Topology of Tiling Dynamical Systems

这项拨款支持Lorenzo Sadun博士对非周期瓷砖的动力学、拓扑和遍历理论的研究。重点是几类比以前研究的更复杂的瓷砖。这包括分层的“融合”铺层，类似于自相似的“替代”铺层，只有在每个长度尺度上结构可以有质的不同。它还包括具有无限局部复杂性的瓷砖（其中瓷砖以无限多的局部图案组合在一起）和非欧几里得空间的瓷砖，其中相关的群行为是不可接受的。在每一种情况下，我们的目标都是理解简单瓷砖的哪些组合、拓扑和遍历性质可以延续，并创建一些其他性质不能延续的例子。进一步的目标是解决广泛的概念问题：铺装空间的遍历和拓扑性质如何与底层铺装的几何性质相关？瓦片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起来的，这些例子仅仅是海洋中可能存在的岛屿。该项目旨在弥合这些孤岛，发展更一般的理论和定性的新例子，这些例子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，该基金还支持培养一名数学科学博士生，从而解决了美国年轻科学家严重短缺的问题。它还支持研究人员的访问，他们将给研究生和本科生讲课。

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.