Mathematical Logic PhD projects

This page provides a (partial) list of specific (and not so specific) PhD projects currently offered around the Mathematical Logic topic.

Identifying an interesting, worthwhile and do-able PhD project is not a trivial task since it depends crucially on the interests and abilities of the student (and the supervisor!) The list below therefore contains a mixture of very specific projects and fairly general descriptions of research interests across the School. In either case you should feel free to contact the potential supervisors to find out more if you're interested.

You should also explore the School's research groups' pages, and have a look around the homepages of individual members of staff to find out more about their research interests. You may also contact us for general enquiries.

Title

Algebraic differential equations and model theory

Group Mathematical Logic
Supervisor
Description

Generally speaking this area is currently my main focus of research. Differential rings and algebraic differential equations have been a crucial source of examples for model theory (more specifically, geometric stability theory), and have had numerous application in number theory, algebraic geometry, and combinatorics (to name a few).

In this project we propose to establish and analyse deep structural results on the model theory of (partial) differential fields. It has been known, for quite some time now, that while the classical notions of 'dimension' differ for differential fields, there is a strong relationship between them. We aim to tackle the following foundational (still open) question of this theory: are there infinite dimensional types that are also strongly minimal? This is somewhat related to the understanding of regular types, which interestingly are quite far from being fully classified. A weak version of Zilber's dichotomy have been established for such types, but is the full dichotomy true?

The above is also connected to the understanding of differential-algebraic groups (or definable groups in differentially closed fields). While the notions of dimension agree for these objects in the 'ordinary' case, the question is still open in the 'partial' case. We expect that once progress is made in the direction of the above problems, we will also be closer to the answer of this question.

There are classical references for all of the above concepts and standing problems, so the interested student should have no problem in learning the background material (and start making progress) in a relatively short period of time.

 

Title

Representations of algebras and interpretations

Group Mathematical Logic
Supervisor
Description

This is an indication of the area in which my current work is focussed, hence the area in which I would expect to supervise a student.

First, it's algebra with input from model theory and category theory.
The area of algebra is module (= representation) theory, especially representations of algebras.

In extremely general terms, the aim is to understand the structure of the category of modules.  This might mean getting a description of some of the most interesting modules and the maps between them or it might mean finding some structure (topological, geometric, algebraic, ...) on a set of these, and investigating that 'larger-scale' structure on (part of) the category of modules.

The input of model theory (part of mathematical logic) in the specific context of the representation theory of finite-dimensional algebras, where interest is typically focussed on finite-dimensional representations, leads us to extend our interest to at least some of the infinite-dimensional representations, even if our eventual applications are back in the context of the finite-dimensional ones.  The same general pattern, of looking at (somewhat) 'large' representations, can be seen over algebras which are not finite-dimensional.

Another input of model theory is the concept of interpretation which, in this context, can be seen as a certain kind of functor between categories of modules.  Understanding how these link categories of modules is another rather general aim.

My website  (www.maths.manchester.ac.uk/~mprest/publications.html)  gives more (too much) information but some flavour of the area can be got by browsing around there.

Any offer of a place will include a description of a broad research problem but a specific project will be determined taking account of a variety of factors, in particular, the current state of knowledge and activity in the area and the interests and development of the student.  It can also be that the direction of the project changes as it develops, in the light of what is discovered.

Title

Interdefinability of abelian functions

Group Mathematical Logic
Supervisor
Description

Recently there has been a great deal of interaction between model theorists and number theorists on topics around `unlikely intersections', see for example [3]. One outcome of this is that there are now various functional transcendence results known for certain covering maps. The original example of this is Ax's functional version [1] of Schanuel's conjecture. This result and its more recent descendants have been used to study interdefinability of Weierstrass elliptic functions [2] and the initial aim of this project is to extend this to abelian functions. This would involve a mixture of model theory, differential algebra and number theory, although these are not all required to get started. It should also lead naturally to further interesting questions in these areas.


[1] Ax, James On Schanuel's conjectures. Ann. of Math. (2) 93 1971 252–268. See  http://www.jstor.org/stable/1970774

[2] Jones, G., Kirby, J. and Servi, T., Local interdefinability of Weierstrass elliptic functions. Journal of the Institute of Mathematics of Jussieu, To appear. See http://dx.doi.org/10.1017/S1474748014000425

[3] Zannier, Umberto, Some problems of unlikely intersections in arithmetic and geometry. Annals of Mathematics Studies, 181. Princeton University Press, Princeton, NJ, 2012

Title

Axiomatic approaches to the Hrushovski Programme

Group Mathematical Logic
Supervisor
Description

The celebrated Hrushovski Programme is aimed at proving  that the group of fixed points of a generic automorphism of a simple group of finite Morley rank behaves as a pseudofinite group and, with some luck, is pseudofinite indeed. The aim of the project is to analyse a few configurations where the assumptions of the Hrushovski Conjecture are strengthened. For example, an interesting case is where the fixed points sets of the automorphism in question have "size" with values in a linearly ordered ring which behaves in a strict analogy with cardinality of finite sets; will in that case the group of fixed points be pseudofinite? This question may perhaps involve some non-trivial model theory of the ring of "sizes" and some abstract versions of the Lang-Weil inequality linking the Morley rank of an invariant definable set and the "size" of  the set of its fixed points.

Title

Development of group theory in the language of internal set theory

Group Mathematical Logic
Supervisor
Description

The internal set theory, as proposed by Edward Nelson in 1977, blurs the line between finite and infinite sets in a very simple, effective and controlled way.

This PhD project is aimed at a systematic development of the theory of finite and pseudofinite groups in the language of the internal set theory. This is motivated by problems in a branch of computational group theory, the so-called black box recognition of finite groups. Its typical object is a group generated by several matrices of large size, say, 100 by 100, over a finite field. Individual elements of such a group can be easily manipulated by a computer; however, the size of the whole group is astronomical, and arguments leading to identification of the structure of the group are being de facto carried out in an infinite object. The internal set theory provides tools that allow us to deal with finite objects and numbers that are, in effect, infinite. This is an exciting, unusual, but accessible topic for study.

Read more at http://www.maths.manchester.ac.uk/~avb/pdf/PhD_Topic_Internal_Set_Theory.pdf.

Prerequisites for the project: university level courses in algebra. Some knowledge of mathematical logic is desirable.

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