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Dr Nick Gill

Profile summary

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:

  1. 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! The proof of the conjecture has been made public and will soon be published.
  2. 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.
  3. 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.


A generalization of Szep's conjecture for almost simple groups (2024-04)
Gill, Nick; Giudici, Michael and Spiga, Pablo
Vietnam Journal of Mathematics, 52 (pp. 325-359)

Can we do inclusive politics in urgent times? (2024)
Hennig, Brett; Gill, Nick and Lord, Thomas
Innovation: The European Journal of Social Science Research ((Early Access))

The binary actions of simple groups with a single conjugacy class of involutions (2024)
Gill, Nick and Guillot, Pierre
Journal of Group Theory ((Early access))

Irredundant bases for finite groups of Lie type (2023-05-22)
Gill, Nick and Liebeck, Martin W.
Pacific Journal of Mathematics, 322(2) (pp. 281-300)

Large minimal invariable generating sets in the finite symmetric groups (2023-03-10)
Garzoni, Daniele and Gill, Nick
Israel Journal of Mathematics, 255 (pp. 581-598)

On the number of conjugacy classes of a primitive permutation group (2023)
Garzoni, Daniele and Gill, Nick
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 153(1) (pp. 115-136)

Nilpotent covers of symmetric and alternating groups (2022-12)
Gill, Nick; Kimeu, Ngwava and Short, Ian
Bulletin of the Belgian Mathematical Society – Simon Stevin, 29(2) (pp. 249-266)

Statistics for Sn acting on k-sets (2022)
Gill, N. and Lodà, B.
Journal of Algebra, 607(Part A) (pp. 286-299)

The character table of a sharply -transitive subgroup of the alternating group of degree 12 (2021-03)
Gill, Nick and Hughes, Sam
International Journal of Group Theory, 10(1) (pp. 11-30)

On the height and relational complexity of a finite permutation group (2021)
Gill, Nick; Lodá, Bianca and Spiga, Pablo
Nagoya Mathematical Journal, 246 (pp. 372-411)

A generalization of a theorem of Rodgers and Saxl for simple groups of bounded rank (2020-06-01)
Gill, N.; Pyber, L. and Szabó, E.
Bulletin of the London Mathematical Society, 52(3) (pp. 464-471)

Binary permutation groups: Alternating and classical groups (2020-02)
Gill, Nick and Spiga, Pablo
American Journal of Mathematics, 142(1) (pp. 1-43)

Cherlin's conjecture for almost simple groups of Lie rank 1 (2019-11)
Gill, Nick; Hunt, Francis and Spiga, Pablo
Mathematical Proceedings of the Cambridge Philosophical Society, 167(3) (pp. 417-435)

Conway Groupoids and Completely Transitive Codes (2018)
Gill, Nick; Gillespie, Neil I. and Semeraro, Jason
Combinatorica, 38(2) (pp. 399-442)

Cherlin’s conjecture for sporadic simple groups (2018)
Dalla Volta, Francesca; Gill, Nick and Spiga, Pablo
Pacific Journal of Mathematics, 297(1) (pp. 47-66)

Abelian covers of alternating groups (2016-08)
Barrantes, Daniel; Gill, Nick and Ramírez, Jeremías
Archiv der Mathematik, 107(2) (pp. 135-150)

Quasirandom group actions (2016-08)
Gill, Nick
Forum of Mathematics, Sigma, 4, Article e24

Transitive projective planes and insoluble groups (2016-05-01)
Gill, Nick
Transactions of the American Mathematical Society, 368(5) (pp. 3017-3057)

Generating groups using hypergraphs (2016-03)
Gill, Nick; Gillespie, Neil I.; Nixon, Anthony and Semeraro, Jason
The Quarterly Journal of Mathematics, 67(1) (pp. 29-52)

Perfect commuting graphs (2016)
Britnell, John R. and Gill, Nick
Journal of Group Theory, 20(1)

On growth in an abstract plane (2015-08)
Gill, Nick; Helfgott, Harald A. and Rudnev, Misha
Proceedings of the American Mathematical Society, 143(8) (pp. 3593-3602)

