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Matthew Tointon

I am the Stokes Research Fellow at Pembroke College, Cambridge, affiliated to the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge.

I tend to be interested in problems and results that involve some combination of algebra, combinatorics and geometry; much of my work lies at the interface between additive combinatorics and geometric group theory.

I completed my PhD in at Magdalene College, Cambridge in 2013 under the supervision of Ben Green. I have held postdoctoral positions at Homerton College, Cambridge, at the Université de Paris-Sud, and at the Université de Neuchâtel, Switzerland.


Book: Introduction to Approximate Groups

My book Introduction to Approximate Groups is now available to pre-order from Cambridge University Press in hardback and paperback.

Papers and preprints

In reverse chronological order:

  1. A finitary structure theorem for vertex-transitive graphs of polynomial growth (with Romain Tessera), submitted, 25pp.     arXiv
  2. Raconte-moi... les groupes approximatifs (in French), Gaz. Math. 160 (2019), 53-59.     arXiv    |    Journal
  3. Polylogarithmic bounds in the nilpotent Freiman theorem, to appear in Math. Proc. Cambridge Philos. Soc., 15pp     arXiv    |    Journal
  4. Probabilistic nilpotence in infinite groups (with Armando Martino, Motiejus Valiunas and Enric Ventura), submitted, 30pp.     arXiv
  5. Scaling limits of Cayley graphs with polynomially growing balls (with Romain Tessera), 37pp.     arXiv
  6. The Mahler conjecture in two dimensions via the probabilistic method, Amer. Math. Monthly 125 (2018), 820-828.     arXiv    |    Journal
  7. Commuting probability of infinite groups, submitted, 20pp.     arXiv
  8. The asymptotic dimension of box spaces of virtually nilpotent groups (with Thiebout Delabie), Discrete Math. 341 (2018) 1036-1040.     arXiv    |    Journal
  9. Properness of nilprogressions and the persistence of polynomial growth of given degree (with Romain Tessera), Discrete Anal. 2018:17, 38 pp.     arXiv    |    Journal
  10. Horofunctions on graphs of linear growth (with Ariel Yadin), C. R. Math. Acad. Sci. Paris 354 (2016), 1151-1154.     arXiv    |    Journal
  11. Approximate subgroups of residually nilpotent groups, Math. Ann. 374 (2019), 499-515.     arXiv    |    Journal
  12. Nilprogressions and groups with moderate growth (with Emmanuel Breuillard), Adv. Math. 289 (2016), 1008-1055.     arXiv    |    Journal
  13. Polynomials and harmonic functions on discrete groups (with Tom Meyerovitch, Idan Perl and Ariel Yadin), Trans. Amer. Math. Soc. 369 (2017), 2205-2229.     arXiv    |    Journal
  14. Characterisations of algebraic properties of groups in terms of harmonic functions, Groups Geom. Dyn. 10 (2016), 1007-1049.     arXiv    |    Journal
  15. Recurrence and non-uniformity of bracket polynomials, Online J. Anal. Comb. 9 (2014), 36pp.     arXiv    |    Journal
  16. Freiman's theorem in an arbitrary nilpotent group, Proc. London Math. Soc. 109 (2014), 318-352.     arXiv    |    Journal


Emmanuel Breuillard, Thiebout Delabie, Tom Meyerovitch, Idan Perl, Romain Tessera, Ariel Yadin.


The following video describes parts of my work to a non-specialist audience.

Unpublished material

An alternative approach to Freiman's theorem in p-groups.

An essay on the Tits alternative, which I wrote in 2009 as part of my Part III studies in Cambridge.

Graduate student and postdoc seminar 2015/16, Université Paris-Sud