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Groups definable in tame expansions of o-minimal structures


Key facts

Type of research degree
Application deadline
Ongoing deadline
Country eligibility
International (open to all nationalities, including the UK)
Competition funded
Source of funding
University of Leeds
Dr Pantelis Eleftheriou
School of Mathematics
Research groups/institutes
Logic, Pure Mathematics
<h2 class="heading hide-accessible">Summary</h2>

This project lies at the nexus of model theory (mathematical logic), group theory and combinatorics. The main objects of study are groups definable in various structures, which can be of topological/geometric nature, such as o-minimal structures and tame expansions of them, or more generally of combinatorial nature, such as structures with NIP (not the independence property). The NIP property is also of interest to statistics and machine learning. <br /> <br /> In the o-minimal setting, definable groups have been fairly understood with perhaps the most notable result being the solution of Pillay&rsquo;s conjecture, which draws an explicit connection between those groups and real Lie groups. Extensions of Pillay&rsquo;s conjecture in one of many possible expansions of o-minimal structures will be investigated in this project. Concrete examples include the expansion of the real field by a predicate for (a) the set of real algebraic numbers, (b) a dense independent set, (c) the set of all rational powers of 2, (d) the set of all integer powers of 2, or (e) any subgroup of the real multiplicative group with the Mann property. Those settings have recently seen the development of model-theoretic tools, which will be used in this project. <br /> <br /> In the more general, NIP setting, fewer tools are available, and concrete questions will involve first the development of an understanding of definable sets in NIP expansions of o-minimal structures, and then its application to the study of definable groups.<br /> <br /> The successful applicant will benefit from a large and exceptionally vibrant research group in mathematical logic, including 7 permanent academic staff with expertise in model theory, set theory, recursion theory, proof theory, categorical logic, and logic in computer science. The logic group consistently also includes several postdocs and PhD students, runs 4 regular seminar series and is a node of several regional and international research networks. The group is an active participant in the MAGIC consortium, which provides specialist lecture courses for mathematics postgraduates at a network of 20 UK Universities.<br />

<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,&nbsp;</strong></em>in the research information section&nbsp;that the research degree you wish to be considered for is&nbsp;<em><strong>Groups definable in tame expansions of o-minimal structures</strong></em> as well as&nbsp;<a href="">Dr Pantelis Eleftheriou</a> as your proposed supervisor&nbsp;and in the finance section, please state clearly&nbsp;<em><strong>the funding that you are applying for, if you are self-funding or externally sponsored</strong></em>.</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 style="margin-bottom:11px"><em>As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</em></p> <p class="MsoNoSpacing">Applications will be considered on an ongoing basis. &nbsp;Potential applicants are strongly encouraged to contact the supervisors for an informal discussion before making a formal application. We also advise that you apply at the earliest opportunity as the application and selection process may close early, should we receive a sufficient number of applications or that a suitable candidate is appointed.</p> <p>Please note that you must provide the following documents in support of your application by the closing date of 3 April 2024 for Leeds Opportunity Research Scholarship or 8 April 2024 for Leeds Doctoral Scholarship:</p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul>

<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 style="margin-bottom:12px"><strong>Self-Funded or externally sponsored students are welcome to apply.</strong></p> <p><strong>UK</strong>&nbsp;&ndash;&nbsp;The&nbsp;<a href="">Leeds Doctoral Scholarships</a>&nbsp;and&nbsp;<a href="">Leeds Opportunity Research Scholarship</a>&nbsp;(open from October 2023)&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 (now closed for 2024/25 entry).<span style="font-size:11.0pt"><span style="line-height:107%"><span style="font-family:&quot;Calibri&quot;,sans-serif"><span style="color:#4a4a4a"> </span></span></span></span>The&nbsp;<a href="">Leeds Marshall Scholarship</a>&nbsp;is available to support US citizens. <a href="">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Important:&nbsp;</strong> Any costs associated with your arrival at the University of Leeds to start your PhD including flights, immigration health surcharge/medical insurance and Visa costs are not covered under this studentship.</p> <p class="MsoNoSpacing">Please refer to the <a href="">UKCISA</a> website for information regarding Fee Status for Non-UK Nationals.</p>

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

<p>For general enquiries about applications, contact Doctoral College Admissions by email to&nbsp;<a href=""></a></p> <p>For questions about the research project, please contact Dr Pantelis Eleftheriou by email to&nbsp;<a href=""></a></p>

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