Difference algebraic groups

Difference algebraic groups can be described as subgroups of the general linear group defined by algebraic difference equations in the matrix entries. They occur naturally as Galois groups in certain Galois theories and have applications to arithmetic problems. The main goal of this project is to further develop the very rich and beautiful theory of these groups and to expand their areas of application.<br /> <br /> Algebraic groups can be described as subgroups of the general linear group defined by algebraic equations in the matrix entries. They play an important role in algebraic geometry and representation theory and have a well-developed structure theory. This project is at the crossroads of the theory of algebraic groups and difference algebra, i.e., the systematic study of difference equations from an algebraic point of view.<br /> <br /> A substantial body of results is already available for difference algebraic groups defined by ordinary difference equations. See e.g., [1]. A main theme of this project is to generalize these results to partial difference equations.<br /> <br /> This project is ideal for students with a good background and interest in algebraic geometry and commutative algebra.<br /> <br /> The successful applicant will benefit from large and vibrant research groups in algebra and logic running several weekly seminars.<br /> <br /> [1] Finiteness properties of affine difference algebraic groups, International Mathematics Research Notices 2022(1), 506-555, 2022

<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>2026/27</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 2026</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>Difference algebraic groups</strong> as well as <a href="https://eps.leeds.ac.uk/maths/staff/14204/dr-michael-wibmer">Dr Michael Wibmer</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 for University or School Scholarships for 2026/27 entry, 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> <p style="margin-bottom:11px">Formal applications for research degree study should be made online through the <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University's website</a>. Please state clearly in the Planned Course of Study section that you are applying for <strong>PhD Pure Mathematics FT </strong>and in the research information section that the research degree you wish to be considered for is <strong>Difference algebraic groups</strong> as well as <a href="https://eps.leeds.ac.uk/maths/staff/14204/dr-michael-wibmer">Dr Michael Wibmer</a> as your proposed supervisor. <strong>Please state in the Finance section the funding source you are applying for, if you are self-funding or are externally sponsored (please include the name of your sponsor).</strong></p> <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>Applications will be considered on an ongoing basis.  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><strong>If you are applying for University or School Scholarships for 2026/27 entry, 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><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><strong>Self-Funded or externally sponsored students are welcome to apply.</strong></p> <p><strong>Scholarship opportunities open from October 2025</strong></p> <p><strong>UK</strong> – The <a href="https://phd.leeds.ac.uk/funding/209-leeds-doctoral-scholarships-2024">Leeds Doctoral Scholarship</a> <strong>(closing date: 1 April 2026)</strong> and <a href="https://phd.leeds.ac.uk/funding/234-leeds-opportunity-research-scholarship-2022">Leeds Opportunity Research Scholarship</a> <strong>(closing date: 1 April 2026)</strong> are available to UK applicants.  <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Non-UK</strong> – The <a href="https://phd.leeds.ac.uk/funding/48-china-scholarship-council-university-of-leeds-scholarships-2021">China Scholarship Council - University of Leeds Scholarship</a> is available to nationals of China <strong>(Now closed for October 2026 entry)</strong>. The <a href="https://phd.leeds.ac.uk/funding/73-leeds-marshall-scholarship">Leeds Marshall Scholarship</a> is available to support US citizens. <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p>Please note that if you are successful in securing an academic offer for PhD study, this does not mean that you have been successful in securing an offer of funding.</p> <p>If you are applying for the Leeds Doctoral Scholarship or the Leeds Opportunity Research Scholarship, you will need to complete a separate application, specific to these scholarships, to be considered for funding.</p> <p>You will be responsible for paying the overtime fee in full in your writing up/overtime year (£340 in Session 2025/26), but the scholarship maintenance allowance will continue to be paid for up to 6 months in the final year of award.</p> <p><strong>Eligibility Criteria</strong></p> <ul> <li>Applications are open to either International or Home (UK) applicants.</li> </ul> <p>Please refer to the <a href="https://www.ukcisa.org.uk/">UKCISA</a> website or our <a href="https://www.leeds.ac.uk/undergraduate-fees/doc/fee-assessment">fee assessment page</a> for information regarding Fee Status for Non-UK Nationals.</p> <p><strong>Important: Please be aware that any expenses related to the relocation of international students to the UK (<a href="https://www.leeds.ac.uk/international-visas-immigration/doc/applying-student-visa">visa, Immigration Health Surcharge</a>, flights, etc) would be their responsibility and is not covered by this award</strong></p> <p><strong>Other Conditions</strong></p> <ul> <li>Candidates who have previously been awarded a PhD or are currently registered on a PhD are excluded from applying. Those who were previously studying for a PhD but did not complete may be considered. </li> <li>Awards must be taken up by 1st October 2026.</li> <li>Applicants must live within a reasonable distance of the University of Leeds whilst in receipt of this scholarship.</li> </ul>

<p>For questions about this research project, please contact Dr Michael Wibmer by email to <a href="mailto:m.wibmer@leeds.ac.uk">m.wibmer@leeds.ac.uk</a></p> <p>For general enquiries about applications, please contact PGR Admissions by email to <a href="mailto:phd@engineering.leeds.ac.uk">phd@engineering.leeds.ac.uk</a></p>