Oleksandra Gasanova: Periodic lozenge tilings of the plane

Abstract.

We start with the tiling of the plane by equilateral triangles. Their vertices form a lattice which we will call \(L_0\). By merging two adjacent triangles of this tiling we obtain a rhombus, also known as a lozenge. It is clear that there exist 3 different orientations of them, and that the plane can be tiled with lozenges. Now let \(L_1\) be a cofinite sublattice of \(L_0\). We are interested in lozenge tilings of the plane which are invariant under the translation by any element in \(L_1\). Since \(L_0/L_1\) is finite, we are using only finitely many lozenges in our tiling (mod \(L_1\)). To each \(L_1\)-periodic tiling one can attach a vector (called the type of the tiling) storing the information about the number of lozenges of each orientation used in the tiling (mod \(L_1\)). This way we can split all the \(L_1\)-periodic tilings into groups of different types.

The main focus of the talk is to address the following questions:
1) For a given cofinite sublattice \(L_1\) of \(L_0\), which types of \(L_1\)-periodic tilings exist?
2) For a given type, can we list all the \(L_1\)-periodic tilings of this type?