Appears in collection : Research School in Discrete Mathematics and Computer Science / École de recherche en mathématiques discrètes et informatique - WEEK 1

The recent discovery of the “hat”, an aperiodic monotile points to the complexity that can be encoded within even a simple shape – might it even be undecidable whether or not a given shape can be used to form a tiling of the plane? We'll survey related this and related questions for tiles and subshifts of finite type in the plane and other geometric spaces.