On periodic tiling conjecture in Z^d.

Abstract

In this talk I will give an introduction to the periodic tiling conjecture (PTC) for tilings of Z^d. Greenfeld-Tao proved that there exists a very large dimension where PTC fails. 

I will show that with some extra structure on the tile set, PTC holds in any dimension. To this effect, I will discuss the notion of joint co-tile and how it allows us to use tools from ergodic theory and dynamical systems (a link attributed to Bhattacharya) to prove results in tiling theory. This is joint work with Tom Meyerovitch (Ben Gurion University of the Negev) and Yaar Solomon (Ben Gurion University of the Negev).