Gaps theorems and statistics of patterns in quasicrystals

我们宇宙中的许多美都出现在复杂系统的秩序中。作为科学家，我们对自然界的描述依赖于我们描述这种秩序的能力。对称性是一种有价值的工具，有时可以让我们简化描述，但事实上，我们试图描述的许多系统并不完全对称。这一主题贯穿于整个科学，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一种名为切割和投影方法的数学结构的模式。这些图案可以被认为是空间的瓦片。它们就像我们每天在墙上、地板和艺术品上看到的瓷砖一样，只是它们通常缺乏平移对称。然而，这些图案存在于自然界、病毒、量子物理中的能态研究以及最近发现的准晶材料中，这是一个事实。我们将主要研究切割和投影集中图案的变形特性和统计。这是一个相对较新的研究领域，我们的研究将围绕我们最近帮助开发的一种联系展开，这种联系涉及数论、拓扑学和动力系统等数学领域的思想的结合。简而言之，为了解释这种联系，我们可以将空间的每一个“无限”瓦片与“有限”拓扑空间联系在一起。从概念上讲，拓扑空间可以被认为是一个甜甜圈，可能有很多(甚至无限多)洞，它表面的每个点都长出了“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上同调工具来理解它。与瓷砖相关的拓扑空间的上同调与我们在瓷砖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，上同源就会检测到这一点，这反过来又会立即告诉我们，我们将在瓷砖中看到的不同配置的瓷砖的数量接近理论上可能的最小数量。这种联系也以另一种方式起作用，也就是说，理解瓷砖中的模式也给了我们关于相关空间的拓扑的信息。对于割集和投影集，瓷砖中的图案可以用动力学系统和产生它们的装置的数论性质来理解。我们对这些问题的处理应该有助于我们发展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现实世界的物理和生物学问题。

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.