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 doable 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 
Model Theory of Fields with Operators 

Group  Mathematical Logic 
Supervisor  
Description 
Model theory is a branch of Mathematical logic that has had several remarkable applications with other areas of mathematics, such as Combinatorics, Algebraic Geometry, Number Theory, Arithmetic Geometry, Complex and Real Analysis, Functional Analysis, and Algebra (to name a few). Some of these applications have come from the study of modeltheoretic properties of fields equipped with a family of operators. For instance, this includes differential/difference fields. In this project, we will look at the model theory of fields equipped with a general class of operators (that unifies other known approaches) and also within certain natural classes of fields (such as real closed fields). Several foundational questions remain open around what is called "modelcompanion", "elimination of imaginaries", and the "trichotomy", this is a small sample of the problems that will be tackled.

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 differentialalgebraic 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. 
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. 
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 nontrivial model theory of the ring of "sizes" and some abstract versions of the LangWeil 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 socalled 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. 