THERMOGAMAS - What is the Right Periodic Tiling Conjecture in ℤ^d? (Yaar Solomon)

Abstract: The periodic tiling conjecture in ℤ^d asserts that if a tile (=finite set) tiles ℤ^d, then it must also tile it periodically. In dimension d=1, an old theorem of Newman shows an even stronger assertion, which is that every tiling of  ℤ is itself periodic. For d=2, Bhattacharya recently proved that the periodic tiling conjecture is true. Namely, any one tile that can tile ℤ² can also tile it periodically! One the other hand, even more recently, Greenfeld and Tao showed that the periodic tiling conjecture is false in a large enough dimension d. In this talk, after giving all the definitions and background, I will briefly explain how (in spite of Greenfeld-Tao counterexample) both Newman's and Bhattacharya's theorems can be extended to any dimension d, with a slightly different statement and setup. This talk is based on a joint work with Tom Meyerovitch and Shrey Sanadhya