Representations of DAHAs and differential and difference equations

 DAHAs (Double Affine Hecke Algebras) is a generalisation of Hecke algebras, invented by Ivan Cherednik in the mid-90’s who used them rather spectacularly in the theory of Macdonald polynomials. Since then they have found many uses in different areas of mathematics. The representation theory of DAHAs has been intensively studied but is far from completion. This project will develop a link between representations of DAHAs and differential and difference equations of special types. The project will suit students interested in algebra, representation theory, algebraic geometry and their connections to mathematical physics.

<p>Hecke algebras are deformations of Weyl groups, with an important role in representation theory and Lie theory in particular.  DAHAs (Double Affine Hecke Algebras) is their generalisation invented by Ivan Cherednik in the mid-90’s who used them rather spectacularly in the theory of Macdonald polynomials. Since then they have found many uses in different areas of mathematics. </p> <p>The representation theory of DAHAs has been intensively studied but is far from completion. This project will develop a link between representations of DAHAs and differential and difference equations of special types. Our departure point is the picture that emerged in works [1,2,3] and which identifies different moduli spaces arising in representation theory and the theory of integrable nonlinear equations. The goal is to bring DAHAs and their representations into this picture and study new moduli spaces of differential and difference equations arising that way.</p> <p>The project is ideal for students interested in algebra, representation theory, algebraic geometry and their connections to mathematical physics.</p> <p>The successful applicant will benefit from vibrant research groups in algebra, geometry and integrable systems running several weekly seminars. </p> <p>[1] G. Wilson: Collisions of Calogero-Moser particles and an adelic Grassmannian (with an Appendix by I. G. Macdonald), Invent. Math. 133, 1--41 (1998)</p> <p>[2] Yu. Berest, G. Wilson: Automorphisms and ideals of the Weyl algebra, Math. Ann. 318(1), 127--147 (2000)</p> <p>[3] P. Etingof, V. Ginzburg: Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147, 243--348 (2002)</p>

<p>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>.</p> <p>Please state clearly in the Planned Course of Study section that you are applying for <em><strong>PhD Pure Mathematics FT</strong></em>, in the research information section that the research degree you wish to be considered for is <em><strong>Representations of DAHAs and differential and difference equation</strong></em> as well as <a href="https://eps.leeds.ac.uk/maths/staff/4012/dr-oleg-chalykh">Dr Oleg Chalykh</a> as your proposed supervisor and <em><strong>in the finance section, please state clearly the funding that you are applying for, if you are self-funding or externally sponsored.</strong></em></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 grading scheme</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul> <p><strong>If English is not your first language, you must provide evidence that you meet the University's minimum English language requirements (below).</strong></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.

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<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-2026">Leeds Opportunity Research Scholarship</a> <strong>(closing date: 1 April 2026)</strong> are available to UK applicants.  Alumni Bursary 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-2026">China Scholarship Council-University of Leeds Scholarship</a> is available to nationals of China <strong>(closing date: 7 January 2026)</strong>. The Leeds Marshall Scholarship is available to support US citizens. Alumni Bursary 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, Leeds Opportunity Research Scholarship, China Scholarship Council-University of Leeds Scholarship or Leeds Marshall 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>Important: Please note that that the award does not cover the costs associated with moving to the UK.  All such costs (visa, Immigration Health Surcharge, 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 UKCISA website for information regarding Fee Status for Non-UK Nationals.</p>

<p>For further information about this project, please contact Dr oleg Chalykh by email to <a href="mailto:EMAIL@leeds.ac.uk">o.chalykh@leeds.ac.uk</a> or by calling +44 (0)113 343 5112</p> <p>For further information about your application, please contact PGR Admissions by email to <a href="mailto:phd@engineering.leeds.ac.uk">phd@engineering.leeds.ac.uk</a></p>