Maximal-closed permutation groups and reducts of first-order structures


Contact Professor H. Dugald Macpherson to disucss this project further informally.

Project description

This project is on the boundary between model theory (mathematical logic) and infinite permutation group theory. Let S be the group of all permutations of a countably infinite set X. Then S is a (metrisable) topological group with respect to a natural topology, the `topology of pointwise convergence’. A subgroup of S is closed in this topology if and only is it is the automorphism group of a first order structure with universe X, is compact if and only if it is closed and has all orbits finite, and is locally compact if and only if the pointwise stabiliser of some finite set is compact. A subgroup G of S is maximal-closed’ in S if it is maximal subject to being a closed subgroup of S; we define notions such as `maximal locally compact’ similarly.

The goal of the project is to develop a theory of maximal-closed and maximal locally compact subgroups of S (or of other groups). It is of great interest simply to find examples – typically, one can only prove a group is maximal-closed through access to a fine structure theory coming from model theory or permutation group theory. There is a close connection between the automorphism group of a structure on X being maximal-closed, and the structure having no proper non-trivial `reducts’. There are many natural examples to explore, and interesting questions on key examples are likely to arise, for example related to totally disconnected locally compact groups.

The project builds on the paper [M. Bodirsky, H.D. Macpherson, `Reducts of structures and maximal-closed permutation groups’, Journal of Symbolic Logic 81 (2016), 1087—1114] and on work in progress with Cheryl Praeger and Simon Smith.

Entry requirements

Applications are invited from candidates with or expecting a minimum of a UK upper second class honours degree or equivalent in Mathematics or a related subject.

If English is not your first language, you must provide evidence that you meet the University’s minimum English Language requirements.

How to apply

Formal applications for research degree study should be made online through the university's website. Please state clearly in the research information section that the PhD you wish to be considered for is 'Maximal-closed permutation groups and reducts of first-order structures' as well as Professor H. Dugald Macpherson as your proposed supervisor.

We welcome scholarship applications from all suitably-qualified candidates, but UK black and minority ethnic (BME) researchers are currently under-represented in our Postgraduate Research community, and we would therefore particularly encourage applications from UK BME candidates.  All scholarships will be awarded on the basis of merit.

If you require any further information please contact the Graduate School Office, e: