This meeting will focus on the generalisation of Burnside rings from finite to profinite groups and their applications in representation theory in particular. In finite group theory, Burnside rings and the functionality of mapping a group to its Burnside ring have led to very useful results, which have also been extended to fusion systems.

The timetable was:

• 13:30-14:30:Brita Nucinkis (Royal Holloway, University of London), An introduction to the Burnside ring as a Mackey functor
• 14:35-15:05:Zac Hall (Lancaster), An approach to the Burnside Ring of pro-fusion systems for Profinite groups
• 15:05-15:15:Break
• 15:15-15:45:Veronica Kelsey (St Andrews), Relational Complexity of Primitive Permutation Groups
• 15:50-16:50:Serge Bouc (Amiens); A functorial resolution of units of Burnside rings

The FCG Research Group is supported by an LMS Joint Research Groups in the UK Scheme 3 grant. Limited funding is available for PhD students, allocated on a first come first served basis.

For UK-based mathematicians with caring duties, the LMS has a Caring Supplementary Grant scheme which allows participants of meetings like ours to apply for help covering caring costs.

Abstracts

Brita Nucinkis (Royal Holloway, University of London), An introduction to the Burnside ring as a Mackey functor

I will give an introduction into the Burnside ring for finite groups and describe some of its properties as a Mackey functor. I will then give a survey of possible extensions for Burnside functors for profinite groups.

Zac Hall (Lancaster), An approach to the Burnside Ring of pro-fusion systems for Profinite groups

Bringing together Burnside rings for fusion systems as discussed by Reeh (2014), pro-fusion systems by Stancu and Symonds (2013) and finally the Burnside ring of a profinite group by Dress and Siebeneicher (1988) an approach will be discussed to extend these to cover the Burnside ring of a pro-fusion system. The connection between these structures seems to be about as 'nice' as you could hope for.

Veronica Kelsey (St Andrews), Relational Complexity of Primitive Permutation Groups

The general concept of relational complexity is a measure of when local symmetries imply the existence of global symmetries. The definition of relational complexity for permutation groups was introduced by Cherlin, Martin and Saracino in 1996. In this talk we begin with an introduction to the topic, and then discuss some recent work on bounding the relational complexity (and other group statistics) for primitive permutation groups.

Serge Bouc (Amiens), A functorial resolution of units of Burnside rings

Most of the structural properties - prime spectrum, species, idempotents, ... - of the Burnside ring of a finite group have been precisely described a few years after its introduction in 1967. An important missing item in this list is its group of units. After a - non-exhaustive - review of this subject, I will present some recent results on the functorial aspects of this group.