Logic Seminar

Please note that the time is PST

The periodic tiling conjecture (PTC) in Zd, that was recently disproved by Greenfeld and Tao, asserts that if a finite set F⊆Zd tiles Zd by translations, then it also tiles by (possibly different) translations that are periodic. On the other hand, the case d=2 has an affirmative answer which was proved earlier by Bhattacharya. The analogous question in R2 is open even for polygonal sets.
Here, the most general result is by Kenyon, who proved that PTC holds for topological disks. In an ongoing work with de Dios Pont, Greenfeld and Madrid, we obtain a structure result about translational tilings by polygonal sets with rational slopes by connecting the ideas from the discrete and continuous case. Our main result states that if such a tiling is topologically minimal, then it is weakly periodic and satisfies a weak version of PTC. In the talk I will discuss the main differences between tilings in Z2 and R2 as well as the main ideas how to apply the structure theory of Greenfeld and Tao in the continuous setting.