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Computing algebraic invariants of symbolic dynamical systems

计算符号动力系统的代数不变量

基本信息

批准号：
EP/V007459/1
负责人：
Reem Yassawi
金额：
$ 42.9万
依托单位：
The Open University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV007459%2F1
关键词：
Computing algebraic invariants symbolic dynamical

项目摘要

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.

欧几里得对称性在自然界中无处不在。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。其他的对称性可以通过电子显微镜观察到，比如晶体的平移对称性，更微妙的是准晶的对称性，在上个世纪的大部分时间里，准晶的存在都受到怀疑。准晶是不具有正常晶体的平移对称性的晶体结构。准晶具有层次结构：在小尺度上出现的图案和结构在越来越大的尺度上再现。第一个准晶的数学模型是由罗杰·彭罗斯爵士在世纪前发现的。彭罗斯镶嵌具有反射对称性，但缺乏平移对称性。二维空间的旋转对称镶嵌必须具有三重、四重或六重旋转对称。但彭罗斯镶嵌具有局部五重旋转对称性，彭罗斯镶嵌只是一个数学模型，并不一定保证在自然界中存在。但在1982年，丹尼尔·谢赫特曼发现，五边形对称实际上出现在自然界中，同时在电子显微镜下研究快速冷却的铝和锰的熔融混合物。由于他的工作，他在2011年获得了诺贝尔奖。自从彭罗斯镶嵌的发现，数学家们已经发现了许多方法来创建这样的安排：有无限多的数学镶嵌的平面不具有平移对称性。由于受限于自然界提供的建筑材料种类，科学家们很难创造或发现这些瓦片。由此产生了两个相辅相成的问题。第一个问题是，什么时候两个数学镶嵌是等价的，第二个问题是，这些数学镶嵌中的哪一个可以在我们周围的世界中实现？研究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镶嵌时，他们可以忽略已知与已经实现的镶嵌等价的镶嵌。数学家使用抽象的代数结构（如对称群）来研究对称性。我们可以通过与称为不变量的代数结构相关联来表征平铺的结构特性。如果两个平铺是等价的，则它们的不变量是相同的。因此，理解平铺的代数不变量可以回答第一个问题。在这个项目中，我们试图更好地理解这些不变量中的一些，如何在它们中表现对称性，以及如何计算它们，以便我们可以在分类数学准晶方面取得进展。