Abstract

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.