Tilings are everywhere around us: brick walls, bee hives, chessboards, etc. The study of tilings is fascinating and has connections to numerous areas in mathematics as well as applications in many areas of science and engineering, such as in the design of materials, the study of quasicrystals and signal processing and in the construction of computer algorithms. This study goes back to ancient Greeks and has remained vibrant up to the present day. Recent works indicate a mysterious divide between “structured” tiling problems, in which the tilings are well behaved, to “wild” tiling problems, where almost anything can happen. One goal of this project is to develop tools to reveal the tiling mystery and, in turn, apply these tools to study related problems. Along with advancing the project’s research goals, the principal investigator also initiates events to the benefit of the community and contribute to synergistic activities related to the topics of this project, such as mentoring young researchers and organizing seminars, summer schools, conferences and workshops. This project investigates central structural problems and conjectures in the areas of discrete analysis, additive combinatorics and symbolic dynamics via exploring interrelationships among them and using tools from various mathematical fields. More specifically, the principal investigator continues to develop tools to advance the study of the structure of translational tilings. This includes the study of continuou