Groups definable in tame expansions of o-minimal structures

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’s conjecture, which draws an explicit connection between those groups and real Lie groups. Extensions of Pillay’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 />

<p>To apply for this project you will need to make a formal application for research degree study through the <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University website</a>. You will need to create a login ID with a username and PIN. </p> <p>•    For <strong>Application type</strong> please select <strong>Research Degrees – Research Postgraduate</strong>. <br /> •    The admission year for this project is <strong>2027/28</strong> Academic Year. <br /> •    You will need to select your <strong>Planned Course of Study</strong> from a drop-down menu. For this project, scroll down and select <strong>PHP Pure Mathematics Full Time.</strong> <br /> •    The project start date for this project is <strong>1 October 2027</strong>, please use this as your <strong>Proposed Start Date of Research</strong>. <br /> •    Please state clearly in the<strong> Research Information Section</strong> that the research degree you wish to be considered for is <strong>Groups definable in tame expansions of o-minimal structures</strong> as well as <a href="https://www1.maths.leeds.ac.uk/~pmtpe/">Dr Pantelis Eleftheriou</a> as your proposed supervisor.</p> <p><strong>Please state clearly in the Finance section</strong>, <strong>the funding that you are applying for, if you are self-funding or externally sponsored</strong>.</p> <p>More information on how to apply is available on our website <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">here</a>. You will be required to provide a personal statement which outlines your interest in the project you are applying for, why you have chosen it and how your skills map onto the requirements of the project.</p> <p>We will assess applications continuously as we receive them. We welcome and strongly encourage any potential applicants to contact the supervisor(s) for an informal discussion, prior to applying, and recommend submitting your application early.</p> <p><strong>If you are applying with external sponsorship or you are funding your own study, please ensure you provide your supporting documents at the point you submit your application:</strong></p> <ul> <li>Full Transcripts of all degree study or if in final year of study, full transcripts to date including the grading scheme</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul> <p>If English is not your first language, you must provide evidence that you meet the University's minimum English language requirements (below).</p> <p><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>

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.

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.

<p>This is a non funded PhD project. Applicants are expected to be self funded or to secure external funding.</p> <p>You can find out more about our Funding Opportunities on our <a href="https://phd.leeds.ac.uk/">Postgraduate Research Opportunities</a> website.</p> <p><strong>Important: </strong>Please note that all costs associated with your arrival at Leeds (<a href="https://www.leeds.ac.uk/international-visas-immigration/doc/applying-student-visa">visa, Immigration Health Surcharge</a>, flights etc) would have to be met by yourself, or you will need to find an alternative funding source.</p>

<p>For questions related to the application process and eligibility, please contact PGR Admissions by email to <a href="mailto:epspgradmissions@leeds.ac.uk">epspgradmissions@leeds.ac.uk</a></p> <p>For questions about this research project, please contact Dr Pantelis Eleftheriou by email to <a href="mailto:p.eleftheriou@leeds.ac.uk">p.eleftheriou@leeds.ac.uk</a></p>