# Algebra PhD projects

#### This page provides a (partial) list of specific (and not so specific) PhD projects currently offered around the Algebra 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 Algebra
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

#### Rational points on algebraic varieties

Group Algebra
Supervisor
Description

Diophantine equations are a classical object of study in number theory. During the course of the 20th century, it was realised that one obtains a more powerful conceptual framework by considering them through a more geometric lens, namely viewing a solution to a Diophantine equation as a rational point on the associated algebraic variety.

Given an algebraic variety over a number field, natural questions are: Is there is a rational point? If yes, are there infinitely many? If also yes, can one obtain a finer quantitative description of the distribution of the rational points?

These problems are very difficult in general, but in this project the aim is to make some progress for some special classes of varieties (in particular solve some new cases of Manin’s conjecture). A popular current research theme is to consider these problems in families, such as studying the distribution of varieties in a family with a rational point, or controlling failures of the Hasse principle in families.

To solve these problems one usually uses a combination of techniques from algebraic geometry and analytic number theory, but the project could be tailored towards the preferences of the student (e.g. for a student without much knowledge of algebraic geometry).

Title

#### Representations of algebras and interpretations

Group Algebra
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

#### Morita equivalences of finite groups

Group Algebra
Supervisor
Description

Most of my current research is focused on the problem of identifying Morita equivalence classes of blocks of finite groups. This is part of the study of the representation theory of finite groups with respect to fields of prime characteristic. Briefly, Morita equivalence is an equivalence of module categories, preserving the structure of modules for an algebra.

This problem is fundamental to the area, and ties in with another of my areas of interest, global-local relationships in finite groups.

Problems range from Donovan's conjecture, which is a finiteness conjecture concerning the number of Morita equivalence classes, to classification of Morita equivalence classes in specific cases.

A tool I use frequently is the classification of finite simple groups, but there is scope for a variety of projects suited to different interests. The precise nature of the project would be open for discussion with the prospective student.

Title

#### Reflection groups in noncommutative algebra

Group Algebra
Supervisor
Description

Finite linear groups generated by reflections arise in many areas of algebra, Lie theory being a prominent example. In the work of Chevalley, Shephard, Todd and Serre, reflection groups are seen to be the groups which have ""good"" rings of invariants when acting on a ring of polynomials.

More recently, reflection groups have been studied in connection with noncommutative rings that are obtained from commutative rings via deformation (or quantisation) construction inspired by quantum mechanics. In particular, this has led to the rich theory of Cherednik algebras.

I am interested in reflection groups, and their quantum analogues, acting on noncommutative rings arising from quantum algebra. Projects might focus on open conjectures in this area, and should be suitable for students with background, and interest, in representation theory and/or quantum groups.

Title

#### Lie algebra actions on noncommutative rings

Group Algebra
Supervisor
Description

My interests in Lie theory are focused on Lie algebra actions on noncommutative rings. An example is the representation of a Lie group, and its Lie algebra, on exterior powers of a finite-dimensional module, or on a Clifford algebra. Many methods of the Chevalley-Kostant theory still apply, but often need to be combined with tools of noncommutative algebra.

A project in this area may suit a student with background in Lie algebras and/or representation theory.

Title

#### Axiomatic approaches to the Hrushovski Programme

Group Algebra
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.