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 Abstract: Translational tiling is a covering of a space using translated copies of a building block, called a "tile'', without any positive measure overlaps. What are the possible ways that a space can be tiled?
The most well known conjecture in this area is the periodic tiling conjecture. It asserts that any tile of an Euclidean space admits a periodic tiling. This conjecture was first posed over 30 years ago and has been intensively studied over the years. In a joint work with Terence Tao, we prove the failure of the periodic tiling conjecture in high dimensions. In the talk, I will motivate this result and discuss our proof.