Combinatorics of Sequences and Tilings and its Applications

自然界令人感兴趣的一个方面是我们周围世界中明显的对称性和秩序，例如晶体的形状，或者在微观层面上，组成晶体的原子的规则排列。令人惊讶的是，我们对秩序和对称性的起源的理解有限；也许更令人惊讶的是，没有明确的数学定义来定义秩序的概念。在晶体的例子中，潜在的顺序在其成分的周期性排列中是显而易见的。特别有趣的是研究缺乏周期性的有序结构，以及仍然显示出明显有序度的无序系统。例如，平面的彭罗斯瓷砖由两个基本形状组成，如果安排得当，这两个基本形状可以实现任意的大瓷砖，但绝不会完全重复。这种结构不仅从数学角度看很吸引人，而且在物理上是以准晶的形式实现的。这些晶体出现在具有错综复杂的非周期原子顺序的金属合金中。由于结构中缺乏周期性，每个原子都有自己独特的环境，如果你环顾周围足够远的地方，没有两个原子会有完全相同的环境。因此，观察这种结构的性质是很有趣的，例如一个原子的平均邻位数，或者在一定距离上的平均原子数。这些量与这些材料的衍射谱有关，这为原子位置的有序性提供了实验证据。在更抽象的环境中，考虑由平铺表示的结构，相应的问题是平铺中距给定顶点一定距离的顶点的平均数，以所有可能的顶点为中心进行平均。这些数字被称为平均炮击次数，它们是本项目中调查的属性类型的一个例子。事实证明，这些数字的计算与某些类型的数字的有趣性质有关，例如将数字分解为素因数，这是数论中感兴趣的一个主题。此外，这些数字和类似的组合性质与物理学和其他科学中感兴趣的模型密切相关。这使得这个项目从许多不同的角度变得有趣，从纯数学到在物理、结晶学和材料科学中的应用。

One of the intriguing aspects of Nature is the symmetry and order apparent in the world around us, for instance in the shape of crystals, or, at a microscopic level, in the regular arrangement of atoms making up the crystal. We have a surprisingly limited understanding of the origin of order and symmetry; and, maybe even more surprisingly, no clear mathematical definition of the concept of order exists. In the example of the crystal, the underlying order is apparent in the periodic arrangement of its constituents. It is particular interesting to investigate ordered structures that lack periodicity, and disordered systems that still show an apparent degree of order. For instance, a Penrose tiling of the plane consist of two basic shapes, which, when arranged properly, allow for an arbitrary large tiling, but never one that exactly repeats itself. Such structures are not only fascinating from a mathematical point of view, but are physically realised in quasicrystals. These are crystals occurring in particular metal alloys which possess an intricate non-periodic order of atoms. Due to the lack of periodicity in the structure, each atom has its own individual environment, and if one looks far enough around, no two atoms will ever have exactly the same surroundings. Therefore, it is interesting to look at properties of such structures, such as the mean number of neighbours of an atom, or mean numbers of atoms at certain distances. Such quantities are related to the diffraction patters of these materials, which provide the experimental proof of the order in the atomic positions. In a more abstract setting, thinking of a structure represented by a tiling, the corresponding question is that of the mean number of vertices in the tiling that are at a certain distance from a given vertex, averaged over all possible vertices as the centres. These numbers are called the averaged shelling numbers, and they are an example of the type of properties that are investigated in this project. The calculation of these numbers turns out to be related to interesting properties of certain types of numbers, such as factorisation of numbers into prime factors, which is a topic of interest in number theory. Moreover, these numbers and similar combinatorial properties are closely related to models of interest in physics and other sciences. This makes this project interesting from a number of different perspectives, ranging from pure mathematics to applications in physics, crystallography and materials science.