The course runs for five weeks from Monday 27 October 2014 until Monday 24 November 2014. The first lecture will take place in the Burnside room, and following lectures in the Cayley room at De Morgan Hourse from 10:50am until 12:50pm each Monday.

This course presents an introduction to the
*Theory of Aperiodic Order* and some of its applications.

**Keywords:**

symbolic dynamics, discrete geometry, tilings and patterns,
inflation tilings, cut and project sets, model sets, almost periodicity,
diffraction

**Syllabus:**

Order phenomena are ubiquitous in nature, with
crystals being one of the most prominent examples. However, there is
no obvious definition of `order', and no complete understanding of
what manifestions of `order' may exist. Partly motivated by the
discovery of quasicrystals, recognised by the award of the 2011
Nobel Prize in Chemistry to Dan Shechtman, systematic investigations
of aperiodically ordered systems have produced a good description of
systems that, in some sense, are close to periodic. The course will
present an introduction to the theory of Aperiodic Order. It will
cover the following main themes.

- Preliminaries: point sets, symmetry, lattices and crystals, crystallographic restriction, number theoretic tools, Minkowski embedding
- Symbolic dynamics: substitutions, fixed point, hull, properties, well-known examples
- Inflation/deflation symmetry: basic construction, examples and properties, repetitivity
- Model sets: one-dimensional model sets, cut and project schemes, cyclotomic model sets
- Diffraction theory: almost periodic functions, Poisson summation formula, diffraction of crystals, simple example of diffraction of aperiodic structure

**Recommended reading:**

- M. Baake, U. Grimm, R. V. Moody,
*What is Aperiodic Order*, arXiv:math.HO/0203252 - M. Baake, U. Grimm,
*Aperidic Order. Volume 1: A Mathematical Invitation*Cambridge University Press, Cambridge 2013

**Additional optional reading:**

- M. Baake, U. Grimm,
*On the notions of symmetry and aperiodicty for Delone sets*, Symmetry 4 (2012) 566-580; arXiv:1210.0157 - U. Grimm, M. Schreiber,
*Aperiodic tilings on the computer*, in*Quasicrystals: An Introduction to Structure, Physical Properties and Applications*, eds. J.-B. Suck, P. Häussler, M. Schreiber, Springer, Berlin (2002), pp. 49-66; arXiv:cond-mat/9903010. - J.C. Lagarias,
*Meyer's concept of quasicrystal and quasiregular sets*, Commun. Math. Phys. 179 (1996) 365-376 - R.V. Moody,
*Model sets: a survey*, in*From Quasicrystals to More Complex Systems*, eds. F. Axel, F. Dénoyer and J. P. Gazeau, EDP Sciences, Les Uis (2000), pp. 145-166; arXiv:math.MG/0002020 - N. Pytheas Fogg,
*Substitutions in Dynamics, Arithmetics and Combinatorics*, LNM 1794, Springer, Berlin (2002).

**Prerequisites:**

undergraduate mathematics; in particular,
basic notions of number theory, group theory, measure theory

**Preliminary reading:**

- M. Baake, U. Grimm, R. V. Moody,
*What is Aperiodic Order*, arXiv:math.HO/0203252 - M. Baake, U. Grimm,
*Aperidic Order. Volume 1: A Mathematical Invitation*, Cambridge University Press, Cambridge 2013 (Chapters 1-3)

Lecture notes and work sheets will be made available below.

- Lecture 1 (27 October 2014)
- Lecture 2 (3 November 2014)
- Lecture 3 (10 November 2014)
- Lecture 4 (17 November 2014)
- Lecture 5 (24 November 2014)

- Worksheet 1 Solutions to Worksheet 1
- Worksheet 2 Solutions to Worksheet 2
- Worksheet 3 Solutions to Worksheet 3
- Worksheet 4 Solutions to Worksheet 4

Please submit the examination by **Monday 15
December**, either by email
to uwe.grimm@open.ac.uk, or by
posting it to my Open University address, which you can find on my
home page. Any format is
accepted as long as it is readable. If you have any questions or find
that you need additional time, please get in touch with me.

Uwe Grimm |