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Equivariant homotopy and fusion systems


Project Description

The p-local structure of a finite group G is given by the set of its p-subgroups and the information about how they are conjugated to each other. This gives rise to the structure of a "fusion system". Many p-local invariants of groups, for example mod-p cohomology, are governed by their fusion systems. In fact, the fusion system of G completely determine the homotopy type of the p-completion of its classifying space.

Abstract fusion systems, not necessarily ones that arise from groups, show up in other contexts in mathematics. In light of the results just mentioned, it is natural to associate to them "intrinsic" classifying spaces. This has been done by Broto, Levi and Oliver who coined the term p-local finite groups for a fusion system together with a suitably defined classifying space. In fact, the classifying space is merely the p-completion of the geometric realisation of some small category called a linking system associated to the fusion system.

It has been shown recently by Chermak that the linking system of a p-local finite group affords the structure of a partial group which, as its name suggests, has close resemblance to a group.

Equivariant topology is a powerful tool in studying classifying spaces of groups. The aim of this project is to exploit the structure of partial groups to develop a parallel theory of equivariant Bredon cohomology theories for p-local finite groups. Apart from being interesting in their own right, these will be key tools in obtaining structural, as well as computational, results on the mod-p cohomology of the classifying spaces of p-local finite groups. We plan to use these tools to settle open problems in the subject, among them Linckelmann’s gluing problem which has strong connections with Alperin’s conjecture, which is one of the longest standing problem in representation theory.

Candidates should have (or expect to achieve) a UK honours degree at 1st class (or equivalent) in Mathematics.

It is essential that the successful applicant has knowledge of Group theory, algebra, topology (undergraduate level or higher)

APPLICATION PROCEDURE:

• Apply for Degree of Doctor of Philosophy in Mathematics
• State name of the lead supervisor as the Name of Proposed Supervisor
• State ‘Self-funded’ as Intended Source of Funding
• State the exact project title on the application form

When applying please ensure all required documents are attached:

• All degree certificates and transcripts (Undergraduate AND Postgraduate MSc-officially translated into English where necessary)
• Detailed CV
• Details of 2 academic referees

Informal inquiries can be made to Dr Assaf Libman () with a copy of your curriculum vitae and cover letter. All general enquiries should be directed to the Postgraduate Research School ()

Funding Notes

This project is advertised in relation to the research areas of the discipline of Mathematics. The successful applicant will be expected to provide the funding for Tuition fees, living expenses and maintenance. Details of the cost of study can be found by visiting View Website. THERE IS NO FUNDING ATTACHED TO THESE PROJECTS.

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