substitutions. Time permitting, we will discuss ongoing work on substitutions on infinite alphabets. This is based on several joint papers, which will be mentioned throughout the talk.

Speaker: Daniel Ratliff

Title: Ordered by Design: Coarse-Graining Tunable Micelles to form Quasicrystals

Abstract: The study of quasicrystals in soft matter systems has expanded greatly in the last few decades, in no small part due to how these problems can leverage the extensive literature of nonlinear pattern forming PDEs. In soft matter problems the linear dispersion relation, a key component in determining whether quasipatterns are feasible, is now determined by the pairwise interactions of the soft matter molecules and forms part of the “input” of the problem. Informed by Lifshitz-Petrich approaches, several molecular interactions have been proposed and studied successfully to explain how quasicrystals (the soft matter representation of a quasipattern) might be stabilised in these systems, but these interactions are theoretical in origin. A natural question therefore arises – is it possible design a soft matter molecule using these insights which forms quasicrystals? Can this then be controlled to improve the molecule’s propensity to form them?

In this talk, we present a “from pasture to plate” pipeline in which we take a micellar molecular structure and undertake various coarse-grainings to explore its crystal forming properties, with the aim to construct a molecule capable of forming dodecagonal quasicrystals. This starts with a Langevin Monte-Carlo method to extract the pair-wise interactions between two molecules, which then informs how we can describe these interactions in a (tractable) functional form. We then analyse this form to determine when two competing lengthscale are present and control their ratio using properties of the polymers comprising the micelle. Surprisingly we find that the micellar pair interactions with less spatial structure are more likely to form quasicrystals, in contrast to Barkan-Engels-Lifshitz potentials, and we are able to explain why this is by using energetic arguments.

Speaker: Petra Staynova & Alastair Rucklidge

Title: A cut-and-project approach to periodic approximants of 12-fold square-triangle-rhombus aperiodic tilings. 

Abstract: When computing properties of quasicrystals and other aperiodic tilings and patterns, it is often necessary to work with periodic approximants in a finite periodic domain. It is therefore useful to be able to construct a sequence of larger and larger periodic approximants that approach the aperiodic tiling in a well-understood manner. Aperiodic tilings can be generated by taking a higher dimension periodic lattice and projecting it (using an irrational projection) onto a lower dimensional plane. We explore periodic approximants to 12-fold square-triangle-rhombus tilings using this cut-and-project approach, with a sequence of rational approximations to the irrational projection

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.