Possible MPhil/PhD Projects in Pure Mathematics

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 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) innite 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:

1. Which (desirable) features of the object determine the operator algebra?
2. 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

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

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