Research

“We adore chaos because we love to produce order.”
– M. C. Escher

My main research focus is in group theory, the mathematical study of symmetry. I am interested in properties of abstract groups, such as generating sets and subgroup structure, and I am also interested in properties of group actions, such as questions concerning derangements and bases of permutation groups. I am particularly happy when these two topics intersect. While group theoretic questions are usually the motivation for my work, some of my projects would be better described as representation theory, geometric group theory, Lie theory or combinatorics. Lately, I have become involved in formalising proofs in Lean. I am also interested in mathematics education and the philosophy of mathematical practice.

A major strand of my work has concerned generating pairs for groups. This began with finite groups, and a recent highlight here is my paper with Tim Burness and Robert Guralnick [Annals of Mathematics (2021)] which answered the question: “In which finite groups does every nontrivial element belong to a generating pair?” More recently, I have been considering the ways that these sorts of results do and do not extend to infinite groups, and this is the subject of my current EPSRC Postdoctoral Fellowship.

I wrote a survey article based on my plenary lecture at Groups St Andrews in 2022 (see item 16 below). This gives an overview of recent work concerning generating pairs for finite and infinite groups.

For a friendly nontechnical introduction to my work, take a look at my poster I presented at STEM for Britain. For further information about my work, but still requiring no specialist information at all, have a read of my leaflet The Art of Measuring Symmetry.

Publications

  1. Average distances on self-similar sets and higher order average distances of self-similar measures
    Joint with D. Allen, H. Edwards, L. Olsen
    Mathematische Zeitschrift 287 (2017) 287–324
    doi:10.1007/s00209-016-1826-3
  2. On the uniform spread of almost simple symplectic and orthogonal groups
    Journal of Algebra 490 (2017) 330–371
    arXiv:1703.09652 | doi:10.1016/j.jalgebra.2017.07.008
  3. On the uniform domination number of a finite simple group
    Joint with T. C. Burness
    Transactions of the American Mathematical Society 372 (2019) 545–583
    arXiv:1710.07113 | doi:10.1090/tran/7593 | code
  4. The distinguishing number of quasiprimitive and semiprimitive groups
    Joint with A. Devillers, L. Morgan
    Archiv der Mathematik 113 (2019) 127–139
    arXiv:1808.08705 | doi:10.1007/s00013-019-01324-7
  5. Permutations with orders coprime to a given integer
    Joint with J. Bamberg, S. P. Glasby, C. E. Praeger
    Electronic Journal of Combinatorics 27 (2020) P.16 (14 pages)
    arXiv:1807.10450 | doi:10.37236/8678
  6. Finite groups, 2-generation and the uniform domination number
    Joint with T. C. Burness
    Israel Journal of Mathematics 239 (2020) 271–367
    arXiv:1810.12076 | doi:10.1007/s11856-020-2050-8
  7. Infinite 3/2-generated groups
    Joint with C. Donoven
    Bulletin of the London Mathematical Society 52 (2020) 657–673
    arXiv:1907.05498 | doi:10.1112/blms.12356
  8. Connectivity of generating graphs of nilpotent groups
    Joint with A. Lucchini
    Algebraic Combinatorics 3 (2020) 1183–1195
    arXiv:2002.03330 | doi:10.5802/alco.132
  9. The spread of almost simple classical groups
    Lecture Notes in Mathematics vol. 2286, Springer, 2021, viii+154
    ISBN: 978-3-030-74099-3 | doi:10.1007/978-3-030-74100-6
    (I have several print copies. Let me know if you'd like one.)
  10. The spread of a finite group
    Joint with T. C. Burness and R. M. Guralnick
    Annals of Mathematics 193 (2021) 619–687
    arXiv:2006.01421 | doi:10.4007/annals.2021.193.2.5
  11. Shintani descent, simple groups and spread
    Journal of Algebra 578 (2021) 319–355
    arXiv:2008.02558 | doi:10.1016/j.jalgebra.2021.02.021
  12. Flexibility in generating sets of finite groups
    Archiv der Mathematik 118 (2022) 231–237
    arXiv:2111.12534 | doi:10.1007/s00013-021-01691-0
  13. The maximal size of a minimal generating set
    Forum of Mathematics, Sigma 11 (2023) e70 1–10
    arXiv:2303.09509 | doi:10.1017/fms.2023.71
  14. Totally deranged elements of almost simple groups and invariable generating sets
    Journal of the London Mathematical Society 109 (2024) e12935 1–38
    arXiv:2304.10213 | doi:10.1112/jlms.12935
  15. Minimal cover groups
    Joint with P. J. Cameron, D. Craven, H. R. Dorbidi and B. Sambale
    Journal of Algebra 660 (2024) 345–372
    arXiv:2311.15652 | doi:10.1016/j.jalgebra.2024.06.038
  16. The spread of finite and infinite groups
    Groups St Andrews 2022 in Newcastle,
    LMS Lecture Note Series vol. 496, CUP, 2024, 74–117
    arXiv:2210.09635 | doi.org/10.1017/9781009563208.004
  17. Thompson's group T is 3/2-generated
    Joint with C. Bleak and R. Skipper
    Israel Journal of Mathematics (to appear)
    arXiv:2206.05316
  18. Kronecker classes, normal coverings and chief factors of groups
    Joint with M. Fusari and P. Spiga
    Bulletin of the Australian Mathematical Society (to appear)
    arXiv:2410.02569
  19. Representations of extensions of simple groups
    Joint with M. W. Liebeck
    Archiv der Mathematik (to appear)
    arXiv:2405.17593
  20. Orbits of permutation groups with no derangements
    Joint with D. Ellis
    preprint
    arXiv:2408.16064
  21. Classifying the groups of order pq in Lean
    Joint with P. Wu
    preprint
    arXiv:2501.09769

Miscellaneous