2018 LTCC Course Aperiodic Order

Time and Venue

The course runs for five weeks from Monday 12 November 2018 until Monday 10 December 2018. The lectures will take place in the Burnside room at De Morgan House starting at 3:10pm each Monday.

Course Description

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.

1. Preliminaries: point sets, symmetry, lattices and crystals, crystallographic restriction, number theoretic tools, Minkowski embedding
2. Symbolic dynamics: substitutions, fixed point, hull, properties, well-known examples
3. Inflation/deflation symmetry: basic construction, examples and properties, repetitivity
4. Model sets: one-dimensional model sets, cut and project schemes, cyclotomic model sets
5. 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

Course Materials

Lecture notes, work sheets and solutions will be made available below. Note that lecture notes will be updated to contain proofs following the lecture.

Solutions to work sheets will be posted before the following lecture.

Lecture Notes

• Lecture 1 (12 November 2018)
• Lecture 2 (19 November 2018)
• Lecture 3 (26 November 2018)
• Lecture 4 (3 December 2018)
• Lecture 5 (10 December 2018)

Work Sheets and Solutions

• Worksheet 1      Solutions to Worksheet 1
• Worksheet 2      Solutions to Worksheet 2
• Worksheet 3      Solutions to Worksheet 3
• Worksheet 4      Solutions to Worksheet 4

End of course examination

• Question paper

Please submit the examination by 25 January 2019, 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 last modified 26 November 2018