We study colorings of the infinite grid that only contain given allowed colored patterns. It is known that some patterns force the coloring to be non-periodic and that there is no general algorithm to determine if a given collection of allowed patterns admits any valid colorings at all. The project studies these questions in the case that the number of given patterns is small. This setup is an abstract mathematical model to the formation of crystals when the number of feasible local arrangements of particles is limited. A goal is to find bounds on the numbers of patterns that can force non-periodicity, depending on the shapes of the patterns and the dimension of the space. The questions are also considered outside the rigid grid structure. For more information on our algebraic approach to tackle these problems, see www.arxiv.org/abs/1905.04183. The research will be conducted at the department of mathematics and statistics at the University of Turku.

2023

2027