APERIODIC
academic meeting

Academic Meeting 

1-2 July 2024
Fry Building, University of Bristol

APERIODIC was a meeting of the Grimm Network, a collection of mathematicians and scientists interested in aperiodic order. 

Monday 1st July

13.30-14.30 Valérie Berthé (CNRS/Paris Diderot)

14.30-15.30 Jorg Thuswaldner (Montanuniversität Leoben)

15.30-16.00 coffee

16.00-17.00 Lizzy Rieth (Universiteit van Amsterdam)

17.30-18.30 public talk by Latham Boyle (Edinburgh)

walk to art gallery for nibbles and drinks

20.00 dinner 

Tuesday 2nd July

09.00-10.00 Reidun Twarock (York)

10.00-11.00 Francesco Turci (Bristol)

11.00-11.30 coffee

11.30-12.30 Yotam Smilansky (Manchester)

12.30-14.30 lunch

14.30-15.30 Francisco Arana-Herrara (University of Maryland)

15.30-16.00 coffee

16.00-17.00 Anu Jagannathan (Paris Saclay)

Titles and abstracts

Valérie Berthé: Low discrepancy  and  aperiodic  order 

The chairman assignment problem can be stated as follows: k states are assumed to form a union and each year a union chairman must be selected so that at any time the cumulative number of chairmen of each state is proportional to its weight. It is closely related to the (discrete) apportionment problem, which has its origins in the question of allocating seats in the house of representatives in the United States, in a proportional way to the population of each state. The richness of this problem lies in the fact that it can be reformulated both as  a sequencing problem in operations research for optimal routing and scheduling, and as a symbolic discrepancy problem, where the discrepancy measures the difference between the number of occurrences of a letter in a prefix of an infinite word and the expected value in terms of frequency of occurrence of this letter. We will see in this lecture how to construct infinite words with values in a finite alphabet having the smallest possible discrepancy, by revisiting a construction due to R. Tijdeman in terms of dynamical systems and cut and project schemes.

This is a collaborative work with O. Carton, N. Chevallier, W. Steiner, R. Yassawi.

Jorg Thuswaldner: Sequences of Matrices and Substitutions

The classical continued fraction algorithm as well as its multidimensional generalizations produce sequences of unimodular matrices. Our aim is to study such sequences and to discuss convergence properties of them. These convergence properties are formulated in terms of hyperbolicity properties. Like for a single unimodular hyperbolic matrix, it is possible to associate Markov partitions with a ``hyperbolic’’ sequence of matrices. To define such ``nonstationary’’ Markov partitions, we relate a sequence of substitutions to a sequence of matrices. This will lead to a symbolic coding of such a sequence.

Lizzy Rieth: Exploring Deconfined Phases in Quantum Dimer Models on Aperiodic Graphs

Confinement in gauge theories remains a significant challenge in theoretical and mathematical physics, as exemplified by the Clay Institute's Millennium Problem on Yang-Mills theory and the mass gap. Noteworthy progress has been made in 2+1D compact gauge theories, particularly through quantum dimer models on planar bipartite graphs. According to Polyakov's theorem, emergent effective field theory (EFT) descriptions formulated on regular graphs with periodic dimer arrangements never exhibit a deconfined phase. [1] However, research suggests that deconfinement may exist in these models over parameter ranges which are homeomorphic to Cantor sets. [2] In this talk, I will present our findings on aperiodic quantum dimer arrangements and our efforts to develop an EFT framework for these systems, potentially leading to new insights into Polyakov's theorem and the discovery of new deconfined phases.

[1] Polyakov AM. Quark confinement and topology of gauge theories. Nuclear Physics B. 1977 Mar 21;120(3):429-58.

[2] Fradkin E, Huse DA, Moessner R, Oganesyan V, Sondhi SL. Bipartite Rokhsar–Kivelson points and Cantor deconfinement. Physical Review B. 2004 Jun 30;69(22):224415.

Latham Boyle (public talk): The Penrose tiling is a quantum error-correcting code

I will begin by introducing Penrose tilings and quantum error correcting codes. A Penrose tiling is a remarkable, intrinsically non-periodic way of tiling the plane whose many beautiful and unexpected properties have fascinated physicists, mathematicians, and geometry lovers of all sorts, ever since its discovery in the 1970s. A quantum error correcting code is a fundamental way of protecting quantum information from noise, by encoding the information with a sophisticated type of redundancy. Such codes play an increasingly important role in physics: in quantum computing (where they protect the delicate quantum state of the computer); in condensed matter physics (where they provide the fundamental underpinning for new states of matter called topologically-ordered phases); and even in quantum gravity (where the so-called "gauge/gravity duality" – the much-studied idea that spacetime is a kind of a hologram – may be understood as such a code).

