The Farey framework for SL2-tilings

该研究方案将开发一个几何框架，汇集sl2 -瓦领域的重要工作，并回答该领域的主要开放性问题。这个提议的主题起源于简单的数字模式，在它们被发现三十年后被证明是一个深刻的数学理论的一小部分。这些数字图案在20世纪70年代首次被研究，并因其重复出现而被命名为friezes。康威和考克斯特设计了一种很有吸引力的方法，将多边形划分为三角形——三角多边形。除此之外，强大的簇代数理论在2000年代得到了发展，在不同的数学领域得到了深入的应用。人们观察到，横条可以由簇代数构成，这导致了新的数字模式的发展，比横条更普遍，称为SL2-tilings。sl2瓷砖领域自成立以来的十年里蓬勃发展，英国利兹和纽卡斯尔的领先团队。发现了与其他数学领域的联系，包括代数组合学、差分方程、射影几何和表示理论。对于sl2平铺类型的分类已经有了很大的关注，通常使用受Conway和Coxeter的三角多边形启发的模型，为数学家提供一种可视化的方式来解释sl2平铺。2015年，人们观察到康威和考克斯特的理论可以用一个被称为Farey复合体的几何对象来优雅地解释，它可以被粗略地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叫做双曲几何学，即物理学狭义相对论的几何学。PI在2020年接过指挥棒，使用Farey综合体为最近的一系列带有整数条目的sl2瓷砖作品提供了统一的方法。该提案推进了这一统一工作，为迄今为止尚未分类的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，这与表示数字有关；两者都是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，它们使用一种算法有时称为时钟算法，在这种算法中，您可以像在时钟上那样进行加减运算。到目前为止，还没有出现这种sl2贴片的模型；它们有点神秘。该建议的一个亮点将是使用鲜为人知的第n层的Farey复合体来模拟第n层的sl2平铺，就像Farey复合体模拟普通的sl2平铺一样。第二个目标是对条目为正数（不一定是整数）的sl2平铺进行分类。在这里，我们必须离开Farey复杂，而是使用双曲几何中的其他工具，包括PI和Beardon在2014年开发的环链。我们将证明已知的正整数sl2 -平铺的分类模型是更一般的正实sl2 -平铺的几何模型的特殊情况。第三个，也是最雄心勃勃的目标是解决众所周知的棘手的野生整数sl2切片。首先，我们将把注意力限制在那些只有有限个零条目的野生整数sl2平铺上。为了解决这些问题，我们在Farey复合体中引入了分岔路径，这是一种适合于该任务的新型几何对象。然后，我们将探索分岔路径可用于对野生整数SL2-tilings的完整集合进行分类的程度。该项目的结果将是一个框架，它包含并推进了sl2瓷砖的大量前沿研究。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发展的领域的世界领先地位。

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.