PROJECT IV 2018-19

PATTERNS VIA TOPOLOGY

Mathematics is very good at describing both phenomena that are very symmetric, and phenomena that are pretty random, but there are a lot of things in between: more explicitly, while group theory is good at categorising symmetric patterns, and probability and statistics for more random or indeterminate systems, the idea of a pattern that is close to being symmetric, but not quite, is better thought about using topology. This project will use powerful ideas from topology to investigate a variety of interesting objects that are in this intermediary area, neither completely symmetric nor completely random.

Examples of the sort of patterns we have in mind might be patterns in the plane such as the Penrose tiling that have rich local structures, but no global symmetries, or infinite sequences of numbers or letters whose finite subsequences keep repeating, but without any global repetition: such objects arise in geometry, number theory and theoretical computing. Yet more examples arise in dynamics, thought of as states of a system that evolve over time: for example, the position of all the planets in the sky today may be a pattern that will never exactly come again (so, does not repeat exactly over time), but something close to today's configuration may occur on many future (and past) occasions.

This project will look at examples of such complex patterns, how their properties can be translated into topological terms, and how tools from topology can be used to examine them.

PREREQUISITES

Topology III - MATH3281 is necessary. Depending on how you chose to develop the project, the module Algebraic Topology IV, MATH4161, might be helpful if it was also taken.

RESOURCES

Good places to start are the texts

L. Sadun, Topology of Tiling Spaces, American Maths Society, University Lecture Series 46 (2008)

M Senechal, Quasicrystals and Geometry, Cambridge University Press (1996).

For some background on complex patterns, try the classic

Branko Grunbaum and G. C. Shephard, Tilings and Patterns: An Introduction, W.H.Freeman & Co (1989)

There are the slides of a nice general talk introducing the subject at

http://www.math.uvic.ca/faculty/putnam/r/UNR_colloquium.pdf

John Hunton (mailto:john.hunton@durham.ac.uk)