Aperiodic Order: The Mathematics of Systems of Approximate Symmetry

Abstract

Symmetry is frequently exploited in Mathematics, but there are many situations in which systems exhibit long-range recurrence without precise periodic repetition. A simple example is given by a coding of an irrational circle rotation. With Shechtman’s discovery of quasicrystals – physical materials with long-range order but also rotational symmetry precluding the standard periodicity of usual crystals – it seems that ‘‘aperiodically ordered’’ patterns can appear in nature too. In this talk I will introduce the field of Aperiodic Order, which investigates intriguing infinite idealisations of such patterns. A prototypical family of examples is given by Penrose’s famous rhomb, or kite and dart tilings. I will then explain what sorts of mathematical structures can be introduced to systemise their study. I will focus on the construction of the tiling space of an aperiodic pattern, through which one may construct fundamental invariants using standard tools from Algebraic Topology.

Date
Location
Cardiff University, United Kingdom (via Zoom)
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Jamie Walton
Assistant Professor