Aperiodic Tiling

一类欧几里得空间的瓷砖，最著名的是平面的彭罗斯瓷砖，已被证明是一个丰富的研究课题，在广泛的方向上产生影响。研究者继续他对这些瓷砖的跨学科研究，强调瓷砖的统计旋转对称性的后果，特别是瓷砖的完全旋转对称性，如平面的风车瓷砖。该方法是遍历理论、环理论和代数拓扑学的结合。空间的平铺是理解大整体中许多小组件之间可能存在的几何关系的重要工具。特别是平铺帮助我们理解几何对称。近年来，人们发现了具有意想不到的对称性的瓷砖，并成功地应用于新型物理材料的原子结构建模。研究者研究这些新的对称性，并利用瓷砖与数学各个部分的接口方式，将对称性作为研究数学各个部分结构的工具。

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.