Workshop on Aperiodic Order

  • Uwe Grimm
  • John Hunton and Mark Lawson
  • Mark Lawson
  • Antoine Julien
  • Alex Clark
  • Jamie Walton
  • Robbert Fokkink
  • John Hunton
  • Antoine Julien
  • Alan Haynes and Jamie Walton
  • Mark Lawson
  • Dan Rust
  • John Hunton and Mark Lawson
  • Alex Clark and Robbert Fokkink
  • Ethan Akin and Mark Lawson
  • Ethan Akin and Mark Lawson
  • Robbert Fokkink, Alex Clark and Jean-baptiste Aujogue
  • Robbert Fokkink, Ethan Akin and Alex Clark
  • Alan Haynes, Johannes Kellendonk and Antione Julien
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  • Alan Haynes
  • Dan Rust
  • Jamie Walton
  • Alex Clark
  • Alex Clark
  • Dan Rust and Greg Maloney
  • Jean Savinien
  • Picture
  • Antoine Julien and Siegfried Beckus
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5th - 9th January 2015, The University of Leicester, England

This was a small, informal workshop focussed on Aperiodic Order.  It included some talks and opportunities for delegates to work together.  See below for details of talks.

 Group photo

 

Hayes, Kellendonk, Julien
Alan Haynes, Johannes Kellendonk and Antoine Julien

 Poster for this event

Talks on Monday 5th January
 
Time
Uwe Grimm
On the relation between diffraction and dynamical spectra
12:00
Siegfried Beckus
Gähler-Anderson-Putnam graphs of 1-dimensional systems
 
 
15:00
John Hunton
17:00

 

Talks on Tuesday, 6 January


Time

Ethan Akin

10:00

Mark Lawson

Pseudogroups, étale groupoids and groups

I shall explain how the classical theory of Stone duality can be generalized to a non-commutative setting using inverse semigroups.

A special case of this theory leads to a connection with a class subgroups of the group of auto-homeomorphisms of the Cantor space.
The origins of this work lie in Kellendonk's construction of C*-algebras from aperiodic tilings, and a reformulation of this work in more algebraic terms by Daniel Lenz.

There are also links to classic work by Kumjian, Paterson and Renault, and the recent papers of Matui.

The theory will be described  from scratch with an emphasis on the simple intuitive ideas that animate it.

The research that underlies it is joint with Ganna Kudryavtseva and Daniel Lenz,  and I shall also touch on some recent joint work with Philip Scott. I have also benefited from conversations with Collin Bleak and Pedro Resende.

12:00

Jamie Walton

16:45

Talks on Wednesday, 7 January


Time

Johannes Kellendonk (Lyon)

Ellis semigroups for tilings

10:00

Antione Julien (Trondheim)

Deformations and homeomorphisms between tiling spaces

The theory of tiling deformation, as developed by Clark and Sadun on the
one hand and Kellendonk on the other hand, describes what happens when
one changes the shape of the tiles in a tiling without changing the
combinatorics of their adjacency. Small deformations of a tiling define
a new tiling space which is homeomorphic to the original one.

In this talk I will show that any homeomorphism between tiling spaces
has an underlying deformation attached to it: deformations do not
describe a sub-family of homeomorphisms, but can be used to describe all
homeomorphisms. I will also try to give a meaning to the sentence "tiling
spaces homeomorphic to a given space Omega are parametrized by the first
cohomology group of Omega". Time depending, I will tell what can be said
additionally when two tiling spaces are diffeomorphic in a suitable sense.

This is joint work with Lorenzo Sadun.

12:00

Jean Savinien (Lorraine)

Metric doubling in subshifts

15:30

Talks on Thursday, 8 January

Time

Alan Haynes (York)

Diophantine approximation and point patterns in cut and project sets

In this talk we will discuss a connection between frequencies of patterns in Euclidean cut and project sets, and gaps problems in Diophantine approximation. Building on work of Antoine Julien, we will explain how the Diophantine approximation properties of the subspace defining a cut and project set can influence the number of possible frequencies of patterns of a given size. Once this connection is established, we will show how techniques from Diophantine approximation can be used to prove that the number of frequencies of patterns of size r, for a typical cut and project set, is almost always less than a power of log r. Furthermore, for a collection of cut and project sets of full Hausdorff dimension we can show that the number of frequencies of patches of size r remains bounded as r tends to infinity. For comparison, the number of patterns of size r in any totally irrational cut and project set of dimension d always grows at least as fast as a constant times r^d. This talk is based on joint work with Henna Koivusalo, Lorenzo Sadun, and James Walton.

10:00

Greg Maloney (Newcastle)

K-theory for C*-algebras associated to multi-substitution tiling spaces

 

14:00

 

Dan Rust (Leicester)

Barge-Diamond approximants for multi-substitution tiling spaces.

Multi-substitution tiling systems (closely related to the well-known S-adic sequences) were recently studied topologically by Gahler & Maloney. They introduced a generalisation of the famous Anderson-Putnam complex for a substitution tiling in order to describe the associated tiling space as an inverse limit of CW complexes. This allows for the calculation of the Cech cohomology of the space. We will see how we can similarly generalise the Barge-Diamond complex of a 1D multi-substitution system and apply this method to a particular set of substitutions on two letters called the Mixed Chacon substitutions. We will see that the set of all mixed Chacon tiling spaces form an uncountable collection of non-homotopy equivalent spaces distinguished by an uncountable collection of non-isomorphic cohomology groups.

 

 

16:00

 

 

Talks on Friday, 9 January


Time

Jean-baptiste Aujogue (Santiago)

Meyer set dynamical systems and cut & project sets

 The aim of this talk is to present a result on the embedding of a Meyer set into a canonically associated cut & project set of R^d.


I will first give a general presentation of what is a Meyer set as well as
its associated dynamical system. Then I will discuss about eigenfunctions
of this system, which will allow me to state the embedding result. I will
end this talk with a discussion on regional proximality for these
particular systems.
 

10:00

Ethan Akin (New York)

On equicontinuity

 

11:30

 

 

 

 

 

 

 

 

 

 

 

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