Executive Summary : | The space of tilings of the Zd lattice is a complex model in statistical physics and natural questions from combinatorics, harmonic analysis, and ergodic theory. However, it is poorly understood, with many fundamental questions remaining unanswered. This project aims to initiate a comprehensive study of these questions and test the hypothesis that these tilings are well-behaved with natural assumptions. The first part of the project aims to prove a large deviations principle for tilings of Zd by dimers, focusing on the set of invariant Gibbs measures on the space of tilings and the statistics of the number of tilings of nice subsets of Zd. The project has been initiated with Scott Sheffield and Catherine Wolfram, and is an important open question in the field. The second part involves fixing a finite set of boxes in Zd satisfying a certain co-primality condition, believing that any ergodic Zd action can be equivariantly embedded into the space of tilings of the Zd lattice by these boxes. This question is similar to the first part but simpler in the case of dimers. The third part focuses on understanding the cohomology of tilings by these boxes, starting with the combinatorial question: Can tilings of two distinct regions be extended to a tiling of the entire lattice? Studying the cohomology of the tiling spaces provides a topological invariant and has played a significant role in progress in the first two questions. In the final part, the project aims to understand tilings by a fixed finite subset of Zd, proving that if the set can tile Zd, it can also tile it periodically. |