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


Key facts

Type of research degree
Application deadline
Ongoing deadline
Project start date
Tuesday 1 October 2024
Country eligibility
International (open to all nationalities, including the UK)
Competition funded
Source of funding
University of Leeds
Dr Joao Goncalves Faria Martins
Additional supervisors
Dr Cristina Anghel
School of Mathematics
Research groups/institutes
Pure Mathematics
<h2 class="heading hide-accessible">Summary</h2>

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.

<h2 class="heading hide-accessible">Full description</h2>

<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 &quot;generating&quot; pieces, and obtain global invariants by combining all of its local values.&nbsp; 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&nbsp; as well as higher category theory do a great job for us.</div> <div>&nbsp;</div> <div>Higher categories provide an effective framework for combining local values of invariants in order to obtain&nbsp; globally defined quantities. This is because higher categories have morphisms of several different dimensions, 0, 1, 2, etc, and moreover&nbsp; 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>&nbsp;</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;">&nbsp;</div>

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

<p style="margin-bottom:11px">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>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="">Joao Faria Martins</a>&nbsp;&nbsp;(also known as <a href="">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&#39;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. &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&nbsp;Leeds Opportunity Research Scholarship and 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 class="MsoNoSpacing"><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;<a href="">Leeds Opportunity Research Scholarship</a>&nbsp;and <a href="">School of Mathematics Scholarships</a>&nbsp;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 (now closed for 2024/25 entry). 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:</strong>&nbsp; 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<em><strong> not</strong></em> covered under this studentship.</p> <p>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 further information about this project, please contact Dr <a href="">Jo&atilde;o Faria Martins</a>&nbsp;by email to&nbsp;<a href=""></a>&nbsp;or by telephone to&nbsp;(+44) (0113) 343 5130.</p> <p>For further information about your application, please contact Doctoral College Admisisons by email to&nbsp;<a href=""></a></p> <p>&nbsp;</p> <p>&nbsp;</p>

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