STEM Faculty

Professional biography

I am Professor of Geometry, Deputy Head of School of Mathematics and Statistics, and REF Lead for B10 Mathematical Sciences. I joined the Open University in 2010 having previously worked at the University of Cambridge and the National University of Ireland, Maynooth. 

Research interests

My research interests include geometry and dynamics, complex analysis, frieze patterns, continued fractions, and group theory. Here is some recent work.

Classifying SL₂-tilings

The Farey framework for SL₂-tilings is a project on classifying SL₂-tilings using the geometric, numeric, and combinatorial properties of the Farey graph. It is funded by the EPSRC grant EP/W002817/1 and is carried out in collaboration with Matty van Son (former postdoc, now at the University of the West Indies), research student Andrei Zabalotskii, former research student Margaret Stanier, and Oleg Karpenkov (University of Liverpool). See Classifying SL₂-tiling, Frieze patterns and Farey complexes, and Classifying integer hypertilings.

SL₂-tiling and corresponding paths in the Farey graph

Iterated function systems of holomorphic maps on hyperbolic Riemann surfaces

Research on this topic includes Iterated function systems of holomophic maps with Marco Abate (University of Pisa). Before that I worked with two research students Argyrios Christodoulou and (earlier) Matthew Jacques. See Stability of the Denjoy–Wolff theorem and Semigroups of hyperbolic isometries.

Forward and backward limit sets

Necessary and sufficient conditions for convergence of continued fractions

In Necessary and sufficient conditions for convergence of continued fractions former research student Margaret Stanier and I describe a simple algorithm for determining whether an integer continued fraction converges, inspired by the geometry of the Farey complex. 

Homotopies in the Farey complex

Nilpotent covers of symmetric and alternating groups

We prove that the symmetric group on n letters has a unique minimal cover by maximal normal subgroups. This cover has striking properties; for example, the number of conjugacy classes in the cover is equal to the number of partitions of n into distinct positive integers. This research was funded in part by a Mentoring African Researchers in Mathematics grant held jointly with Nick Gill and carried out in collaboration with (former) research student Kimei Arphaxad Ngwava from Moi University, Kenya. See Nilpotent covers of symmetric and alternating groups.

Forward limit sets of substitution semigroups

We consider the dynamics of finitely-generated semigroups of substitutions of finite alphabets, which is closely related to the theory of S-adic substitutions. We examine the forward limit set of a semigroup; this is the collection of all S-adic limits with the same generating set. This is joint work with former research student Ibai Aedo and our fine colleague Uwe Grimm, who died in 2021 during preparation of the paper.

First-letter graph for some substitutions

Teaching interests

I teach at all levels of the OU mathematics curriculum. I authored several chapters of the level 1 modules MST124 Essential mathematics 1 and MST125 Essential mathematics 2, I chaired the level 3 module M337 Complex analysis (2016–2021), and I chaired the MSc Mathematics module M829 Analytic number theory II (2012–2022, 2024–2026). Additionally, I chaired production of the following modules. 

M337 Complex analysis

I chaired the rewrite of M337, completed in 2018. The revised books for M337 have approximately 1150 pages, well over 1000 figures, and a wealth of exercises and solutions. 

M840 Complex function theory

I chair the M840 dissertation topic Complex function theory, released in 2021, and formerly called Riemann surfaces. This is a sequel to M337 for those students of the MSc Mathematics degree who might wish to pursue research in the field.

Closed curves on a compact Riemann surface of genus 3

Impact and engagement

Aperiodic tilings – a meeting and mathematical art exhibition in honour of Uwe Grimm

In 2022 with Charlotte Lighter and Reem Yassawi I organised the three-day event Aperiodic tilings, which brought together mathematicians, engineers, scientists, and artists to share research and communicate artwork in the subject of aperiodic tilings. The exhibition included painting, sculptures, 3D printing, musical art, computer-generated images, and hands-on activities. It was open to the public and attracted 750 visitors. An online gallery displays artwork from the event. The event celebrated the research and public engagement work of Uwe Grimm.

Artwork from Aperiodic tilings

Navigating by numbers

I lead the Navigating by numbers outreach programme. The purpose of the programme is to communicate an attractive network of geometric and numeric ideas centred around the Farey graph. This involves Coxeter's frieze patterns, SL₂-tilings, continued fractions, dessins d'enfants, hyperbolic geometry, and a host of other mathematical concepts. This programme complements my EPSRC project EP/W002817/1 on SL₂-tilings. 

The Farey graph