Computing algebraic invariants of symbolic dynamical systems

欧几里得对称在自然界中无处不在。其中一些对称是肉眼可见的，比如蝴蝶翅膀的两侧对称。其他的对称性可以通过电子显微镜观察，例如晶体的平移对称性。更难以描述的是准晶体的对称性，在上个世纪的大部分时间里，准晶体的存在一直受到怀疑。准晶体是不具有正常晶体平动对称性的晶体结构。准晶体具有层次结构：在小尺度上出现的图案和结构可以在越来越大的尺度上重现。半世纪前，罗杰·彭罗斯爵士发现了第一个准晶体的数学模型。彭罗斯瓷砖具有反射对称，但缺乏平移对称。二维空间的平移对称平铺必须具有三倍、四倍或六倍旋转对称。但彭罗斯瓷砖具有局部五重旋转对称。彭罗斯的瓷砖只是一个数学模型，它不一定保证在自然界中存在。但在1982年，丹尼尔·谢赫特曼（Daniel Schechtman）在电子显微镜下研究快速冷却的铝和锰的熔融混合物时，发现了五角形对称性实际上在自然界中出现。由于他的工作，他在2011年获得了诺贝尔奖。自从彭罗斯平铺法被发现以来，数学家们已经发现了许多创造这种排列的方法：有无限多的平面的数学平铺法不具有平移对称性。由于受限于自然界提供的各种积木，科学家们很难创造或发现这些瓷砖。这就产生了两个相辅相成的问题。第一个问题是，什么时候两种数学拼贴在某种程度上是等价的，第二个问题是，哪些数学拼贴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调查第二个问题，因为在试图实现数学平铺时，他们可以忽略已知的与已经实现的平铺相等的平铺。数学家用抽象的代数结构如对称群来研究对称性。我们可以通过将称为不变量的代数结构与瓷砖相关联来表征其结构性质。如果两个平铺是等价的，它们的不变量是相同的。因此，对平铺的代数不变量的理解将引出第一个问题的一些答案。在这个项目中，我们试图更好地理解这些不变量中的一些，对称性是如何在它们中表现出来的，以及如何计算它们，这样我们就可以在分类数学准晶体方面取得进展。

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.