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Dr Nick Gill

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.


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

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)

A generalization of Szep's conjecture for almost simple groups
Gill, Nick; Giudici, Michael and Spiga, Pablo
Vietnam Journal of Mathematics ((In Press))

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 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