Undecidability of tiling the space with a fixed number of tiles

Recently, Greenfeld and Tao disproved the conjecture that translational tilings of a single tile can always be periodic (Ann. Math. 200(2024), 301-363). In another paper (to appear in J. Eur. Math. Soc.), they also show that if the dimension $n$ is part of the input, the translational tiling for subsets of $Z^n$ with one tile is undecidable. These two results are solid evidence for the conjecture that translational tiling of $Z^n$ with a monotile is undecidable, for some fixed $n$. In general, we study the decidability or undecidability of the following problem: let $k$ and $n$ be fixed positive integers, and a tile is a finite subset of $Z^n$, given a set $S$ of $k$ tiles in $Z^n$, is there an algorithm to decide whether $Z^n$ can be tiled by translated copies of tiles in the set $S$? In this talk, we report some recent progress on the undecidability of this problem based on the joint work with Zhujun Zhang.