Nilpotent covers and non-nilpotent subsets of finite groups of Lie type (2015)
Azad, Azizollah; Britnell, John R. and Gill, Nick
Forum Mathematicum, 27(6)

Growth in solvable subgroups of GLr(Z/pZ) (2014-10)
Gill, Nick and Helfgott, Harald Andrés
Mathematische Annalen, 360(1-2) (pp. 157-208)

Bounds on the diameter of Cayley graphs of the symmetric group (2014-08)
Bamberg, John; Gill, Nick; Hayes, Thomas P.; Helfgott, Harald A.; Seress, Akos and Spiga, Pablo
Journal of Algebraic Combinatorics, 40(1) (pp. 1-22)

A note on the Weiss conjecture (2013-12)
Gill, Nick
Journal of the Australian Mathematical Society, 95(3) (pp. 356-361)

On the product decomposition conjecture for finite simple groups (2013-11)
Gill, Nick; Pyber, László; Short, Ian and Szabó, Endre
Groups, Geometry, and Dynamics, 7(4) (pp. 867-882)

Orientably regular maps with Euler characteristic divisible by few primes (2013-08)
Gill, Nick
Journal of the London Mathematical Society, 88(1) (pp. 118-136)

Conjugacy in Thompson's group $F$ (2013)
Short, Ian and Gill, Nick
Proceedings of the American Mathematical Society, 141 (pp. 1529-1538)

(2,m,n)-groups with Euler characteristic equal to $-2^as^b$ (2013)
Gill, Nick
Electronic Journal of Combinatorics, 20, Article 8(3)

Real and strongly real classes in SLn(q) (2011-05)
Gill, Nick and Singh, Anupam
Journal of Group Theory, 14(3) (pp. 437-459)

Real and strongly real classes in PGLn(q) and quasi-simple covers of PSLn(q) (2011-05)
Gill, Nick and Singh, Anupam
Journal of Group Theory, 14(3) (pp. 461-489)

Growth of small generating sets in SLn(ℤ/pℤ) (2011)
Gill, Nick and Helfgott, Harald Andres
International Mathematics Research Notices, 18 (pp. 4226-4251)

Reversible maps and composites of involutions in groups of piecewise linear homeomorphisms of the real line (2010-03)
Gill, Nick and Short, Ian
Aequationes Mathematicae, 79(1-2) (pp. 23-37)

Reversibility in the group of homeomorphisms of the circle (2009-10)
Gill, Nick; O'Farrell, Anthony G. and Short, Ian
Bulletin of the London Mathematical Society, 41(5) (pp. 885-897)

Large dimensional classical groups and linear spaces (2008-11-15)
Camina, Alan R.; Gill, Nick and Zalesski, A. E.
Bulletin of the Belgian Mathematical Society (Simon Stevin), 15(4) (pp. 705-731)

Transitive projective planes (2007)
Gill, Nick
Advances in Geometry, 7(4) (pp. 475-528)

$PSL(3,q)$ and line-transitive linear spaces (2007)
Gill, Nick
Beiträge zur Algebra und Geometrie, 48(2) (pp. 591-620)

Polar spaces and embeddings of classical groups (2007)
Gill, Nick
New Zealand Journal of Mathematics, 36 (pp. 175-184)

Linear spaces with significant characteristic prime (2006)
Gill, Nick
Innovations in Incidence Geometry, 3 (pp. 109-119)

Nilpotent Singer groups (2006)
Gill, Nick
Electronic Journal of Combinatorics, 13(1) (R94)

Cherlin’s Conjecture for Finite Primitive Binary Permutation Groups (2022)
Gill, Nick; Liebeck, Martin W. and Spiga, Pablo
Lecture Notes in Mathematics
ISBN : 978-3-030-95956-2 | Publisher : Springer | Published : Cham

Conway’s groupoid and its relatives (2017)
Gill, Nick; Gillespie, Neil; Praeger, Cheryl and Semeraro, Jason
In: Bhargava, Manjul; Guralnick, Robert; Hiss, Gerhard; Lux, Klaus and Tiep, Pham Huu eds. Finite Simple Groups: Thirty Years of the Atlas and Beyond. Contemporary Mathematics (pp. 91-110)
Publisher : American Mathematical Society