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


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Key facts

Type of research degree
Application deadline
Ongoing deadline
Country eligibility
International (open to all nationalities, including the UK)
Competition funded
Professor H Dugald Macpherson
School of Mathematics
Research groups/institutes
Pure Mathematics
<h2 class="heading hide-accessible">Summary</h2>

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&rsquo;. 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&rsquo; in S if it is maximal subject to being a closed subgroup of S; we define notions such as `maximal locally compact&rsquo; similarly.<br /> <br /> 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 &ndash; 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&rsquo;. 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.<br /> <br /> The project builds on the paper [M. Bodirsky, H.D. Macpherson, `Reducts of structures and maximal-closed permutation groups&rsquo;, Journal of Symbolic Logic 81 (2016), 1087&mdash;1114] and on work in progress with Cheryl Praeger and Simon Smith.

<h2 class="heading">How to apply</h2>

<p>Formal applications for research degree study should be made online through the&nbsp;<a href="">University&#39;s website</a>. Please state clearly in the Planned Course of Study section that you are applying for <em><strong>PHD Pure Mathematics FT</strong></em> and in the research information section&nbsp;that the research degree you wish to be considered for is <em><strong>Maximal-closed permutation groups and reducts of first-order structures</strong></em>&nbsp;as well as&nbsp;<a href="">Professor H. Dugald Macpherson</a> as your proposed supervisor.</p> <p>If English is not your first language, you must provide evidence that you meet the University&#39;s minimum English language requirements (below).</p> <p>&nbsp;</p>

<h2 class="heading heading--sm">Entry requirements</h2>

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

<h2 class="heading heading--sm">English language requirements</h2>

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.

<h2 class="heading">Funding on offer</h2>

<p><strong>Self-Funded or externally sponsored students are welcome to apply.</strong></p> <p><strong>UK&nbsp;</strong>&ndash;&nbsp;The&nbsp;<a href="">Leeds Doctoral Scholarships</a>, <a href="">Akroyd &amp; Brown</a>, <a href="">Frank Parkinson</a> and <a href="">Boothman, Reynolds &amp; Smithells</a> Scholarships are available to UK applicants. &nbsp;<a href="">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Non-UK </strong>&ndash; The&nbsp;<a href="">China Scholarship Council - University of Leeds Scholarship</a>&nbsp;is available to nationals of China. The&nbsp;<a href="">Leeds Marshall Scholarship</a>&nbsp;is available to support US citizens.&nbsp; <a href="">Alumni Bursary</a> is available to graduates of the University of Leeds.</p>

<h2 class="heading">Contact details</h2>

<p>For further information regarding your application, please contact Doctoral College Admissions by&nbsp;email:&nbsp;<a href=""></a>, or by telephone: +44 (0)113 343 5057.</p> <p>For further information regarding the project, please contact Professor H Dugald MacPherson by email:&nbsp;&nbsp;<a href=""></a></p>

<h3 class="heading heading--sm">Linked research areas</h3>