Topology of Tiling Dynamical Systems

平铺动力系统的拓扑

• 批准号: 1101326
• 负责人: Lorenzo Sadun
• 金额: $ 18.7万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101326&HistoricalAwards=false
• 关键词: Topology; Tiling; Dynamical; Systems

This grant supports the research of Dr. Lorenzo Sadun on the dynamics, topology and ergodic theory of aperiodic tilings. The focus is on several classes of tilings that are more complicated than previously studied. This includes hierarchical "fusion" tilings, akin to self-similar "substitution" tilings, only where the structure can be qualitatively different at each length scale. It also includes tilings with infinite local complexity (where tiles fit together in infinitely many local patterns) and tilings of non-Euclidean spaces where the relevant group action is not amenable. In each setting, the goal is to understand which combinatorial, topological and ergodic properties of simpler tilings carry over, and to create examples where other properties do not. A further goal is to address the broad conceptual question: how do ergodic and topological properties of a tiling space relate to geometric properties of the underlying tilings?Tiling theory has many applications within science and engineering and has been used to model materials such as quasicrystals, to code information on computers, and to solve abstract problems in logic. However, much of the theory is built around a small set of classic examples, mere islands in the sea of possible tilings. This project aims to bridge these islands, developing both a more general theory and qualitatively new examples, examples that will allow for a still broader range of applications. For example, tilings of curves spaces may aid in the design of efficient computer networks. In addition to supporting basic research, the grant supports the training of a PhD student in the mathematical sciences, thereby addressing a severe shortage of young American scientists. It also supports visits by researchers who will give lectures to graduate and undergraduate students.

该补助金支持洛伦佐·萨顿博士对非周期性平铺的动力学、拓扑学和遍历理论的研究。重点是几类平铺比以前研究的更复杂。这包括类似于自相似“替代”镶嵌的分层“融合”镶嵌，仅在结构可以在每个长度尺度上定性地不同的情况下。它还包括具有无限局部复杂性的平铺（其中平铺以无限多个局部模式拟合在一起）和相关群作用不服从的非欧几里德空间的平铺。在每一种设置中，目标是了解更简单的平铺的组合，拓扑和遍历属性，并创建其他属性不存在的例子。 进一步的目标是解决广泛的概念问题：如何遍历和拓扑性质的平铺空间相关的几何性质的基础平铺？平铺理论在科学和工程中有许多应用，并已被用于模拟材料，如准晶体，在计算机上编码信息，并解决逻辑中的抽象问题。 然而，大部分理论都是围绕着一小部分经典例子建立的，仅仅是可能的平铺大海中的岛屿。该项目旨在将这些岛屿联系起来，开发更一般的理论和定性的新实例，这些实例将允许更广泛的应用。 例如，曲线空间的平铺可以帮助设计高效的计算机网络。除了支持基础研究外，该赠款还支持培养数学科学博士生，从而解决美国年轻科学家严重短缺的问题。它还支持研究人员访问，他们将为研究生和本科生讲课。