Henna Koivusalo

The Tales of Aperiodic Order

The term `aperiodic order' describes discrete point sets (or tilings), which have no translational period but feature some signs of long-range organisation. The tale of the study of aperiodic order is fundamentally intertwined with physics, but as a field of mathematics also lies in the deep shadow of logic. My take on this story will cover the past 60-odd years in approximate chronological order, beginning with first examples of aperiodic tilesets, the Nobel prize-winning discovery of quasicrystal materials, and the quest to find wild quasicrystals, and ending with the unbelievable story, from just a couple of years ago, of finding the first aperiodic monotile.

Time permitting, I will explain in further detail some results on my favourite patterns with aperiodic order, the cut and project sets, which are obtained by taking an irrational slice through a lattice and projecting it to a lower dimensional subspace. The definition of cut and project sets allows for many interpretations and generalisations, and they can naturally be studied in the context of dynamical systems, discrete geometry, harmonic analysis, or Diophantine approximation, for example, depending on one's own tastes and interests.