The Farey framework for SL2-tilings

这项研究计划将开发一个几何框架，汇集了一个重要的机构的工作在该领域的SL 2-tilings和回答主要的开放问题，在该领域。这一建议的主题起源于简单的数字模式，30年后，他们的发现被证明是一个深刻的数学理论的一小部分。这些数字模式首次在20世纪70年代被研究，并因其重复的外观而被命名为中楣。康威和考克斯特设计了一种有吸引力的方法，使用划分为三角形的多边形（三角形多边形）对中楣进行分类。与此无关，在2000年代发展了强大的簇代数理论，在不同的数学领域找到了深入的应用。人们注意到可以从簇代数中构造出中楣，这导致了新的数字模式的发展，比中楣更一般，称为SL 2-tilings。SL 2-tilings领域自成立以来的十年中蓬勃发展，在英国的利兹和纽卡斯尔有领先的研究小组。发现了与其他数学领域的联系，包括代数组合学、差分方程、射影几何和表示论。SL 2-tilings的分类一直是一个重要的焦点，通常是受到Conway和Coxeter的三角形多边形的启发，为数学家提供一种直观的方式来解释SL 2-tilings。在2015年，有人观察到Conway和Coxeter的理论可以用一个叫做Farey复体的几何对象来优雅地解释，它可以被松散地认为是一个无限三角形多边形。PI在2020年接过了接力棒，使用Farey复形为最近的一系列关于SL 2-tilings的整数项工作提供了统一的方法。该建议推进了这一统一工作，为迄今为止抵制分类的SL 2-tilings类提供几何模型。为了实现这一点，我们将应用双曲几何和连分数领域的技术，这与表示数有关;两者都是PI的专业领域。有三个主要目标，如下所示。第一个目标是分类SL 2-模n拼接，是SL 2的集合体，使用一种有时称为时钟算法的算法的平铺，在这种算法中，您可以像在时钟上一样进行加减运算。到目前为止，还没有出现这种SL 2贴片的模型;它们是一个谜。该提案的一个亮点是使用鲜为人知的n级Farey复形来建模n级的SL 2-tilings，就像Farey复形建模正常的SL 2-tilings一样。第二个目标是对具有正数（不一定是整数）条目的SL 2-tilings进行分类。在这里，我们必须离开Farey复形，而是使用双曲几何的其他工具，包括PI和Beardon在2014年开发的horocycles链。我们将证明，已知的模型分类正整数SL 2-tilings的几何模型的特殊情况下，更一般的正真实的SL 2-tilings. Third，最雄心勃勃的目标是解决臭名昭著的棘手类野生整数SL 2-tilings。首先，我们将注意力限制在那些只有1000个零元素的野生整数SL 2-平铺上。为了接近这些，我们引入分叉路径的Farey复杂，一种新的类型的几何对象适合的任务。然后，我们将探讨在何种程度上分叉路径可以用来分类的野生整数SL 2-tilings.The项目的结果将是一个框架，其中包括和推进了大量的前沿研究SL 2-tilings. The收集。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大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.