# Possible MPhil/PhD Projects in Pure Mathematics

Suggested projects for postgraduate research in pure maths.

## Areas of expertise

In pure mathematics our main areas of research are analysis and algebra.

Within analysis we work on:

- complex analysis
- functional analysis
- operator theory
- operator algebras
- topological homology

Within algebra we work on:

- geometric and combinatorial group theory
- homological algebra
- representation theory

If you're applying for a MPhil/PhD project in one of these areas, please provide the titles of up to three projects from the list below, in order of preference. Applicants are invited to apply online.

For further information, please contact the PG tutor/selector in pure mathematics: David Kimsey

The first part of the thesis is to establish a connection between higher-rank graph C*-algebras and building C*-algebras. Then, combining known results from the theory of buildings, theory of graph C*-algebras and geometry group theory, develop new methods of K-theory computations.

Supervisor: Dr. Alina Vdovina

The project is to use operator theory to discover and prove results about the geometry and function theory of domains in ${\mathbb{C}}^n$.

Students should have a basic knowledge of complex analysis in one variable and of linear operators on Hilbert space. A typical task is to construct an analytic function from a disc in ${\mathbb{C}}$ to a concrete set in ${\mathbb{C}}^n$ meeting some specifications. The project concerns operators on Hilbert space, function theory and $H^\infty$ control. The function-theoretic problems are related to control engineering; these problems can lead to questions of complex geometry and interpolation in several complex variables.

Supervisor: Dr Zinaida Lykova and Prof Nicholas Young

The project is to solve questions on the structure and properties of topological and operator algebras from the viewpoint of homological algebra.

Students should have a basic knowledge of functional analysis and algebra. I am particularly interested in the homological problems that arise in noncommutative geometry.

A typical task is to study homological properties and cyclic-type cohomology of topological algebras. The theory has applications in many branches of mathematics, including the theory of de Rham homology in differential geometry, automatic continuity theory and K-theory.

Supervisors: Dr Zinaida Lykova and Dr Michael White

Moment problems are classical in nature and intertwine many different topics in mathematics, eg, spectral theory of self-adjoint operators, orthogonal polynomials, probability theory and optimisation theory. In this project, we will investigate noncommutative/multidimensional analogues of moment problems, in the sense that the given sequence are matrices, bound linear operators, quaternions or Clifford numbers. Connections with algebraic geometry will also be emphasised.

Supervisor: Dr David Kimsey

In the past 50 years, a major trend in Operator Theory focuses on the use of operator algebras for encoding geometrical and topological objects. Operator algebras may be considered as algebras of (bounded in norm) innite matrices with complex entries. A central aspect of the program is to explore the passage from intrinsic properties of the object into properties of the associated operator algebras, and use invariants of the latter to classify the former. There are two interrelated questions that orient our study:

- Which (desirable) features of the object determine the operator algebra?
- What is the (desirable) level of equivalence for classifying objects?

Examples of examined objects so far include tilings, tangles, graphs, dynamical systems, groups, semigroups, varieties, homogeneous ideals, factorial languages and stochastic matrices. The research so far incorporates operator algebras (both selfadjoint and non-selfadjoint) in terms of: representation theory, dilation theory, ideal structure, KMS-states theory, C*-envelopes, re exivity, and hyperrigidity; for objects related to: C*correspondences, product systems, subproduct systems, semigroup actions on operator algebras, and homogeneous ideals.

For this project we will examine operator algebras arising from such geometric structures in this endeavour.

Supervisor: Dr Evgenios Kakariadis

Download pdf for full description Problems in Artin Groups (pdf 28KB)

Supervisor: Prof Sarah Rees

Given a normed function algebra, how do we describe those elements which are in the unit ball of the algebra?

There turn out to be a number of ways (collected under the umbrella of a realisation theorem), and the various descriptions allow us to explore important properties of the algebra.

What we discover is that there is a subtle interplay between such algebras and operator theory - operator theory yields information about the algebra, while the algebras associated to operators can help to untangle their structure.

Though this will be a pure mathematics project, it is worth mentioning that realizations are also important in engineering applications - indeed the transfer function representation of an element of the unit ball of the algebra which forms a key parts of a realisation theorem initially came from engineers studying feedback control.

This project will explore realisation theorems and some of their recent generalisations, as well as their application. In particular, we'll consider in this way rational dilation problems, which attempt to gather information about an operator by viewing it as a piece of some simpler (in this case, normal) operator.

Supervisor: Dr Michael Drischel

Classical representation theory studies how groups and rings act on vector spaces by linear maps. Representation theory hence encodes symmetry, and as such it appears in many areas of mathematics and mathematical physics. In a groundbreaking development in the 1980s mathematical physicists combined the notions of symmetry and quantisation and invented quantum groups. Quantum groups have played a fundamental role in representation theory ever since.

