Translational tiling is a covering of a space (e.g., Euclidean space) using translated copies of a building block, called the "tile'', without any positive measure overlaps.
Which are the sets that tile by translations? What can be said about the structure of the tilings? These questions have been extensively studied from different perspectives by researchers from various fields.
In a joint work with Terence Tao, we construct the first aperiodic translational tiling, disproving a long standing conjecture in the area. In the talk, I will survey the study of translational tilings and discuss our recent progress.