Although at first sight, Penrose tilings and quantum error correcting codes might seem completely unrelated, I will explain how Penrose tilings give rise to (or, in a sense, *are*) a new type of quantum error correcting code in which any errors in any spatially localized region (no matter how large) of the quantum state space, may be diagnosed and corrected; how variants of the code (based on beautiful cousins of the Penrose tiling, called the Ammann-Beenker and Fibonacci tilings), can be defined on discrete, finite quantum computers, in an arbitrary number of spatial dimensions; and (more speculatively!) what this *might* be hinting about the underlying quantum-mechanical structure of spacetime itself.

Reidun Twarock: Viral Tiling Theory: From Viral Geometry to Polyhedral Packings 

The geometric principles underpinning virus structure act like a mathematical microscope, providing a key to understanding viral infections. Most viruses have protein shells, called viral capsids, that surround, and thus protect, their genomes. Viral Tiling Theory enables classification of capsid architecture in terms of surface lattices that encode the positions of individual capsid proteins and the interactions between them. By combining these geometric, and related topological, descriptors of virus architecture with stochastic simulations, I will demonstrate how viral geometry provides insights into viral life cycles that pave the way to innovation in antiviral therapy and virus nanotechnology. I will also show how the mathematical concepts underpinning viral infections can be used to address mathematical problems in  polyhedral packings.  

Francesco Turci: Making sense of local structure in glasses

Glasses are solids that differ from crystals because they are fundamentally disordered. Or are they? In this talk, we will explore how ideas around the role of local structure for the properties of supercooled liquids and glasses have evolved in recent years. We will consider competing theoretical frameworks such as thermodynamic theories, aspects of geometrical frustration, and dynamical facilitation approaches and discuss the role of local structure from these different perspectives. We will focus on model systems where optimal icosahedral packings play an important role and show how theory and experiments can probe their heterogeneous structural distribution. Finally, we will propose a unifying structural-dynamical scenario reinterpreting glassy behaviour from the point of view of a nonequilibrium phase transition.

Yotam Smilansky: Hyperbolic multiscale tilings and their orbits

Substitution rules provide a classical method for constructing aperiodic tilings via a substitution-inflation procedure. However, when multiple distinct scales are allowed a different approach is required, and new geometric objects emerge. In my talk I will introduce multiscale substitution tilings, sequences and partitions, and describe how they can be lifted into tilings of the hyperbolic upper-half space. These hyperbolic tilings extend constructions previously considered by, among others, Penrose, Kakutani and Kamae (and essentially illustrated by Escher), and provide a unified framework for the study of the aforementioned Euclidean constructions. I will illustrate some recent results about individual hyperbolic tilings and about their orbits under the geodesic and horospheric actions, and conclude with a prime orbit theorem for the geodesic flow. Based on joint work with Yaar Solomon. 

Francisco Arana Herrera: Ergodic theory of rational billiards

The ergodic theory of rational billiards, i.e., billiards on a table whose angles are rational multiples of pi, has been a subject of intense study in dynamical systems. We survey the history of the field and discuss recent progress in the study of weak mixing, i.e., mixing modulo a negligible set of exceptions. This is joint work in progress with Jon Chaika and Giovanni Forni.

Anuradha Jagannathan: Quantum quasiperiodic matter 

Quasiperiodic tilings, in addition to being fascinating from the esthetic and the mathematical points of view are an invaluable resource for constructing models of physical properties of quasicrystals. Indeed, ever since quasicrystals were discovered and proven to be a new type of organization of matter, a recurring question has been: do these structures possess any distinctive and unique properties, compared to periodic crystals ? As I will discuss in this overview, tight-binding models on tilings are a good starting point for investigations of the unique electronic properties of quasicrystals. To start, I will illustrate the nature of electronic states in a two-dimensional tiling by considering an exactly soluble wavefunction due to Kalugin and Katz. I will explain how this type of critical state could explain experimental observations of quantum critical phenomena and of superconductivity in quasicrystals. Another fascinating aspect of quasicrystals that has been studied in recent years concerns their topological character. I will discuss the relation between quasicrystals and Quantum Hall problems, for some simple cases, and describe ways to observe nontrivial topologies using photonic waveguides and cold atoms.