PROJECT IV 2015-16
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: similarly,
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 uses 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 symmetry: such objects arise in geometry,
number theory and theoretical computing. 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
EMAIL
John
Hunton (mailto:john.hunton@durham.ac.uk)