Schedule for: 25w5437 - Directions in Aperiodic Order

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

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

After reviewing the construction of the Hat family of tilings, we compute the first cohomology, which governs shape deformations and homeomorphisms of tiling spaces. This group is as small as it possibly could be, indicating that all versions of the Hat tilings are topologically conjugate, up to a linear transformation. We then focus on a particular example of a Hat tiling, the self-similar CAP tiling, and see that it is a cut-and-project tiling with a window that we describe.  All other Hat tilings, including the original shape, are essentially reprojections of the CAP. I'll only talk a little about Spectre tilings. Their story is qualitatively the same, with essentially the same answer but with calculations that are more involved. This is joint work with Michael Baake and Franz Gähler. 

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

When computing properties of quasicrystals and other aperiodic tilings and patterns, it is often helpful 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.

In this talk, we present a family of metallic mean Wang tiles. This is a family of aperiodic sets of Wang tiles (unit squares with labeled edges) whose dynamics involves the positive root of the polynomial $x^2-nx-1$. This root is sometimes called the $n$-th metallic mean, and in particular, the golden ratio when $n=1$ and the silver ratio when $n=2$. The metallic mean Wang shifts are self-similar, aperiodic, minimal and uniquely ergodic. When n=1, the set contains 16 tiles which are equivalent to the Ammann set of Wang tiles deduced from the Ammann A2 L-shapes.

I aim to relate the three notions in the title by studying a space filling curve through the half-hex substitution. We'll see that we naturally obtain the R1-rule of the Taylor-Socolar tile along with another tile that Jamie Walton and I studied. I hope to mention some new directions and interesting fractal objects.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Schrödinger operator plays a fundamental role in mathematical quantum mechanics and operator theory. After discussing background information about Schrödinger operators, spectral theory, and quantum dynamics, we will explore some properties of discrete Schrödinger operators on aperiodic graphs, with an emphasis on graphs that arise from substitution tilings of the plane.

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!

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Over the past century, the field of symbolic dynamics has developed to study sequences with entries from a finite set. Coven and Meyerowitz gave sufficient conditions for a subset of the integers (a {\em digit set}) to tile all of $\mathbb Z$ via translation. This “filling in” is used in extremely fundamental ways in a number of arguments. For example, Cassaigne gives a calculation of the complexity function that relies on the study of special factors, subwords that can be extended to the right (or left) in more than one way. This relies inextricably on the contiguous nature of the underlying infinite sequence. Thus, it is natural to ask what can be said in the case that a subset of the integers does not tile all of $\mathbb Z$. In joint work with Frank and Yang, we explore this question through an examination of {\em gapped digit tilings}, a fun variant on constant-length substitutions.

We explore the spectrum of continuum Schrödinger operators with a potential defined on unit intervals according to the Fibonacci substitution.  When this potential is piecewise constant, the spectrum of periodic approximations can be formulated exactly in terms of a finite dimensional nonlinear eigenvalue problem.  We will describe this formulation, which follows in the tradition of the "dynamic stiffness matrix" approach in the mechanics literature, and briefly discuss some ways numerical analysts approach such problems.  In addition to their application to the mathematics of aperiodic order, these problems can form interesting test cases for numerical algorithms.  (This work is a collaboration with David Damanik, Jake Fillman, Anton Gorodetski, and May Mei.)

The recently discovered Spectre tilings [1] are quasiperiodic [2]. A physical realization of a density corresponding to a decoration of the Spectre prototile therefore possesses continuous symmetries associated with the relative phases of incommensurate density waves, and the spontaneous formation of physical structure with these symmetries would require the relaxation long wavelength variations in these relative phases. In the context of a tiling model, this phason relaxation process must proceed via a sequence of local rearrangements of the tiles. In the familiar cases of quasicrystalline tilings consisting of rhombic (or rhombohedral) prototiles, phason rearrangements can be modeled as flips of local clusters that have the effect of moving a mismatch in the tiling along a linear (or planar) structure known as a worm. For the Spectre tiling, the analogous rearrangements of tiles and the structure of worms are more complex. I will describe the structure of phason rearrangements in the Spectre tiling, present a model that allows for simulation of phason relaxation through local rearrangements, and present results on the dynamics of the relaxation process. Acknowledgements: I thank P. Mostert for helpful exchanges within tiling@googlegoups.com. [1] D. Smith, J.S. Myers, C.S. Kaplan, C. Goodman-Strauss, Comb. Theory, 4(2) (2024). [2] M. Baake, F. Gähler, J. Mazáč, and L. Sadun, arXiv:2411.15503 (2024).Top of Form

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Aperiodic tight binding operators have richer topological invariants than periodic ones. In particular the gap-labelling group, which is a bulk invariant, has more structure. We are interested in other (physical) interpretations of these invariants, for instance as edge invariants, that is, as topological invariants associated to the edge states of the model with a boundary. However, if the tight binding operator has finite local complexity then the usual formulation of the bulk edge correspondence leads to trivial edge invariants, as the hull is then totally disconnected. This talk is about ways to overcome this difficulty. We will see, in particular models, that the integrated density of states below a gap of the model on Z is related partly to the spectral flow through the gap of the eigenvalues of the edge states of the restriction of an augmented model to N; but there is also other spectral flow.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk we will study measure-theoretical rigidity and partial rigidity for classes of Cantor dynamical systems including Toeplitz systems. The use of Bratteli-Vershik diagrams enables us to control the structure of its ergodic invariant measures. Among other things, we will analyze different Toeplitz systems for their rigidity, show that there exist Toeplitz systems which have zero entropy and are not partially measure theoretically rigid with respect to any of its invariant measures and construct a Toeplitz system which has countably infinitely many ergodic invariant probability measures that are rigid with the same rigidity sequence. This talk is based on joint work with Henk Bruin, Olena Karpel and Piotr Oprocha.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

We define (yet) another notion of the fundamental group, and show that it is related to a different fundamental group that was introduced a few years ago. Time permitting, we explain how it can be used towards Cantor deformations. This is joint work with Lorenzo Sadun.