Periodicity and decidability of tilings of $\mathbb{Z}^{2}$
  We prove that any finite set $F\subset {\mathbb{Z}^2}$ that tiles ${\mathbb{Z}^2}$ by translations also admits a periodic tiling. As a consequence, the problem whether a given finite set $F$ tiles ${\mathbb{Z}^2}$ is decidable. 
Siddhartha Bhattacharya
math.CO math.DS