Combinatorial identities from interacting particle systems

This PhD project will investigate the connections between interacting particle systems and combinatorial identities. Recently, many old and new identities related to integer partitions, modular forms, and representation theory have been (re)proven using relationships between important interacting particle systems. Examples of this include the Jacobi triple product identity, Euler's identity, and the q-binomial theorem, as well as some interesting and surprising new identities. There is still a lot of work to be done to understand these identities, especially in the case of new identities where the corresponding relationships between modular forms and representation theory is mostly unknown. There are also many natural avenues towards additional identities. This work will help to elucidate links between many seemingly disparate areas of mathematics.

<p style="text-align:start; margin-bottom:16px">Many important and widely studied spin or particle systems in probability and statistical mechanics can be related to each other through interesting mappings. Being able to equate interesting quantities for one type of system with quantities for another system may allow a whole new range of tools and techniques to be applied. This has led to many great successes and is a popular approach. What is perhaps less well known is that these equalities may tell us something interesting about a seemingly unrelated area of Mathematics.<br />  <br /> We have recently seen that studying quite natural distributions for well known particle systems such as the Asymmetric Simple Exclusion Process (ASEP) and the Asymmetric Zero Range Process (AZRP) can lead to new proofs of classical identities from combinatorics and number theory, such as the Jacobi triple product identity, and to entirely new and non-obvious identities. Interestingly, some important identities such as Rojers-Ramanujan do not yet have a corresponding proof coming from probability/particle systems.<br />  <br /> Very recent work has shown that surprising identities for particular types of integer partitions, known as (Generalised) Frobenius Partitions ((G)FP) can be obtained from considering the famous Ising model in one dimension. A key property of this Ising model is that the interaction and external field are inhomogeneous (i.e. they look different depending on the position). This inhomogeneity is so that the measure concentrates on blocking configurations, that is spin configurations that are all up sufficiently far to the right (i.e. i>>0) and all down sufficiently far to the left (i.e. i<<0). Using previously known identities it was also possible to give a surprising product for the partition function of this Ising model.</p> <p>As a student working on this model there are several interesting and challenging future directions of investigation:</p> <ul type="disc"> <li>Extend the work mentioned above to multiple classes of spin (more than just up or down) as has been done for the case of ASEP. Spins will be 1st, 2nd, 3rd,… class, with spins of a higher class having elevated status. You can then attempt to derive combinatorial identities.</li> <li>Extend the work to more than two types of spin, each type having equivalent status. This gives the well known Potts model. It would be very nice to compare identities obtained in this case to indentities for the Ising case. It may also be possible to find a product form for the partition function.</li> <li>Consider an Ising model with finitely many disagreements. In this case there are four possibilities for what value the spins take far to the left and right (each up or down). The corresponding identities will include (G)FPs with restricted total size, and with restricted maximum size of parts. Such partitions are important areas of research. A product form for the partition function in this case would be very nice.</li> <li>For the algebra/number theory minded students, there are connections between the identities mentioned above (such as the Jacobi triple product identity) and identities between modular forms and representations of Lie algebras. Understanding these identities in the cases mentioned above would add an important missing part of the picture. It may also lead to new recipes for generating combinatorial identities from particle systems.</li> <li>An ambitious project is to consider mappings between different quantum spin systems (such as the Jordan-Wigner transform) and try to obtain identities for this case. It is likely that the corresponding identities between representations of Lie algebras will be slightly easier in this case.</li> </ul> <p><strong>References</strong></p> <ol> <li>B. Lees and J. Jay <a data-clk="hl=en&sa=T&ei=GhS-ZZOyPK6Ly9YP2rWz8Ac&authuser=1" href="https://arxiv.org/abs/2401.16311">Combinatorial identities from an inhomogeneous Ising chain</a> (arXiv preprint)</li> <li>M. Balázs and R. Bowen:<a href="https://link.springer.com/article/10.1007/s40687-022-00342-2"> Product blocking measures and a particle system proof of the Jacobi triple product.</a> Ann. Inst. H. Poincaré Probab. Stat. <strong>54</strong>(1), 514–528 (2018)</li> <li>D. Adams, M. Balázs, and J. Jay <a data-clk="hl=en&sa=T&ei=wRS-ZZ_3OdfKy9YPvvmrqAM&authuser=1" href="https://arxiv.org/abs/2305.16769">ASEP proofs of some partition identities and the blocking stationary behaviour of second class particles</a> (arXiv preprint).</li> <li>M. Balázs, D. Fretwell, and J. Jay <a data-clk="hl=en&sa=T&ei=6hS-ZbvME_WAy9YP64aqmAE&authuser=1" href="https://link.springer.com/article/10.1007/s40687-022-00342-2">Interacting particle systems and Jacobi style identities</a>. Research in the mathematical sciences Volume 9, article number 48, (2022)</li> </ol>

<p style="text-align:start">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 Statistics FT</strong></em> and in the research information section that the research degree you wish to be considered for is <em><strong>Combinatorial identities from interacting particle systems</strong></em> as well as <a href="https://eps.leeds.ac.uk/faculty-engineering-physical-sciences/staff/13180/dr-ben-lees">Dr. Benjamin Lees</a> as your proposed supervisor and in the finance section, please state clearly <em><strong>the funding that you are applying for, if you are self-funding or externally sponsored (including 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>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.</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 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 students are welcome to apply.</strong></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> <strong>(closing date: Monday 3 February 2025)</strong>, <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 <em><strong>not</strong></em> 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 Benjamin Lees by email to <a href="mailto:b.t.lees@leeds.ac.uk">b.t.lees@leeds.ac.uk</a></p> <p>For further information about your application, please contact PGR Admissions by email to <a href="mailto:maps.pgr.admissions@leeds.ac.uk">maps.pgr.admissions@leeds.ac.uk</a></p>