Schedule for: 22w2232 - Almost Periodicity in Aperiodic Order

Note: the Lecture rooms are available after 16:00.

Beverages and a small assortment of snacks are available in the lounge on a cash honour system.

A buffet breakfast is served daily between 7:00am and 9:00am in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

This talk is about a unique algebraic structure that arises out of the exceptional non-crystallographic reflection group of the icosahedron in 3D-space and the corresponding group of the 600-cell in 4D-space. This structure is both a root system and a countably infinite group that arises arises in iterative steps and discretizes SU(2). We determine the remarkable structure of this group and its connections to the ring generated by the golden ratio and the prime 2. This is joint work with Jun Morita. The talk is essentially self-contained.

The floor function defines a binary relation on positive integers $\mathbb N^+$, saying $d$ is a floor quotient of $n$ if $d=\lfloor \frac{n}{k} \rfloor$ for some integer $k$. The set $X_Q= \{ (\pm n, \pm d), (\pm d, \pm n)\} \subseteq \mathbb Z^2$ is an aperiodic pattern. The floor quotient relation defines a partial order on the positive integers. The talk describes the internal structure of this partial order, and properties of its order zeta function and M$\ddot{\rm o}$bius function. (Joint work with David Harry Richman (U. Washington)).

A buffet lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In this talk, we present a family of substitutions on compact alphabets which realise any number greater than 2 as their inflation factor. We show that they admit a unique strictly positive length function (up to scaling) which assigns a positive length to each letter and turns the substitution into an inflation rule. As a corollary, we prove that there are substitution tilings with transcendental inflation factors, and we will give a sketch of the proof of how to obtain an explicit substitution with such property. This is joint work with Dirk Frettloeh (Universitaet Bielefeld) and Alexey Garber (Univ. of Texas Rio Grande Valley).

In this talk, which is on joint work with Alvaro Bustos Gajardo and Neil Mañibo, I will discuss S-adic shifts generated by sequences of morphisms that are constant-length. We call a sequence of constant-length morphisms torsion-free if any prime divisor of one of the lengths is a divisor of infinitely many of the lengths. We show that torsion- free directive sequences generate shifts that enjoy the property of quasi-recognizability which can be used as a substitute for recognizability. Indeed quasi-recognizable directive sequences can be replaced by a recognizable directive sequence. With this, we give a finer description of the spectrum of shifts generated by torsion-free sequences defined on a sequence of alphabets of bounded size, in terms of extensions of the notions of height and column number. We will give some examples which illustrate the subtleties that arise.

For euclidean pure point diffractive Delone sets of finite local complexity and with uniform patch frequencies it is well known that the patch counting entropy computed along the closed centred balls is zero. We consider such sets in the setting of sigma-compact locally compact Abelian groups and show that the topological entropy of the associated Delone dynamical system is zero. We furthermore construct counterexamples, which show that the patch counting entropy of such sets can be non-zero in this context. Other counterexamples will show that the patch counting entropy of such a set can not be computed along a limit and even be infinite in this setting.

In this talk, we first briefly summarize the properties of 1D Fibonacci substitution, and then we recall the definition of the Fibonacci direct product variation tilings in a two-dimensional space. Based on their description as cut-and-project sets, we show a classification based on the shape of the windows. Further, we prove that all Fibonacci DPVs with a polygonal window are topologically conjugate. It is joint work with Michael Baake and Franz Gähler.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

Yet another application of tiling cohomology! Given two strongly pattern-equivariant mass distributions on an aperiodic tiling, when can you do a bounded transport from one to the other? When can you do it in a (weakly or strongly) pattern-equivariant way? These questions boil to questions about the cohomology class in $\check H^d$ of the difference between the mass distributions and lead to some interesting topological questions.

2-day workshop participants are welcome to use BIRS facilities (TCPL) until 15:00 on Sunday, although participants are still required to checkout of the guest rooms by 11 M. There is no coffee break service on Sunday afternoon, but self-serve coffee and tea are always available in the 2nd floor lounge, Corbett Hall.

In this talk we will explore some properties of dynamical systems that are Kakutani equivalent to a system with discrete spectrum. Joint work with Dominik Kwietniak