Translational monotilings

I will survey results on translational monotilings of the integer lattice \(Z^d\) and Euclidean space \(R^d\). 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 \(Z^2\) by Bhattacharya but was recently disproved in high dimensions by Greenfeld and Tao. A key tool underlying several positive results is the magical dilation lemma. I will present a version of this lemma due to Greenfeld and Tao and discuss some of its consequences.