Combinatorial and higher-categorical techniques in low dimensional topology and topological quantum field theory

This PhD project addresses the construction of topological invariants in low dimensional geometric topology, arising from combinatorial representation theory, and (possibly) also their applications to physics, in particular to mathematical models of topological phases of matter and topological quantum computing.

<div class="x_elementToProof" style="text-align: left; margin: 0px; background-color: rgb(255, 255, 255) !important;">Our framework for understanding, constructing and computing topological invariants is to decompose topological objects (e.g. knots, manifolds, knotted surfaces, etc.) into smaller "generating" pieces, and obtain global invariants by combining all of its local values.  The latter local values cannot be arbitrarily given: typically, compatibility relations must be satisfied in order for the end-result of combining local values not to depend on the way objects were decomposed. Formulating topological invariants is here mainly inspected through determining how to map topological generators in order that appropriate relations hold between them: this is where both combinatorial representation theory  as well as higher category theory do a great job for us.</div> <div> </div> <div>Higher categories provide an effective framework for combining local values of invariants in order to obtain  globally defined quantities. This is because higher categories have morphisms of several different dimensions, 0, 1, 2, etc, and moreover  several different ways to combine those higher order morphisms, along different directions. This repertoire of different ways to compose higher order morphisms gives a way to book-keep the multitude of different ways chunks of topological objects can be combined along different directions, and possibly in different orders.</div> <div> </div> <div>This PhD project can pursue a number of different paths for finding new topological invariants. From applying homotopy-theoretical techniques, to following a differential geometric framework, or even a purely combinatorial/algebraic flavor. There may also be opportunities to explore applications to modelling topological phases of matter and to the ensuing paradigms for topological quantum computing.</div> <div style="text-align: left; margin: 0px; background-color: rgb(255, 255, 255) !important;"> </div>

<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 <em><strong>PHD Pure Mathematics FT </strong></em>and in the research information section that the research degree you wish to be considered for is <em><strong>Combinatorial and higher-categorical techniques in low dimensional topology and topological quantum field theory</strong></em> and in the bulk as well as Dr <a href="https://eps.leeds.ac.uk/maths/staff/4021/dr-joao-faria-martins">Joao Faria Martins</a>  (also known as <a href="https://eps.leeds.ac.uk/maths/staff/4021/dr-joao-faria-martins">Joao Goncalves Faria Martins</a>) as your proposed supervisor. Please state in the Finance section <em><strong>the funding source you are applying for, if you are self-funding or are externally sponsored (please include the name of your sponsor).</strong></em></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><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>Applications will be considered after the closing date.  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 class="MsoNoSpacing"><strong>Please note that you must provide the following documents in support of your application by the closing date of Monday 6 January 2025 if applying for the China Scholarship Council-University of Leeds Scholarship, Monday 3 February 2025 if applying for Leeds Doctoral Scholarship or Tuesday 1 April 2025 for Leeds Opportunity Research Scholarship.</strong></p> <p><strong>If you are applying for the School of Mathematics Scholarship 2025/26, or 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</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul>

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 style="margin-bottom:12px"><strong>Self-Funded or externally sponsored </strong>students are welcome to apply.</p> <p><strong>UK</strong> – The <a href="https://phd.leeds.ac.uk/funding/138-leeds-doctoral-scholarship-2025-faculty-of-engineering-and-physical-sciences#:~:text=Key%20facts&text=One%20Leeds%20Doctoral%20Scholarship%20is,rata%20for%20part%2Dtime%20study.">Leeds Doctoral Scholarship</a> (closing date: Monday 3 February 2025), <a href="https://phd.leeds.ac.uk/funding/234-leeds-opportunity-research-scholarship-2022">Leeds Opportunity Research Scholarship</a> <strong>(closing date: Tuesday 1 April 2025)</strong> and <a href="https://phd.leeds.ac.uk/funding/55-school-of-mathematics-scholarship-2025-26">School of Mathematics Scholarship 2025/26</a> <strong>(open from October 2024)</strong> are available to UK applicants.</p> <p><strong>Non-UK</strong> – <a href="https://phd.leeds.ac.uk/funding/55-school-of-mathematics-scholarship-2025-26">School of Mathematics Scholarship 2025/26</a> <strong>(open from October 2024) </strong>are available to all International applicants.  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> <strong>(closing date: Monday 6 January 2025)</strong> is available to nationals of China. 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>You will be responsible for paying the overtime fee in full in your writing up/overtime year (£320 in Session 2024/25), but the scholarship maintenance allowance will continue to be paid for up to 6 months in the final year of award.</p> <p><strong>Important: </strong>Please note that that the award does <strong><em>not</em></strong> cover the costs associated with moving to the UK.  All such costs (<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>Please refer to the <a href="https://www.ukcisa.org.uk/">UKCISA</a> website for information regarding Fee Status for Non-UK Nationals.</p>

<p>For further information about this project, please contact Dr <a href="http://www1.maths.leeds.ac.uk/~pmtjfa/index.html">João Faria Martins</a> by email to <a href="mailto:j.fariamartins@leeds.ac.uk">j.fariamartins@leeds.ac.uk</a> or by telephone to (+44) (0113) 343 5130.</p> <p>For further information about your application, please contact PGR Admisisons by email to <a href="mailto:maps.pgr.admissions@leeds.ac.uk">maps.pgr.admissions@leeds.ac.uk</a></p> <p> </p> <p> </p>