CARMA Research Group

Symmetry

About us

Leader: George Willis

Symmetry is accounted for mathematically through the algebraic concept of a 'group'. Our research focusses on '0-dimensional', aka 'totally disconnected, locally compact', groups, which arise as symmetries of graphs (in the sense of networks). We aim to analyse the structure of these groups and thus to understand the types of symmetries that graphs may possess. Whereas connected locally compact groups are well understood, much remains to be done in the totally disconnected case and we are filling significant gaps in knowledge. The research also has significant links with harmonic analysis, number theory and geometry — ideas from these fields are used in our research and our results feed back to influence these other fields. Another part of our research aims to develop computational tools for working with the groups and for visualising the corresponding symmetries.

We collaborate with researchers in Europe, Asia and North and South America in this work, which is being supported by Australian Research Council funds of $2.8 million in the period 2018-22.

More information about us and our research can be at https://zerodimensional.group. Feel free to contact us at contact@zerodimensional.group.

People

Members of this research group:

  • George Willis
  • Michal Ferov
  • Alejandra Garrido
  • Colin Reid
  • David Robertson
  • Stephan Tornier

Events

  • Symmetry in Newcastle

    5:30 pm, Monday, 1st Nov 2021
    , Online
    "Optimal regularity of mapping class group actions on the circle"
       — Prof Sang-Hyun Kim

    Abstract:
    We prove that for each finite index subgroup $H$ of the mapping class group of a closed hyperbolic surface, and for each real number $r>0$ there does not exist a faithful $C^{1+r}$--action of $H$ on a circle. (Joint with Thomas Koberda and Cristobal Rivas)
    "Uniform discreteness of arithmetic groups and the Lehmer conjecture"
       — Dr François Thilmany

    Abstract:
    The famous Lehmer problem asks whether there is a gap between 1 and the Mahler measure of algebraic integers which are not roots of unity. Asked in 1933, this deep question concerning number theory has since then been connected to several other subjects. After introducing the concepts involved, we will briefly describe a few of these connections with the theory of linear groups. Then, we will discuss the equivalence of a weak form of the Lehmer conjecture and the "uniform discreteness" of cocompact lattices in semisimple Lie groups (conjectured by Margulis). Joint work with Lam Pham.
  • 65th Annual Meeting of the Australian Mathematical Society

    Tue, 7th Dec 2021 — Fri, 10th Dec 2021
    , Online

    The 65th Annual Meeting of the Australian Mathematical Society will be hosted, online, by CARMA and the School of Information and Physical Sciences. Please visit the conference web page for more information.