Professor Ian Short

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.

: "Research meeting for the event Aperiodic tilings – a meeting and mathematical art exhibition in honour of Uwe Grimm".

EPSRC Research Associates in Mathematical Sciences have offered £100,000 for research associate positions with the M&S department to start in October 2021.

Characterising aperiodically ordered structures and understanding their properties has been a major area of activity since the discovery of quasicrystals in the 1980s. There are three types of spectral quantities which are commonly studied in this context: (1) the diffraction measure, which is the Fourier transform of the autocorrelation measure and is directly related to the outcome of diffraction experiments; (2) the dynamical spectrum, which is related to the action of translations on the system; (3) the Schroedinger spectrum, which is the energy spectrum of a (discrete) Schroedinger operator describing the motion of electrons in an aperiodic system. While a good understanding of the relationshop between diffraction and dynamical spectra has recently emerged, the relation between these two spectra and those of the corresponding (discrete) Schroedinger operator has remained elusive. This project aims at investigating this relationship, building on recent results that point to a key role played by Lyapunov exponents of matrix product cocycles for the diffraction spectrum.

A grant for £1700 from the London Mathematical Society to host the One Day Function Theory Meeting at De Morgan House, London, on 12 September 2016.