WORKSHOP: Computational and Combinatorial aspects of Tilings

几何是数学中最多样化的领域之一。在克莱因的埃尔兰根计划之后，几何在20世纪主要将自己定义为研究一组对称下物体不变量的性质，并导致了许多重要的结果，从群论到微分方程。从另一个角度来看，考克斯特的作品说明了通过研究简单的几何物体，如多面体，可以揭示出美丽的结构。瓷砖的研究将这两个几何主题结合在一起。平铺理论的核心问题是一组形状能否平铺平面。这个问题可以追溯到古希腊人，他们当然知道等边三角形、正方形和正六边形是唯一能平铺平面的正多边形。开普勒也研究了瓷砖问题和伊斯兰艺术家的艺术作品。数学工作主要集中在周期性排列上，但这项研究是群论发展的一部分。首先是空间群，平面上和高维空间上的周期性平铺对称群以及偶发有限单群\cite{Conway:SPLAG}。周期性和对称结构的研究继续发展。今年发表了一项关于该地区的综合研究\cite{Conway:ST}，其中包括对康威的新证明，证明了飞机上空间群的特征。一般的问题出现在希尔伯特的第18个问题中，但在1961年，王浩首先明确地提出了多米诺骨牌问题：是否存在一种算法，可以告诉我们一组给定的瓷砖是否会铺在平面上。伯杰在1964年在欧几里得平面上给出了否定的答案（因此在所有大于1维的欧几里得空间中，这个问题是微不足道的）。伯杰指出，这个问题是无法确定的。这样做的一个结果是，存在不周期性平铺的形状集，称为非周期性原集。伯杰发现了这样一个非周期原集，但它包含了近2万块瓦片。这个数字逐渐下降，直到发现了著名的彭罗斯瓷砖，只有2块瓷砖。这些新的结果结合了对非周期性但有序的物理结构的简单模型的需求（例如准晶体结构的模型），导致了对平铺理论的大量研究。然而，在大多数情况下，研究是在非常接近应用领域的地方进行的。本次研讨会是继去年在美国北卡罗来纳州戴维森举行的AMS区域会议的卫星会议之后举行的。那次会议出席的人很多，引起了很多人的兴趣。本次研讨会的主要目标是在这一成功的基础上：在不同的研究人员之间建立更强的联系，并将几个领域令人兴奋的新发展与其他研究人员交流。

Geometry is one of the most diverse of all areas of mathematics. Following Klein's Erlangen program, geometry defined itself in the twentieth century predominantly as the study of the properties of an object invariant under a group of symmetries, and led to many important results ranging from group theory to differential equations. From a different angle, the work of Coxeter illustrates the beautiful structures that can be revealed from studying simple geometric objects like polytopes. The study of tilings brings together these two themes of geometry.The central question in the theory of tiling is whether a set of shapes can tile the plane or not. This question goes back to the Ancient Greeks, who were certainly aware that the equilateral triangle, square and regular hexagon are the only regular polygons to tile the plane. Tiling questions were also investigated by Kepler and the artistic work of Islamic artists. Mathematical work concentrated on periodic tilings, but this study was part of the development of group theory. Initially with the space groups, the groups of symmetries of periodic tilings on the plane and in higher dimensions and also in the discovery of sporadic finite simple groups \cite{Conway:SPLAG}. The study of periodic and symmetric structures continues to be developed. This year sees the publication of a comprehensive study of the area~\cite{Conway:ST} which includes a new proof of Conway of the characterisation of space groups on the plane.The general question emerges in Hilbert's 18th problem, but was first stated explicitly by Hao Wang in 1961 as the Domino Problem: Does an algorithm exist that tells if a given set of tiles will tile the plane. This problem was answered in the negative on the euclidean plane (and thus in all euclidean space of dimension greater than 1, where the problem is trivial) by Berger in 1964. Berger showed the the problem was undecidable. One consequence of this is that there exist sets of shapes that do not tile periodically, called aperiodic protosets. Berger found such an aperiodic protoset, but it contained nearly 20,000 tiles. This number was gradually brought down until the discovery of the famous Penrose tiling with just 2 tiles. These new results combined with a need for simple models of non-periodic, but ordered, structures from physics (for examples as models for the structure of quasicrystals) have led to a great deal of research in tiling theory. In most cases however the research has been carried out very close to the area of application. The present workshop follows on from a satellite meeting at the regional AMS meeting in Davdison, NC, USA last year. That meeting was well attended and generated a lot of interest. The main goal of this workshop is to build on that success: forging stronger links between different researchers working with tiling and communicating the exciting new developments in several areas to other researchers in tilings and its applications.