WORKSHOP: Computational and Combinatorial aspects of Tilings

几何学是所有数学领域中最多样化的一个。继克莱因的埃尔兰根计划，几何定义本身在20世纪主要是作为研究性质的对象不变下的一组对称，并导致了许多重要的结果，从群论微分方程。从另一个角度来看，考克斯特的工作说明了美丽的结构，可以从研究简单的几何对象，如多面体揭示。镶嵌理论的研究将这两个几何主题结合在一起，镶嵌理论的中心问题是一组形状是否可以镶嵌平面。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的正多边形。开普勒和伊斯兰艺术家的艺术作品也对瓷砖问题进行了研究。数学工作集中在定期tilings，但这项研究的一部分，发展的群论。最初与空间群，群的对称性周期平铺在平面上，并在更高的维度，也在发现零星有限简单的群体\cite{康威：SPLAG}。对周期性和对称性结构的研究继续得到发展。这一年看到出版的全面研究的面积~\cite{康威：ST}，其中包括一个新的证明康威的表征空间群的平面上。一般的问题出现在希尔伯特的第18个问题，但首先明确指出，由王浩在1961年作为多米诺骨牌问题：是否存在一个算法，告诉如果一组给定的瓷砖将瓷砖的平面。这个问题是回答在否定的欧几里德平面（因此在所有欧几里德空间的维数大于1，其中的问题是微不足道的）由伯杰在1964年。伯杰证明了这个问题是不可判定的。这样做的一个结果是，存在不定期平铺的形状集，称为非周期性原型集。伯杰发现了这样一个非周期性的原型集，但它包含了近20，000块瓷砖。这个数字逐渐下降，直到发现着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（例如准晶结构的模型），导致了大量的平铺理论研究。然而，在大多数情况下，研究都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Davdison举行的区域AMS会议的卫星会议之后举行的。这次会议的与会者很多，引起了很大的兴趣。本次研讨会的主要目标是建立在这一成功的基础上：在从事瓷砖工作的不同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传达给瓷砖及其应用领域的其他研究人员。

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.