In this talk, I will survey results about translational monotilings of \(\mathbb{Z}^d\) and \(\mathbb{R}^d\), with a particular focus on the case \(d \leq 2\). One of the central questions in this area is the decidability of the tiling problem. Closely related is the periodic tiling conjecture (PTC), which has been confirmed in \(\mathbb{Z}^2\) by Bhattacharya but was recently disproved in high dimensions by Greenfeld and Tao. For \(\mathbb{R}^2\), analogous questions remain open, even for polygonal sets. The most general result here is due to Kenyon, who established that PTC holds for topological disks. In ongoing work with de Dios Pont, Greenfeld, and Madrid, we show that translational monotilings by axis-parallel polygonal sets satisfy a weaker version of PTC and derive a decidability result in this context.
Discrete and continuous translational monotilings
16.01.2025 15:00 - 15:50
Organiser:
KGRC
Location: