Topological and Geometric Aspects of Tiling Dynamical Systems

平铺动力系统的拓扑和几何方面

• 批准号: 0401655
• 负责人: Lorenzo Sadun
• 金额: $ 11.1万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401655&HistoricalAwards=false
• 关键词: Topological; Geometric; Aspects; Tiling; Dynamical

Dr. Lorenzo Sadun will analyze several problems on the topology and dynamics of tiling spaces. One problem concerns computing the Cech cohomology groups of tiling spaces. Another problem is to find criteria for when the natural translation action on a substitution tiling space (or a deformation of such an action) is topologically mixing. A third problem is to consider tilings of hyperbolic space, where the natural group action is nonabelian and in that the group is not amenable.These problems are motivated by quasicrystals, a class of solids that are highly ordered but are not crystals. Quasicrystals are modeled by aperiodic tilings, and ergodic theory relates the bulk properties of a quasicrystal (such as its ability to diffract light, its electrical conductance, and its rigidity) to certain averages taken over a space of possible quasicrystals, or equivalently a space of tilings. The better we understand the abstract mathematical properties of tiling spaces, the better we understand the down-to-earth physical properties of quasicrystals and of other materials.

Lorenzo Sadun博士将分析有关平铺空间的拓扑和动力学的几个问题。 一个问题涉及计算平铺空间的Cech上同调群。 另一个问题是找到当替换平铺空间上的自然平移动作（或这种动作的变形）是拓扑混合时的标准。 第三个问题是考虑双曲空间的平铺，其中自然群作用是非交换的，并且群是不顺从的。这些问题的动机是准晶体，一类高度有序但不是晶体的固体。 准晶是用非周期性的镶嵌来模拟的，遍历理论将准晶的整体性质（如对光的吸收能力、电导率和刚性）与可能的准晶空间（或等价的镶嵌空间）上的某些平均值联系起来。 我们对平铺空间的抽象数学性质理解得越好，就越能理解准晶体和其他材料的实际物理性质。