Funded PhD Studentship in Mathematical Logic

Summary

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).

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.

We are delighted to offer a fully funded PhD studentship and applications are invited from strongly motivated and academically excellent candidates for PhD study in Mathematical Logic, within these strategic priority Research areas:

Computability Theory and Proof Theory: Dr Paul Shafer, Professor Michael Rathjen. 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 p.e.shafer@leeds.ac.uk or Professor Michael Rathjen by email to m.rathjen@leeds.ac.uk.

Model Theory: Dr Pantelis Eleftheriou, Professor Dugald MacPherson, Dr Vincenzo Mantova, Dr Michael Wibmer. 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 p.eleftheriou@leeds.ac.uk, Professor Dugald MacPherson by email to h.d.macpherson@leeds.ac.uk, Dr Vincenzo Mantova by email to v.l.mantova@leeds.ac.uk or Dr Michael Wibmer by email to m.wibmer@leeds.ac.uk.

Set Theory: Dr Andrew Brooke-Taylor, Dr Asaf Karagila. 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 A.D.Brooke-Taylor@leeds.ac.uk or Dr Asaf Karagila by email to a.karagila@leeds.ac.uk.

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.