EPSRC Reference: 
EP/W002817/1 
Title: 
The Farey framework for SL2tilings 
Principal Investigator: 
Short, Professor I 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Faculty of Sci, Tech, Eng & Maths (STEM) 
Organisation: 
The Open University 
Scheme: 
New Investigator Award 
Starts: 
01 July 2022 
Ends: 
30 June 2025 
Value (£): 
375,376

EPSRC Research Topic Classifications: 
Algebra & Geometry 
Logic & Combinatorics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


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Panel History: 

Summary on Grant Application Form 
This research programme will develop a geometric framework to bring together a significant body of work in the field of SL2tilings 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 SL2tilings.
The field of SL2tilings 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 SL2tilings, usually with models inspired by Conway and Coxeter's triangulated polygons, to give mathematicians a visual way of interpreting SL2tilings.
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 SL2tilings with integer entries. This proposal advances this unifying work to offer geometric models for classes of SL2tilings 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 SL2tilings modulo n, which are collections of SL2tilings 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 SL2tilings; they were something of a mystery. A highlight of the proposal will be the use of the littleknown Farey complex of level n to model SL2tilings of level n, just as the Farey complex models normal SL2tilings.
The second objective is to classify SL2tilings 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 SL2tilings are special cases of geometric models for more general positive real SL2tilings.
The third, most ambitious objective is to tackle the notoriously thorny class of wild integer SL2tilings. First we will restrict our attention to those wild integer SL2tilings 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 SL2tilings.
The outcome of the project will be a framework which encompasses and advances a substantial body of cuttingedge research in SL2tilings. Each of the three objectives introduces distinct, new techniques. The research will strengthen the UK's worldleading profile in this rapidly expanding field.

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