Complexity of random substitution tilings

随机替换平铺的复杂性

• 批准号: EP/Y023358/1
• 负责人: Tony Samuel
• 金额: $ 10.44万
• 依托单位: University of Birmingham
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY023358%2F1
• 关键词: Complexity; random; substitution; tilings

The discovery of quasicrystals - honoured with the 2011 Nobel Prize in Chemistry - came as a surprise to physicists, chemists and materials scientists. Quasicrystals are structures which possess long-range order but lack translational symmetry and have found a wealth of applications, from catalysts to low friction coatings, over to theoretical applications in quantum computing. As with many structures, most physical quasicrystals are not perfect, and exhibit local defects. Thus, good theoretical models for quasicrystals should exhibit features of both long-range order and local disorder. Such models are provided by tiling spaces of random substitutions. Here, we will initiate the development of a much-needed and substantial hierarchical classification of these models.Topological entropy is an invariant of dynamical systems which quantifies their complexity. Throughout this project, we will develop a robust theory of topological entropy for random substitution tiling spaces, leveraging recent developments for symbolic random substitution systems in one-dimension, combined with the rich theory of deterministic substitution tiling spaces. Moreover, we will consider a novel approach to complexity of such tiling spaces in dimensions two and higher, by quantifying the complexity of lower-dimensional subsystems induced by slicing the original system along a hyperplane. This will provide a method of discerning tilings which have the same topological entropy. Thus, creating a new means of distinguishing and ordering random substitution tiling spaces based on their complexity, and hence, taking the initial steps towards establishing a hierarchical classification.The themes of this project can be found in many mathematical fields outside of aperiodic order, including dynamical systems, fractal geometry, geometric measure theory, harmonic analysis and number theory. Further, as random substitution tiling spaces provide theoretical models for physical phenomena, such as quasicrystals and fractal percolation, we expect that the techniques we develop will have a lasting impact beyond mathematics.

准晶的发现让物理学家、化学家和材料科学家大吃一惊--2011年诺贝尔化学奖获得者。准晶是一种具有长程有序但缺乏平移对称性的结构，从催化剂到低摩擦涂层，再到量子计算中的理论应用，准晶都有广泛的应用。与许多结构一样，大多数物理准晶都不是完美的，并且表现出局部缺陷。因此，好的准晶理论模型应该同时表现出长程有序和局域无序的特征。这样的模型是由随机替换的平铺空间提供的。在这里，我们将启动对这些模型的迫切需要和实质性的分层分类的发展。拓扑熵是动力系统的不变量，它量化了它们的复杂性。在整个项目中，我们将利用一维符号随机替换系统的最新发展，结合确定性替换平铺空间的丰富理论，建立随机替换平铺空间的拓扑熵的稳健理论。此外，我们将考虑一种新的方法，通过量化沿超平面分割原始系统所引起的低维子系统的复杂性，来研究这类二维及更高维平铺空间的复杂性。这将提供一种识别具有相同拓扑熵的平铺的方法。因此，创建了一种新的方法来区分和排序基于其复杂性的随机替换瓷砖空间，从而朝着建立层次分类的方向迈出了第一步。本项目的主题可以在非周期序之外的许多数学领域找到，包括动力系统、分形几何、几何测度论、调和分析和数论。此外，由于随机替代瓷砖空间为准晶和分形渗流等物理现象提供了理论模型，我们预计我们开发的技术将产生超越数学的持久影响。