Shrey Sanadhya - Periodicity of joint co-tiles in Z^d - Hatfest скачать в хорошем качестве

Shrey Sanadhya - Periodicity of joint co-tiles in Z^d - Hatfest

Given a finite set (a tile) $F \subset \mathbb{Z}^d$, we say that a set $A \subset \mathbb{Z}^d$ is a co-tile of $F$ if the collection of sets $F+a$, for $a \in A$, forms a tiling of $\mathbb{Z}^d$. For finitely many tiles $F_1,...,F_k$, we say that $A$ is a joint co-tile if $A$ is a co-tile of each $F_i$. In this talk, we will discuss the structure of joint co-tiles in $\mathbb{Z}^d$, particularly their periodicity. We will discuss the connections of this notion to the periodic tiling conjecture (PTC), whose $\mathbb{Z}^2$ case was resolved by Bhattacharya, and a counterexample in high dimension was recently given by Greenfeld-Tao. Our setup extends a theorem of Newman (every co-tile in $\mathbb{Z}$ is periodic) to higher dimensions. It provides characterization for the periodicity of a co-tile for all $d$ greater than 2. This is joint work with Tom Meyerovitch and Yaar Solomon [arXiv:2301.11255]. A talk from Hatfest at the Mathematical Institute, University of Oxford. See https://sites.google.com/view/thegrimmnetw... for more information.