Computing algebraic invariants of symbolic dynamical systems

在自然界中，欧几里得对称性无处不在。其中一些对称性是肉眼可见的，例如蝴蝶翅膀的两侧对称性。其他对称性可以通过电子显微镜观察，例如晶体的平移对称性。更微妙的描述是准晶的对称性，上个世纪的大部分时间里，准晶的存在一直受到怀疑。准晶是不具有正常晶体的平移对称性的晶体结构。准晶具有等级结构：在小尺度上出现的图案和结构在越来越大的尺度上被复制。准晶的第一个数学模型是罗杰·彭罗斯爵士在半个世纪前发现的。彭罗斯瓷砖具有反射对称性，但缺乏平移对称性。二维空间的平移对称平铺必须具有三重、四重或六重旋转对称。但彭罗斯瓷砖具有局部的五重旋转对称性。彭罗斯瓷砖只是一个数学模型，并不一定保证在自然界中存在。但在1982年，丹尼尔·谢克特曼在电子显微镜下研究快速冷却的铝和锰的熔融混合物时，发现了自然界中实际上存在的五角形对称。由于他的工作，他在2011年获得了诺贝尔奖。自从彭罗斯瓷砖的发现以来，数学家们已经发现了许多方法来创造这种排列：平面上有无限多的数学瓷砖不具有平移对称性。受限于大自然提供的各种积木，科学家更难创造或发现这些瓷砖。出现了两个相辅相成的问题。第一个是，什么时候两个数学平铺在某种程度上是等价的，第二个是，这些数学平铺中的哪一个可以在我们周围的世界中实现？回答第一个问题可以指导科学家研究第二个问题，因为在试图实现数学平铺时，他们可以忽略已知与已经实现的平铺等价的平铺。数学家使用对称群等抽象代数结构来研究对称。我们可以通过将称为不变量的代数结构与瓷砖联系起来来表征瓷砖的结构性质。如果两个平铺相等，则它们的不变量相同。因此，对瓷砖的代数不变量的理解将导致第一个问题的一些答案。在这个项目中，我们试图更好地了解其中的一些不变量，对称性如何在其中表现出来，以及如何计算它们，以便我们能够在数学准晶分类方面取得进展。

Euclidean symmetries are all around us in the natural world. Some of these symmetries are visible to the naked eye, such as the bilateral symmetry of a butterfly's wings. Other symmetries can be viewed via an electron microscope, such as the translation symmetries of a crystal.More subtle to describe are the symmetries of quasicrystals, the existence of which was doubted for much of the last century. Quasicrystals are crystalline structures which do not have the translational symmetry of a normal crystal. Quasicrystals have a hierarchical structure: patterns and structures which appear on small scales are reproduced on larger and larger scales.The first mathematical model of a quasicrystal was discovered by Sir Roger Penrose half a century ago. The Penrose tiling has reflectional symmetry, but it lacks a translational symmetry. A translationally symmetric tiling of two dimensional space must have either three-, four- or six-fold rotational symmetry. But the Penrose tiling has local five-fold rotational symmetry.Penrose's tiling is simply a mathematical model, which is not necessarily guaranteed to exist in the natural world. But in 1982, Daniel Schechtman discovered that pentagonal symmetry actually appears in nature, while studying a rapidly chilled molten mixture of aluminium and manganese under an electron microscope. For his work, he received the Nobel prize in 2011.Since the discovery of the Penrose tilings, mathematicians have discovered many ways to create such arrangements: There are infinitely many mathematical tilings of the plane which do not have translational symmetry. Confined to the kinds of building-blocks provided by nature, it is harder for scientists to create, or discover, these tilings.Two questions arise, which are complementary to one another. The first is, when are two mathematical tilings somehow equivalent, and the second is, which of these mathematical tilings can be realised in the world around us? Answering the first question can guide scientists investigating the second question, for then, in trying to realise a mathematical tiling, they can ignore tilings known to be equivalent to ones that have already been realised.Mathematicians study symmetry using abstract algebraic structures such as symmetry groups. We can characterize the structural properties of a tiling by associating to it algebraic constructions called invariants. If two tilings are equivalent, their invariants are the same. So, an understanding of the algebraic invariants of a tiling leads to some answers to the first question. In this project, we seek to gain a better understanding of some of these invariants, how symmetries manifest in them, and how to compute them, so that we can make progress in classifying mathematical quasicrystals.