Aperiodic Tiling

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A class of tilings of Euclidean spaces, the best known being the Penrose tilings of the plane, have proven to be a fertile subject of research with impact in a broad range of directions. The investigator continues his interdisciplinary research of these tilings, emphasizing consequences of the statistical rotational symmetry of the tilings, in particular the full rotational symmetry of tilings such as the pinwheel tilings of the plane. The methods are a combination of ergodic theory, ring theory and algebraic topology.Tilings of space have been an important tool to understand the geometric relationships that are possible between many small components within a larger whole. In particular tilings help us understand geometric symmetries. In recent years tilings with unexpected symmetries have been discovered, and used successfully in modeling the atomic structure of new types of physical materials. The investigator studies these new symmetries, and, making use of the way tilings interface with various parts of mathematics, uses the symmetries as a tool to investigate structures in diverse parts of mathematics.