The Farey framework for SL2-tilings

SL2-tilings 的 Farey 框架

• 批准号: EP/W002817/1
• 负责人: Ian Short
• 金额: $ 47.83万
• 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: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW002817%2F1
• 关键词: Farey; framework; SL2; tilings

This research programme will develop a geometric framework to bring together a significant body of work in the field of SL2-tilings and to answer major open questions in the field.The subject of this proposal originated in simple number patterns which thirty years after their discovery proved to be a small part of a profound mathematical theory. These number patterns were first studied in the 1970s and named friezes because of their repetitive appearance. Conway and Coxeter devised an attractive way to classify friezes using polygons divided into triangles - triangulated polygons. Independent of this, the powerful theory of cluster algebras was developed in the 2000s, which found deep applications in diverse mathematical fields. It was observed that friezes could be constructed from cluster algebras, and this led to the development of new number patterns, more general than friezes, called SL2-tilings.The field of SL2-tilings flourished in the decade since their inception, with leading groups in the UK at Leeds and Newcastle. Connections were uncovered to other mathematical fields, including algebraic combinatorics, difference equations, projective geometry, and representation theory. There has been significant focus on classifying types of SL2-tilings, usually with models inspired by Conway and Coxeter's triangulated polygons, to give mathematicians a visual way of interpreting SL2-tilings.In 2015, it was observed that Conway and Coxeter's theory can be explained elegantly using a geometric object called the Farey complex, which can be thought of loosely as an infinite triangulated polygon. It has a geometry associated to it known as hyperbolic geometry, the geometry of special relativity from physics.The PI took up the baton in 2020, using the Farey complex to offer a unified approach to a host of recent works on SL2-tilings with integer entries. This proposal advances this unifying work to offer geometric models for classes of SL2-tilings that have thus far resisted classification. To achieve this, we will apply techniques from hyperbolic geometry and the field of continued fractions, which is concerned with representing numbers; both are fields of expertise of the PI.There are three primary objectives, as follows.The first objective is to classify SL2-tilings modulo n, which are collections of SL2-tilings that use a type of arithmetic sometimes called clock arithmetic in which you add and subtract in the way you do on a clock. Until now no models have emerged for these SL2-tilings; they were something of a mystery. A highlight of the proposal will be the use of the little-known Farey complex of level n to model SL2-tilings of level n, just as the Farey complex models normal SL2-tilings.The second objective is to classify SL2-tilings with entries that are positive numbers, not necessarily integers. Here we must leave the Farey complex and instead use other tools from hyperbolic geometry, including chains of horocycles developed by the PI and Beardon in 2014. We will demonstrate that known models for classifying positive integer SL2-tilings are special cases of geometric models for more general positive real SL2-tilings.The third, most ambitious objective is to tackle the notoriously thorny class of wild integer SL2-tilings. First we will restrict our attention to those wild integer SL2-tilings with only finitely many zero entries. To approach these, we introduce bifurcating paths in the Farey complex, a new type of geometric object suitable to the task. We will then explore the extent to which bifurcating paths can be used to classify the full collection of wild integer SL2-tilings.The outcome of the project will be a framework which encompasses and advances a substantial body of cutting-edge research in SL2-tilings. Each of the three objectives introduces distinct, new techniques. The research will strengthen the UK's world-leading profile in this rapidly expanding field.

这项研究计划将开发一个几何框架，将SL2-Tabling领域的大量工作聚集在一起，并回答该领域的主要悬而未决的问题。这项建议的主题起源于简单的数字模式，在他们发现30年后，事实证明，这只是一个深刻的数学理论的一小部分。这些数字模式在20世纪70年代首次被研究，并因其重复出现而被命名为Flaeze。康威和科克塞特设计了一种吸引人的方法，用分成三角形的多边形--三角多边形--对薯条进行分类。与此无关的是，强大的簇代数理论在本世纪头十年得到了发展，它在不同的数学领域得到了深入的应用。人们观察到，褶皱可以从簇代数构造而来，这导致了新的数字模式的发展，这种模式比褶皱更普遍，称为SL2切片。SL2切片领域在其诞生后的十年中蓬勃发展，英国的主要小组在利兹和纽卡斯尔。还发现了与其他数学领域的联系，包括代数组合学、差分方程式、射影几何和表示论。人们非常关注SL2-瓷砖的分类，通常是受康威和科克塞特的三角多边形模型的启发，为数学家提供一种解释SL2-瓷砖的直观方式。2015年，人们观察到，康威和科克塞特的理论可以用一个名为Farey Complex的几何对象来优雅地解释，它可以松散地被认为是一个无限的三角多边形。它有一种与之相关的几何学，称为双曲几何学，这是来自物理学的狭义相对论几何学。PI在2020年接过接力棒，使用Farey复合体为最近关于SL2的一系列工作提供了一种统一的方法-整数分割法。这项建议推进了这项统一的工作，为到目前为止一直拒绝分类的SL2-瓷砖类别提供几何模型。为了实现这一点，我们将应用双曲几何和连分式领域的技术，这两个领域都是PI的专业领域。有三个主要目标如下。第一个目标是对以n为模的SL2-平铺进行分类，这是SL2-平铺的集合，它使用一种有时被称为时钟算法的算法，在这种算法中，你可以像在时钟上做的那样进行加减。到目前为止，还没有出现这些SL2瓷砖的模型；它们是某种神秘的东西。该方案的一个亮点将是使用鲜为人知的n级Farey复形来模拟n级的SL2平铺，就像Farey复形模拟正常的SL2平铺一样。第二个目标是用正数而不一定是整数的条目来对SL2平铺进行分类。在这里，我们必须离开Farey Complex，转而使用双曲几何中的其他工具，包括PI和Beardon在2014年开发的星轮链。我们将证明，已知的用于分类正整数SL2-平铺的模型是更一般的正实数SL2-平铺的几何模型的特例。第三，也是最雄心勃勃的目标是解决臭名昭著的野生整数SL2-平铺的棘手类。首先，我们将只关注那些只有有限多个零条目的野整数SL2平铺。为了解决这些问题，我们在Farey复形中引入了分支路径，Farey复形是一种适合于该任务的新型几何对象。然后，我们将探索分支路径可以在多大程度上用于对完整的野生整数SL2切片集合进行分类。该项目的结果将是一个框架，该框架包含并推进了大量SL2切片的前沿研究。这三个目标中的每一个都引入了不同的新技术。这项研究将加强英国在这一迅速扩张的领域的世界领先地位。