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Homological aspects of diagrammatic categories related to algebraic groups

PGR-P-1950

Key facts

Type of research degree
PhD
Application deadline
Friday 28 June 2024
Project start date
Tuesday 1 October 2024
Country eligibility
UK only
Funding
Competition funded
Source of funding
Doctoral training partnership
Supervisors
Professor Paul Martin and Dr Alison Parker
Additional supervisors
Dr Amit Hazi
Schools
School of Mathematics
Research groups/institutes
Algebra, geometry and integrable systems, Pure Mathematics
<h2 class="heading hide-accessible">Summary</h2>

This PhD project concerns diagrammatic methods in the representation theory of algebraic groups in positive characteristic. These methods are fuelled by a well-developed theory of diagrammatic categorification, as well as more recent advances connecting these categorifications to classical diagram algebras such as the Temperley-Lieb and (generalised) blob algebras.<br /> <br /> Central to this approach is the diagrammatic Hecke category, which lifts the combinatorics of (affine) Weyl groups to the level of a monoidal category. This has the advantage of making computations less characteristic-dependent and much more feasible (e.g. Williamson's celebrated counterexamples to Lusztig's conjecture). <br /> <br /> This project will investigate complexes in the diagrammatic Hecke category and related categories, such as Rouquier complexes. These complexes live in the so-called &quot;mixed perverse&quot; Hecke category, which can be thought of as Ringel dual to the usual Hecke category. The project's main aim is to make progress towards finding minimal Rouquier complexes, or equivalently, minimal tilting resolutions of Weyl modules for reductive groups, using p-Jones-Wenzl idempotents. The ultimate goal will be to use these complexes to discover diagrammatic analogues of classical constructions from the representation theory of reductive groups, such as the Frobenius twist and p-filtrations.<br />

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

<p>Formal applications for research degree study should be made online through the&nbsp;<a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">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</strong></em> and in the research information section&nbsp;that the research degree you wish to be considered for is <em><strong>Homological aspects of diagrammatic categories related to algebraic groups</strong></em>&nbsp;as well as&nbsp;<a href="https://eps.leeds.ac.uk/maths/staff/4070/dr-alison-parker">Dr Alison Parker</a>, <a href="https://eps.leeds.ac.uk/maths/staff/14560/dr-amit-hazi">Dr Amit Hazi</a>&nbsp;and <a href="https://eps.leeds.ac.uk/maths/staff/4061/professor-paul-martin">Professor Paul Martin</a> as your proposed supervisors. <em><strong>Please state in the Finance section that the funding source you are applying for is&nbsp;EPSRC Doctoral Training Partnership Studentship (Algebra)</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 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. &nbsp;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 28 June 2024:</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">A highly competitive EPSRC Doctoral Training Partnership Studentship (Algebra) offering the award of fees, together with a tax-free maintenance grant of &pound;19,237 per year for 3.5 years.&nbsp; Training and support will also be provided.</p> <p>This opportunity is open to UK applicants only.&nbsp; All candidates will be placed into the EPSRC Doctoral Training Partnership Studentship Competition and selection is based on academic merit.</p> <p>Please refer to the&nbsp;<a href="https://www.ukcisa.org.uk/">UKCISA</a>&nbsp;website for&nbsp;information regarding Fee Status for Non-UK Nationals.</p>

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

<p>For general enquiries about applications, please contact our Postgraduate Admissions Team (<a href="mailto:maps.pgr.admissions@leeds.ac.uk">maps.pgr.admissions@leeds.ac.uk</a>).<br /> For questions about the research project, please contact Dr Alison Parker (<a href="mailto:A.E.Parker@leeds.ac.uk">A.E.Parker@leeds.ac.uk</a>) or Dr Amit Hazi (<a href="mailto:A.Hazi@leeds.ac.uk">A.Hazi@leeds.ac.uk</a>).</p>


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