Mathematical Sciences: The Mathematical Physics of Crystals

Radin将研究与低温物质晶体对称成因有关的数学问题，以及与欧几里得空间独特遍历平铺相关的动力系统分类。所采用的方法是将传统的遍历理论与非周期瓷砖的新技术相结合。旋转对称瓷砖的最新研究成果将为材料结构研究和动力系统分类开辟新的研究方向。这个项目涉及遍历理论的研究。遍历理论一般关注理解系统的平均行为，其动力学过于复杂或混乱，无法在微观细节上进行跟踪。在“动力学”的标题下，可以放置关于抽象变换如何作用于光滑空间的现代理论。通过这种方式，遍历理论在寻求对均匀空间上的流动进行分类时与几何联系在一起。

Radin will study mathematical problems relating the causes of crystalline symmetry in low temperature matter and classification of the dynamical systems associated with uniquely ergodic tilings of Euclidean space. The methods to be used are a combination of traditional ergodic theory and new techniques from aperiodic tilings. It is expected that the recent results on rotation symmetric tilings will open significant new research in the study of material structure and in the classification of dynamical systems. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.