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.

洛伦佐·萨杜（Lorenzo Sadun）博士将分析有关平铺空间拓扑和动态的几个问题。 一个问题涉及计算CECH共同体的瓷砖空间组。 另一个问题是找到何时自然翻译作用在替代空间上的自然翻译作用（或这种动作的变形）是拓扑混合的。 第三个问题是考虑双曲线空间的瓷砖，其中自然的群体行动是非亚伯式的，并且该群体不适合。 准晶体是通过多个斜纹块建模的，厄基德理论将准晶体（例如其衍射光的能力，其电导及其刚性的能力）与在可能的准晶体或等于瓷砖的空间中所取得的某些平均值相关联。 我们越理解平铺空间的抽象数学特性，我们就越好了解准晶体和其他材料的脚踏实地的物理特性。