The theory of quantum groups has a younger sibling, the theory of quantum symmetric pairs, which is as rich in structure theory as quantum groups themselves. This theory is much less developed. The finite dimensional simple representation of these algebras, for example, have only been classified in a few special cases. The aim of this project is to develop aspects of a general representation theory of quantum symmetric pairs.

Supervisor: Dr Stefan Kolb

In the early part of the 20th Century, the simple Lie algebras over the complex numbers were classified by Killing and Cartan. These are algebraic structures which arise by differentiating the action of smooth groups of symmetries, called Lie groups. It turns out that each of the simple Lie algebras found can be defined over the integers; thus they are defined for any ring, and in particular, any field of any positive characteristic. Lie algebras over such fields are called modular Lie algebras. It turns out that for algebraically closed fields one can again classify the simple Lie algebras, a programme completed by Premet—Strade in 2006. (There are many more isomorphism types than over the complex numbers.)

The next step is to understand these algebras better; for example, their representation theory and their subalgebra structure. A particularly important idea connecting these two areas is that of G-complete reducibility. This was originated by Serre in the context of spherical buildings and algebraic groups, but similar notions exist for Lie algebras. In a recent paper with Adam Thomas, I found tight bounds on the characteristic of the field which would permit non-G-completely reducible simple subalgebras to exist in classical Lie algebras. As part of this project, one could go further, classifying the conjugacy classes of non-G-cr subalgebras in classical Lie algebras and considering analogous ideas for the Lie algebras of Cartan type.

Supervisor: Dr David Stewart

One-relator group theory is a classical and central topic in geometric and combinatorial group theory. It originates from work of Dehn and Magnus in the first half of the 20th century. The idea is to begin to generalise the successful theory of free groups by adding one relator at a time. Eventually we will understand all well behaved finitely presented groups. However, although there is a rich theory of groups with one relator, this theory doesn't easily extend to two or more relators. Therefore we look for other ways to generalise the theory.

In this project, instead of starting with free groups and adding one relator, the idea is to start with a class of groups known as "partially commutative" groups. We will see what sort of theory is obtained by adding an arbitrary relator. (The class of partially commutative groups contains free groups, is simple to describe, and is now one of the central planks of geometric group theory.)

There are some initial results to work with, but there is much to be investigated in this area.

Supervisor: Dr Andrew Duncan

Operator semigroups are families of bounded linear operators which are parameterised by a time variable and describe the solutions of certain differential equations, such as the heat equation or the (damped) wave equation. There is a rich and beautiful abstract theory, motivated primarily by questions about the rate of energy decay in damped waves, which makes it possible to predict precisely the quantitative large-time asymptotic behaviour of operator semigroups given suitable information about the spectral properties of the so-called generator of the semigroup. This theory has recently reached a state of perfection in the case where the underlying Banach space is a Hilbert space, which is to say that its norm is particularly well behaved, and it is known that without any restrictions at all on the underlying space one in general obtains less sharp results.

The more theoretical part of this project would involve investigating the quantitative asymptotic behaviour of operator semigroups on Banach spaces which are not necessarily Hilbert spaces but nevertheless have some reasonably nice geometric properties (such as non-trivial (Fourier) type). However, there would also be scope to consider applications of the theory to important concrete problems, and in particular to certain coupled systems arising in the study of damped waves, in mathematical control theory and elsewhere. Depending on the student's background it may also be possible to investigate some aspects of the theory numerically.

This project will combine aspects of functional analysis, harmonic analysis and complex analysis, so a reasonably strong background in analysis is essential. Ideally, candidates will also have had some exposure to the modern theory of partial differential equations (and in particular to Sobolev spaces), or at the very least they will be willing to pick this up quickly.

Supervisor: Dr David Seifert

In recent years many papers have studied analytic functions of several variables by means of Hilbert space models of functions. We propose to describe a generalization of the notion of a Hilbert space model of a function in the Schur-Agler class of the polydisc. This generalization can be adapted to the investigation of boundary behaviour at a mild singularity of the function on the d-torus. This topic is important for the theories of analytic functions in several complex variables and in operator theory.

Supervisor: Dr Zinaida Lykova and Prof Nicholas Young

Let G be a linear algebraic group defined over an algebraically closed field k, such as the complex numbers. This means that G can be identified as a Zariski closed subgroup of the k-group GL(n) of invertible n × n matrices; that is, G is the vanishing locus of a set of polynomials on the entries of matrices in GL(n). A good example is the group SL(n), since it is the vanishing locus of the polynomial det X -1.

An element of G is called unipotent if it satisfies (X-1)^n=0. It is known, through deep results of Lusztig (or painstaking case-by-case analysis) that reductive groups have finitely many conjugacy classes of unipotent elements, for k of arbitrary characteristic. This important result is the foundation of an enormous amount of theory.

Little attention has been paid to investigating those non-reductive groups with finitely many unipotent classes. The main problem is to describe all the linear algebraic groups with finitely many unipotent classes.

Supervisor: Dr David Stewart