The course runs for five weeks from Monday 12 November 2018 until Monday 10 December 2018. The lectures will take place at De Morgan Hourse.

This course will present an introduction to the mathematical theory of structures such as sequences, point sets, tilings and patterns that exhibit non-periodic order. A prominent example is Penrose's fivefold tiling. This topic combines elements from discrete geometry, (symbolic) dynamics, number theory and harmonic analysis with applications in crystallography.

**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, D. Damanik, U. Grimm,
*What is Aperiodic Order?*Notices of the AMS 63(6) (2016) 647-650 [arXiv:1512.05104] - M. Baake, U. Grimm, R. V. Moody,
*What is Aperiodic Order*, arXiv:math.HO/0203252 - M. Baake, D. Damanik, U. Grimm,
*Aperiodic order and spectral properties*, Snapshots of modern mathematics from Oberwolfach 3/2017 [arXiv:1506.04978]

**Additional optional reading:**

- M. Baake, U. Grimm,
*Aperidic Order. Volume 1: A Mathematical Invitation*Cambridge University Press, Cambridge 2013 - U. Grimm,
*Aperiodic Order*, in*Dynamical and Complex Systems*, LTCC Advanced Mathematics Series volume 5, edited by Shaun Bullett, Tom Fearn and Frank Smith, World Scientific Publishing Europe, London (2017) pp. 41-80. - 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]

**Prerequisites:**

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

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 **TBC**, 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 |