Skip to main content

Elliptic Discrete Integrable Systems

PGR-P-196

Key facts

Type of research degree
PhD
Application deadline
Ongoing deadline
Country eligibility
International (open to all nationalities, including the UK)
Funding
Competition funded
Supervisors
Professor Frank Nijhoff
Schools
School of Mathematics
Research groups/institutes
Applied Mathematics
<h2 class="heading hide-accessible">Summary</h2>

The term &amp;quot;Integrable Systems&amp;quot; refers to a wide class of very special models, described by nonlinear differential (in the &amp;quot;continuous&amp;quot; case) or difference (in the &amp;quot;discrete&amp;quot; case) equations possessing a number of remarkable properties. One of the outstanding features is that these equations are exactly solvable in the sense that, rather than having to rely on numerical techniques or approximations, these equations allow for exact (albeit highly nontrivial) methods for their solution. Examples of these are the well-known soliton solutions of certain partial differential equations in this class. In the discrete case, the theory behind these model difference equations has been steadily developing, mostly from the early 1990s onward, together with the mathematical theories which had to be developed alongside (as they were largely non-existent in the discrete case). <br /> <br /> The project focuses on &amp;quot;elliptic&amp;quot; integrable systems, which are those cases where coefficients and generating quantities for these equations are given in terms of elliptic functions. The latter generalizations of trigonometric functions have a rich mathematical structure, some aspects of which are still being explored (e.g. they play a role in Fermat&#039;s last theorem), and the integrable models which are defined through them are in a sense &amp;quot;at the top of the food chain&amp;quot; of models: they form the richest and most general class of equations. To a large extent the solution structure of those elliptic models has yet to be unraveled, and that will form the core of the project. In the project the student will investigate specific examples of such elliptic discrete integrable systems, which will entail not only to try and apply some well-tested techniques to these more complex cases for inding explicit solutions, but also to develop some methods for generating novel examples of such systems. As motivation, these models are expected to have relevance not only for creating novel mathematics, but also potentially for finding new models of fundamental physics. <br /> <br /> The project is embedded in the activities of a wider research group in Integrable Systems within the School of Mathematics, comprising several permanent staff, postdocs and postgraduate students. The group runs its own weekly seminar, and intertains close connections with other research groups in the School, e.g. in Algebra, Geometry and Analysis, as well as with the Quantum Information group in Physics.

<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/info/130206/applying/91/applying_for_research_degrees">University&#39;s website</a>. Please state clearly in the research information section&nbsp;that the research degree you wish to be considered for is &lsquo;Elliptic Discrete Integrable Systems&rsquo; as well as&nbsp;<a href="https://physicalsciences.leeds.ac.uk/staff/68/professor-frank-nijhoff">Prof Frank Nijhoff</a> as your proposed supervisor.</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>We welcome applications from all suitably-qualified candidates, but UK black and minority ethnic (BME) researchers are currently under-represented in our Postgraduate Research community, and we would therefore particularly encourage applications from UK BME candidates. All scholarships will be awarded on the basis of merit.</em></p>

<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><strong>Self-Funding Students are welcome to apply.</strong></p> <p><strong>UK&nbsp;students</strong>&nbsp;&ndash;&nbsp;The&nbsp;<a href="https://phd.leeds.ac.uk/funding/138-leeds-doctoral-scholarships-2021-january-deadline">Leeds Doctoral Scholarship (January deadline)</a>&nbsp;and the&nbsp;<a href="https://phd.leeds.ac.uk/funding/142-william-wright-smith-scholarship-2021">William Wright Smith Scholarship&nbsp;</a>are available to UK applicants. &nbsp;<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 students</strong>&nbsp;&ndash; The&nbsp;<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>&nbsp;is available to nationals of China. The&nbsp;<a href="https://phd.leeds.ac.uk/funding/73-leeds-marshall-scholarship">Leeds Marshall Scholarship</a>&nbsp;is available to support US citizens.&nbsp; <a href="https://phd.leeds.ac.uk/funding/60-alumni-bursary">Alumni Bursary</a> is available to graduates of the University of Leeds.</p>

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

<p>For further information please contact Doctoral College Admissions by&nbsp;email:&nbsp;<a href="mailto:EMAIL@leeds.ac.uk">m</a><a href="mailto:maps.pgr.admissions@leeds.ac.uk">aps.pgr.admissions@leeds.ac.uk</a>, or by telephone: +44 (0)113 343 5057</p>


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