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
Activities
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"Generalised Polygons and their Symmetries"
James Evans
Abstract:
Generalised polygons were first introduced by Jacque Tits in 1959, in the context of studying geometric realisations of the finite simple groups of Lie type. Thus, the study of their symmetry groups and symmetry properties is a rich area of research. My work has focused on studying the point-primitive quadrangles. In my talk I will describe a computer program for testing whether a particular group can act point-primitively on a generalised quadrangle and its application to analysing the almost simple sporadic groups. My work on this program motivated the discovery of a new result dubbed the Line Orbit Lemma, which in turn inspired the conjecturing of the Hemisystem Conjecture, both of which could prove very useful in the analysis of point-primitive quadrangles.