Combinatorics of Sequences and Tilings and its Applications

自然的一个有趣的方面是我们周围世界明显的对称性和秩序，例如晶体的形状，或者在微观层面上，构成晶体的原子的规则排列。令人惊讶的是，我们对秩序和对称起源的理解非常有限；也许更令人惊讶的是，秩序的概念并没有明确的数学定义。在晶体的例子中，基本的顺序在其成分的周期性排列中是明显的。研究缺乏周期性的有序结构和仍然表现出明显有序程度的无序系统是特别有趣的。例如，平面的彭罗斯瓷砖由两个基本形状组成，当它们排列得当时，允许任意大的瓷砖，但永远不会完全重复自己。这样的结构不仅从数学的角度来看是迷人的，而且在准晶体中是物理实现的。这些是发生在特殊金属合金中的晶体，具有复杂的非周期原子顺序。由于结构中缺乏周期性，每个原子都有自己独立的环境，如果观察得足够远，就会发现没有两个原子有完全相同的环境。因此，研究这种结构的性质是很有趣的，比如一个原子的平均邻接数，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模式有关，这为原子位置的顺序提供了实验证明。在更抽象的情况下，考虑用平铺图表示的结构，相应的问题是平铺图中与给定顶点有一定距离的顶点的平均数量，以所有可能的顶点为中心取平均值。这些数字被称为平均炮击数，它们是本项目所研究的属性类型的一个例子。这些数字的计算结果与某些类型的数字的有趣性质有关，例如将数字分解为质因数，这是数论中感兴趣的主题。此外，这些数字和类似的组合性质与物理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来看都很有趣，从纯数学到物理学、晶体学和材料科学的应用。

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.