Dr Nicholas Gill
Senior Lecturer In Pure Mathematics
School of Mathematics & Statistics
Biography
Professional biography
I joined the OU in October 2021 as a Lecturer in Pure Mathematics. Prior to that I held academic positions at the University of South Wales, the University of Costa Rica, the University of Bristol and the Institute for Mathematical Sciences in Chennai, India.
I work at the OU 2 days a week; for the remainder of the week I am a Project Manager at the Sortition Foundation.
Research interests
I am interested in group theory which is an area of algebra concerned with the mathematical study of symmetry. Some particular interests are:
- Relational complexity: This is a statistic associated with finite permutation groups and originating in model theory. I have spent a lot of time trying to understand when finite permutation groups have relational complexity as large as possible or as small as possible. For the latter situation there is a famous conjecture of Cherlin which describes those primitive groups with RC=2. Martin Liebeck, Pablo Spiga and I held an EPSRC grant aimed at proving this conjecture: we were successful! A full account of the proof of the conjecture has now been published. In April 2024, Martin, Pablo and I, together with Pierre Guillot, will embark on new work, also funded by the EPSRC, to push this work further.
- Growth in groups: The study of how sets grow inside groups turns out to have applications in all sorts of areas of mathematics, from number theory to graph theory. I worked with Harald Helfgott to prove a particular case of the Helfgott-Lindenstrauss conjecture. This conjecture gives conditions under which subsets of finite linear groups grow rapidly. I also held an EPSRC grant to work on a related conjecture due to Liebeck, Nikolov and Shalev. This conjecture has now been proved in joint work with Lifshitz, Pyber and Szabo.
- Groups and finite geometry: Finite groups are connected to finite geometries in many ways: most obviously they can occur as automorphism groups and I have studied this situation in the case where the geometry in question is a finite projective plane. I have also studied how finite geometries can be used to "generate" groups and, with Neil Gillespie and Jason Semeraro, discovered some new ways to generate certain finite simple groups.
Publications
Book
Cherlin’s Conjecture for Finite Primitive Binary Permutation Groups (2022)
Book Chapter
Journal Article
The binary actions of simple groups with a single conjugacy class of involutions (2025)
A Generalization of Szep’s Conjecture for Almost Simple Groups (2024)
A generalization of Szep's conjecture for almost simple groups (2024)
The binary actions of simple groups of Lie type of characteristic 2 (2024)
Can we do inclusive politics in urgent times? (2024)
Irredundant bases for finite groups of Lie type (2023)
Large minimal invariable generating sets in the finite symmetric groups (2023)
On the number of conjugacy classes of a primitive permutation group (2023)
Nilpotent covers of symmetric and alternating groups (2022)
Statistics for Sn acting on k-sets (2022)
The character table of a sharply -transitive subgroup of the alternating group of degree 12 (2021)
On the height and relational complexity of a finite permutation group (2021)
A generalization of a theorem of Rodgers and Saxl for simple groups of bounded rank (2020)
Binary permutation groups: Alternating and classical groups (2020)
Cherlin's conjecture for almost simple groups of Lie rank 1 (2019)
Cherlin’s conjecture for sporadic simple groups (2018)
Conway Groupoids and Completely Transitive Codes (2018)
Quasirandom group actions (2016)
Abelian covers of alternating groups (2016)
Transitive projective planes and insoluble groups (2016)
Generating groups using hypergraphs (2016)
Perfect commuting graphs (2016)
On growth in an abstract plane (2015)
Nilpotent covers and non-nilpotent subsets of finite groups of Lie type (2015)
Growth in solvable subgroups of GLr(Z/pZ) (2014)
Bounds on the diameter of Cayley graphs of the symmetric group (2014)
A note on the Weiss conjecture (2013)
On the product decomposition conjecture for finite simple groups (2013)
Orientably regular maps with Euler characteristic divisible by few primes (2013)
(2,m,n)-groups with Euler characteristic equal to $-2^as^b$ (2013)
Conjugacy in Thompson's group $F$ (2013)
Real and strongly real classes in PGLn(q) and quasi-simple covers of PSLn(q) (2011)
Real and strongly real classes in SLn(q) (2011)
Growth of small generating sets in SLn(ℤ/pℤ) (2011)
Reversibility in the group of homeomorphisms of the circle (2009)
Large dimensional classical groups and linear spaces (2008)
$PSL(3,q)$ and line-transitive linear spaces (2007)
Polar spaces and embeddings of classical groups (2007)
Transitive projective planes (2007)
