Funded PhD Studentship in Mathematical Logic

Mathematical logic is a young subject which has developed over the last 30 years into an amalgam of fast-moving disciplines. These are linked by profound common concerns, around definability, decidability, effectiveness and computability, the nature of the continuum, and foundations. Some branches are highly multidisciplinary and force the researcher to be conversant with other fields (e.g. algebra, computer science).<br /> <br /> The Logic Group at the University of Leeds is one of the largest and most active worldwide, with a long uninterrupted tradition dating back to 1951, when its founder Martin Löb moved to Leeds. It has an international reputation for research in most of the main areas of mathematical logic - computability theory, model theory, set theory, proof theory, and in applications to algebra, analysis, combinatorics, topology, number theory and theoretical computer science. We host a very large and lively group of postgraduate researchers and post-doctoral fellows. The group runs weekly a logic seminar and 2-3 other more specialised seminars and a `Postgraduate Logic Seminar’, with other informal reading groups and postgraduate courses often arranged. There are close connections to the group in Algebra. <br />

<p class="MsoNoSpacing">We are delighted to offer a fully funded PhD project and applications are invited from strongly motivated and academically excellent candidates for fully funded PhD study in Mathematical Logic, within these strategic priority Research areas:</p> <p><strong>Computability Theory and Proof Theory: <a href="https://eps.leeds.ac.uk/maths/staff/4078/dr-paul-shafer">Dr Paul Shafer</a>, <a href="https://eps.leeds.ac.uk/maths/staff/4073/professor-michael-rathjen">Professor Michael Rathjen</a></strong>. Classical computability theory studies which numerical and symbolic functions can and cannot be defined by algorithms that compute their values. More generally, computability theory concerns hierarchies of relative definability among sets and functions. The key questions are: What sets of natural numbers are definable from a given set? and How complicated must these definitions be? This analysis is closely related to the strength of axiomatic theories because proving the existence of complicated sets requires the use of strong axioms. Leeds research in proof theory also emphasises reverse mathematics, along with intuitionism and constructive mathematics, non-classical set theory, cut elimination for infinitary proof systems, ordinal analysis, philosophy of mathematics. Possible PhD research topics include computational reducibility notions and the arising degree structures, reverse mathematics, and computable structure theory. Reducibility notions organize sets and functions by computational strength. Reverse mathematics combines computability-theoretic and proof-theoretic ideas in order to determine precisely which axioms are required to prove a given theorem. Computable structure theory studies the computational aspects of countable or separable mathematical structures. Please contact Dr Paul Shafer by email to <strong><a href="mailto:p.e.shafer@leeds.ac.uk">p.e.shafer@leeds.ac.uk</a> </strong>or Professor Michael Rathjen by email to <strong><a href="mailto:m.rathjen@leeds.ac.uk">m.rathjen@leeds.ac.uk</a></strong>.</p> <p><strong>Model Theory: <a href="https://eps.leeds.ac.uk/maths/staff/9082/dr-pantelis-eleftheriou">Dr Pantelis Eleftheriou</a>, <a href="https://eps.leeds.ac.uk/maths/staff/4056/professor-dugald-macpherson">Professor Dugald MacPherson</a>, <a href="https://eps.leeds.ac.uk/maths/staff/4058/dr-vincenzo-l-mantova">Dr Vincenzo Mantova</a>, <a href="https://eps.leeds.ac.uk/maths/staff/14204/dr-michael-wibmer">Dr Michael Wibmer</a></strong>. Model theory concerns objects in mathematics (graphs, partial orders, groups, rings, fields, metric spaces, etc.) viewed as structures in first-order logic (or sometimes other logical languages). A central concept is that of a first order theory – the collection of all first-order sentences true of a particular structure. One key theme is the identification of dividing lines between `tame’ and `wild’ theories: wild theories might be those where Gödel incompleteness phenomena arise, or where certain combinatorial configurations can be embedded in structures. Tame structures might be those where there is a good theory of independence and dimension generalising those in vector spaces, or where `definable sets’ (solutions sets of first order formulas) have good combinatorial and geometric properties. Often model theorists work with very concrete mathematical structures (such as the real field equipped with an exponential function), and the subject relates closely to other branches of mathematics, such as algebra, geometry, number theory, and combinatorics. Likely PhD themes concern connections of model theory to algebra (e.g. group theory), combinatorics, number theory and differential Galois theory, and in particular, aspects of o-minimality. Please contact Dr Pantelis Eleftheriou by email to <strong><a href="mailto:p.eleftheriou@leeds.ac.uk">p.eleftheriou@leeds.ac.uk</a></strong>, Professor Dugald MacPherson by email to <strong><a href="mailto:h.d.macpherson@leeds.ac.uk">h.d.macpherson@leeds.ac.uk</a></strong>, Dr Vincenzo Mantova by email to <strong><a href="mailto:v.l.mantova@leeds.ac.uk">v.l.mantova@leeds.ac.uk</a></strong> or Dr Michael Wibmer by email to <strong><a href="mailto:m.wibmer@leeds.ac.uk">m.wibmer@leeds.ac.uk</a></strong>.</p> <p><strong>Set Theory: <a href="https://eps.leeds.ac.uk/maths/staff/4010/dr-andrew-brooke-taylor">Dr Andrew Brooke-Taylor</a>, <a href="https://eps.leeds.ac.uk/maths/staff/10607/dr-asaf-karagila">Dr Asaf Karagila</a></strong>. The key and central theme is studying set theoretic foundations of mathematics, specifically the Zermelo-Fraenkel set theory axioms and large cardinal axioms, as well as fragments of the Axiom of Choice, and models of these systems. Other major topics of research in set theory include infinite combinatorics, and regularity properties for definable sets of reals and related structures. Particular themes in Leeds, with potential for PhD supervision, include: applications of set theory to category theory, algebraic topology, and other related areas; large cardinal axioms and forcing; the Axiom of Choice, both within set theory (as it interacts with the theory of forcing, as well as large cardinal axioms, and more general combinatorial concepts) and outside of set theory (in algebra, functional analysis, and more); combinatorial principles and their consequences. Please contact Dr Andrew Brooke-Taylor by email to <strong><a href="mailto:A.D.Brooke-Taylor@leeds.ac.uk">A.D.Brooke-Taylor@leeds.ac.uk</a></strong> or Dr Asaf Karagila by email to <strong><a href="mailto:a.karagila@leeds.ac.uk">a.karagila@leeds.ac.uk</a></strong>.</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>. Please state clearly in the Planned Course of Study section that you are applying for <em><strong>EPSRC DTP Engineering & Physical Sciences</strong></em> and in the research information section that the research degree you wish to be considered for is <em><strong>Funded PhD Studentship in Mathematical Logic.</strong></em> Please include the name of your preferred supervisor as listed.  Please state clearly in the Finance section that the funding source you are applying for is <em><strong>EPSRC Doctoral Landscape Award 2025/26: Pure Mathematics.</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><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>Applications will be considered after the closing date.  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>Please note that you must provide the following documents in support of your application by the closing date of Friday 28 February 2025:</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 class="MsoNoSpacing" style="text-align:start; margin-bottom:24px">A highly competitive EPSRC Doctoral Landscape Award providing full academic fees, together with a tax-free maintenance grant at the standard UKRI rate (£19,237 in academic session 2024/25) for 3.5 years.  Training and support will also be provided.</p> <p>This opportunity is open to all applicants.  All candidates will be placed into the EPSRC Doctoral Landscape Award Competition and selection is based on academic merit.</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 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> <p>Please contact individual supervisors listed for information about projects.</p>