Quasiperiodic tilings and orbit equivalence of dynamical systems

School of Mathematics Colloquium
Thursday, October 17, 2013 - 11:00am
1 hour (actually 50 minutes)
Skyles 006
Norwegian University of Sciences and Technology Trondheim, Norway
In this talk, my goal is to give an introduction to some of the mathematics behind quasicrystals. Quasicrystals were discovered in 1982, when Dan Schechtmann observed a material which produced a diffraction pattern made of sharp peaks, but with a 10-fold rotational symmetry. This indicated that the material was highly ordered, but the atoms were nevertheless arranged in a non-periodic way. These quasicrystals can be defined by certain aperiodic tilings, amongst which the famous Penrose tiling. What makes aperiodic tilings so interesting--besides their aesthetic appeal--is that they can be studied using tools from many areas of mathematics: combinatorics, topology, dynamics, operator algebras... While the study of tilings borrows from various areas of mathematics, it doesn't go just one way: tiling techniques were used by Giordano, Matui, Putnam and Skau to prove a purely dynamical statement: any Z^d free minimal action on a Cantor set is orbit equivalent to an action of Z.