Random Permutations and Integer Partitions with Structural Constraints

Permutations and integer partitions are the basic combinatorial structures that appear in numerous areas of mathematics and its applications. Modern statistical approach is to treat such structures as a random ensemble endowed with a suitable probability measure. Structures with certain constraints on their components are mathematically challenging. The main thrust of this PhD project is to study properties of big structures, focusing on macroscopic features such as limit shape.

<p>Permutations and integer partitions appear in numerous areas of mathematics and its applications — from number theory, algebra and topology to quantum physics, statistics, population genetics, IT & cryptology (e.g., Alan Turing used the theory of permutations to break the Enigma code during World War II). This classic research topic dates back to Euler, Cauchy, Cayley, Lagrange, Hardy and Ramanujan. The modern statistical approach is to treat such combinatorial structures as a random ensemble endowed with a suitable probability measure. The uniform (equiprobable) case is well understood but more interesting models (e.g., with certain weights on the components) are mathematically more challenging.</p> <p>The main thrust of this PhD project is to tackle open and emerging problems about asymptotic properties of "typical" structures of big size, especially under certain structural constraints on the consitituent components. The focus will be on macroscopic features of the random structure, such as its limit shape.It is also important to study extreme values, in particular the possible emergence of a giant component which may shed light on the Bose–Einstein condensation of quantum gas, predicted in 1924 but observed only recently (Nobel Prize in Physics 2001).</p> <p>A related direction of research is the exploration of a deep connection with different quantum statistics; specifically, the ensemble of uniform integer partitions may be interpreted as the ideal gas of bosons (in two dimensions), whereas partitions with distinct parts correspond to fermions. In this context, an intriguing problem is to construct suitable partition classes to model the so-called anyons obeying fractional quantum statistics (also in 2D!). Furthermore, an adventurous idea may be to look for suitable partition models to mimic the unusual properties of graphene (Nobel Prize in Physics 2010), a newly discovered 2D quantum structure with certain hidden symmetries.</p> <p>Successful candidates should have an excellent degree in mathematics and/or statistics, with a good background and research interests in one or more of the following areas: probability; combinatorics; mathematical statistics; analysis; physics.</p> <p>You will be based within a strong research group in <a href="https://eps.leeds.ac.uk/maths-statistics/doc/probability-financial-mathematics">Probability and Financial Mathematics</a>. Our research focuses on the study and modelling of systems and processes featured by uncertainty and/or complexity, using advanced theoretical, simulation and numerical methods. It covers a vast variety of modern topics both in probability (including theory of random processes and stochastic analysis) and in a wide range of applications in mathematical and other sciences, spanning from nonlinear dynamical systems and mathematical physics through mathematical biology and complexity theory to mathematical finance and economics.</p> <p><strong>References</strong></p> <ol> <li>Arratia, R., Barbour, A.D. and Tavaré, S. <em>Logarithmic Combinatorial Structures: a Probabilistic Approach</em>. European Math. Soc., Zürich, 2003. (<a href="https://doi.org/10.4171/000">doi:10.4171/000</a>)</li> <li>Bogachev, L.V. Unified derivation of the limit shape for multiplicative ensembles of random integer partitions with equiweighted parts. <em>Random Structures and Algorithms</em>, <strong>47</strong> (2015), 227–266. (<a href="https://doi.org/10.1002/rsa.20540">doi:10.1002/rsa.20540</a>)</li> <li>Bogachev, L.V. Limit shape of random convex polygonal lines: Even more universality. <em>Journal of Combinatorial Theory A</em>, <strong>127</strong> (2014), 353–399. (<a href="doi.org/10.1016/j.jcta.2014.07.005">doi:10.1016/j.jcta.2014.07.005</a>)</li> <li>Bogachev, L.V. and Yakubovich, Yu.V. Limit shape of minimal difference partitions and fractional statistics.<em> Communications in Mathematical Physics</em>, <strong>373</strong> (2020),1085–1131. (<a href="https://doi.org/10.1007/s00220-019-03513-5">doi:10.1007/s00220-019-03513-5</a>)</li> <li>Bogachev, L.V. and Zeindler, D. Asymptotic statistics of cycles in surrogate-spatial permutations. <em>Communications in Mathematical Physics</em>, <strong>334</strong> (2015), 39–116. (<a href="http://doi.org/10.1007/s00220-014-2110-1">doi:10.1007/s00220-014-2110-1</a>)</li> <li>Lerda, A. <em>Anyons: Quantum Mechanics of Particles with Fractional Statistics</em>. Springer, Berlin, 1992.</li> <li>Vershik, A.M. Asymptotic combinatorics and algebraic analysis. In: <em>Proceedings of the International Congress of Mathematicians (August 3–11, 1994, Zürich, Switzerland)</em>, Vol. 2. Birkhäuser, Basel, 1995, pp. 1384–1394. (<a href="https://doi.org/10.1007/978-3-0348-9078-6_133"><span id="doi-url">doi:10.1007/978-3-0348-9078-6_133</span></a>)</li> </ol>

<p>To apply for this project you will need to make a formal application for research degree study through the <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">University website</a>. You will need to create a login ID with a username and PIN. </p> <p>•    For <strong>Application type</strong> please select <strong>Research Degrees – Research Postgraduate</strong>. <br /> •    The admission year for this project is <strong>2027/28</strong> Academic Year. <br /> •    You will need to select your <strong>Planned Course of Study</strong> from a drop-down menu. For this project, scroll down and select <strong>PHP Statistics Full Time.</strong> <br /> •    The project start date for this project is <strong>1 October 2027</strong>, please use this as your <strong>Proposed Start Date of Research</strong>. <br /> •    Please state clearly in the<strong> Research Information Section</strong> that the research degree you wish to be considered for is <strong>Random Permutations and Integer Partitions with Structural Constraints</strong><em><strong> </strong></em>as well as <a href="https://eps.leeds.ac.uk/maths/staff/4008/professor-leonid-bogachev">Professor Leonid Bogachev</a> as your proposed supervisor.</p> <p><strong>Please state clearly in the Finance section</strong>, <strong>the funding that you are applying for, if you are self-funding or externally sponsored</strong>.</p> <p>More information on how to apply is available on our website <a href="https://www.leeds.ac.uk/research-applying/doc/applying-research-degrees">here</a>. You will be required to provide a personal statement which outlines your interest in the project you are applying for, why you have chosen it and how your skills map onto the requirements of the project.</p> <p>We will assess applications continuously as we receive them. We welcome and strongly encourage any potential applicants to contact the supervisor(s) for an informal discussion, prior to applying, and recommend submitting your application early.</p> <p><strong>If you are applying 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 the grading scheme</li> <li>Personal Statement outlining your interest in the project</li> <li>CV</li> </ul> <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><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 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.

<p>This is a non funded PhD project. Applicants are expected to be self funded or to secure external funding.</p> <p>You can find out more about our Funding Opportunities on our <a href="https://phd.leeds.ac.uk/">Postgraduate Research Opportunities</a> website.</p> <p><strong>Important: </strong>Please note that all costs associated with your arrival at Leeds (<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>For further information about your application, including how to apply, please contact PGR Admissions by email to <a href="mailto:epspgradmissions@leeds.ac.uk">epspgradmissions@leeds.ac.uk</a></p> <p>For further information about this project, please contact Professor Leonid Bogachev by email to <a href="mailto:L.V.Bogachev@leeds.ac.uk">L.V.Bogachev@leeds.ac.uk</a></p>