## Matthew TointonI 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 GroupsMy book |

In reverse chronological order:

- Sharp relations between volume growth, isoperimetry and resistance in vertex-transitive graphs (with Romain Tessera), 35pp. arXiv
- A finitary structure theorem for vertex-transitive graphs of polynomial growth (with Romain Tessera), submitted, 25pp. arXiv
- Raconte-moi... les groupes approximatifs (in French),
*Gaz. Math.***160**(2019), 53-59. arXiv | Journal - Polylogarithmic bounds in the nilpotent Freiman theorem, to appear in
*Math. Proc. Cambridge Philos. Soc.*, 15pp arXiv | Journal - Probabilistic nilpotence in infinite groups (with Armando Martino, Motiejus Valiunas and Enric Ventura), submitted, 30pp. arXiv
- Scaling limits of Cayley graphs with polynomially growing balls (with Romain Tessera), 37pp. arXiv
- The Mahler conjecture in two dimensions via the probabilistic method,
*Amer. Math. Monthly***125**(2018), 820-828. arXiv | Journal - Commuting probability of infinite groups, to appear in
*J. London Math. Soc.*, 18pp. arXiv - The asymptotic dimension of box spaces of virtually nilpotent groups (with Thiebout Delabie),
*Discrete Math.***341**(2018) 1036-1040. arXiv | Journal - Properness of nilprogressions and the persistence of polynomial growth of given degree (with Romain Tessera),
*Discrete Anal.*2018:17, 38 pp. arXiv | Journal - Horofunctions on graphs of linear growth (with Ariel Yadin),
*C. R. Math. Acad. Sci. Paris***354**(2016), 1151-1154. arXiv | Journal - Approximate subgroups of residually nilpotent groups,
*Math. Ann.***374**(2019), 499-515. arXiv | Journal - Nilprogressions and groups with moderate growth (with Emmanuel Breuillard),
*Adv. Math.***289**(2016), 1008-1055. arXiv | Journal - 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 - Characterisations of algebraic properties of groups in terms of harmonic functions,
*Groups Geom. Dyn.***10**(2016), 1007-1049. arXiv | Journal - Recurrence and non-uniformity of bracket polynomials,
*Online J. Anal. Comb.***9**(2014), 36pp. arXiv | Journal - Freiman's theorem in an arbitrary nilpotent group,
*Proc. London Math. Soc.***109**(2014), 318-352. arXiv | Journal

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

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