Gaps theorems and statistics of patterns in quasicrystals

准晶体中的间隙定理和模式统计

• 批准号: EP/M023540/1
• 负责人: Alan Haynes
• 金额: $ 41万
• 依托单位: University of York
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2015
• 资助国家: 英国
• 起止时间: 2015 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FM023540%2F1
• 关键词: Gaps; theorems; statistics; patterns; quasicrystals

Much of the beauty in our universe arises in the emergence of order from complex systems. As scientists, our description of the natural world relies on our ability to describe this order. Symmetry is a valuable tool which sometimes allows us to simplify our description, but in truth many of the systems which we seek to describe are not perfectly symmetrical. This theme runs throughout the sciences and it also appears in many important problems in pure mathematics.The research we are developing in this project will help us to understand patterns which come from a mathematical construction called the cut and project method. These patterns can be thought of as tilings of space. They are like the tilings that we see every day on walls, floors, and in artwork, except that they typically lack translational symmetry. Nevertheless, it is a fact that these patterns occur in the natural world, in viruses, in the study of energy states in quantum physics, and in recently discovered materials known as quasicrystals.We will primarily be studying deformation properties and statistics of patterns in cut and project sets. This is a relatively new line of research, and our study will center around a connection which we have recently helped to develop, which involves a combination of ideas from the mathematical fields of number theory, topology, and dynamical systems.To explain this connection in brief, to every `infinite' tiling of space we can associate a `finite' topological space. The topological space can be thought of conceptually as a donut, possibly with many (or even infinitely many) holes, with `fractal hair' growing out of every point on its surface. Even for mathematicians, this is a strange type of space, but we can understand something about it by using a tool from algebraic topology called cohomology. The cohomology of the topological space associated to a tiling is directly related to the complexity of the patterns which we see in the tiling. For example, if it turns out that our donut has two holes in it then the cohomology will detect this, and this in turn will tell us right away that the number of different configurations of tiles which we will see in our tiling is close to as small as theoretically possible. This connection also works the other way, which is to say that understanding patterns in the tiling also gives us information about the topology of the associated space. For cut and project sets the patterns in the tiling can be understood in terms of dynamical systems and number theoretic properties of the setup which produces them.Our approach to these problems should help us to develop new mathematical methods to describe naturally occurring asymmetrical patterns. It is our hope that these methods will eventually find applications to real world problems in physics and biology.

我们宇宙中的许多美都来自于复杂系统的秩序。作为科学家，我们对自然世界的描述依赖于我们描述这种秩序的能力。对称性是一种很有价值的工具，它有时能使我们简化描述，但事实上，我们试图描述的许多系统并不是完全对称的。这一主题贯穿于所有科学领域，也出现在纯数学的许多重要问题中。我们在这个项目中开展的研究将帮助我们理解来自称为切割和投影方法的数学构造的模式。这些图案可以被认为是空间的平铺。它们就像我们每天在墙上、地板上和艺术品中看到的瓷砖，只是它们通常缺乏平移对称性。然而，这些模式确实存在于自然界中，存在于病毒中，存在于量子物理学中的能态研究中，也存在于最近发现的称为准晶体的材料中。我们将主要研究切割集和投影集中模式的变形性质和统计。这是一个相对较新的研究方向，我们的研究将围绕着我们最近帮助发展的一个联系，它涉及数论、拓扑学和动力系统等数学领域的思想的结合，为了简单地解释这种联系，我们可以将一个“有限”拓扑空间与空间的每一个“无限”平铺联系起来。拓扑空间在概念上可以被认为是一个甜甜圈，可能有许多（甚至无限多）孔，表面上的每一点都长出“分形头发”。即使对数学家来说，这也是一种奇怪的空间类型，但我们可以通过使用代数拓扑学中的一种称为上同调的工具来理解它。与镶嵌相关联的拓扑空间的上同调与我们在镶嵌中看到的模式的复杂性直接相关。例如，如果我们的甜甜圈上有两个洞，那么上同调将检测到这一点，这反过来又会告诉我们，我们在平铺中看到的不同构型的瓦片数量接近理论上可能的最小值。这种联系也以另一种方式起作用，也就是说，理解平铺中的模式也为我们提供了有关相关空间拓扑的信息。对于切割和项目集的模式在瓷砖可以理解的动力系统和数论性质的设置，产生them.Our的方法，这些问题应该帮助我们开发新的数学方法来描述自然发生的不对称图案。我们希望这些方法最终能应用于物理学和生物学中的真实的世界问题。

项目成果

Higher dimensional Steinhaus and Slater problems via homogeneous dynamics

通过齐次动力学解决高维 Steinhaus 和 Slater 问题

• 作者: Haynes A

On Diophantine transference principles

论丢番图移情原则

• DOI: 10.1017/s0305004118000014
• 发表时间: 2018
• 期刊: Mathematical Proceedings of the Cambridge Philosophical Society
• 影响因子: 0.8
• 作者: GHOSH A

Diophantine approximation for products of linear maps - logarithmic improvements

线性映射乘积的​​丢番图近似 - 对数改进

• DOI: 10.1090/tran/6953
• 发表时间: 2017
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Gorodnik A

A characterization of linearly repetitive cut and project sets

线性重复剪切和项目集的表征

• DOI: 10.1088/1361-6544/aa9528
• 发表时间: 2018
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: Haynes A

Reductive pairs arising from representations

• DOI: 10.1016/j.jalgebra.2016.02.006
• 发表时间: 2014-12
• 期刊: arXiv: Group Theory
• 作者: Oliver Goodbourn