Applications to join the department's thriving PhD programme are welcome from students with interests in analysis, probability theory or mathematical physics. The area in which I work, non-commutative probability, is an exciting combination of these three subjects. Knowledge of all of them is not necessary, but an interest to discover more is.

At present I'm particularly interested in the following topics.

- Quantum random walks. A classical random walk consists of repeatedly flipping a coin and moving left or right accordingly. This simple model illustrates many important ideas in probability theory. Its quantum generalisation corresponds to a system interacting with a sequence of identical particles; limit theorems have been obtained, but many interesting questions remain unanswered.
- Non-commutative stopping times. Stopping times are random times which, at any given moment, are known to have occurred or not. The time of the first rainfall this week is a stopping time; the time of the last rainfall is not. The theory of stopping times is vital for developing classical theories of stochastic integration. The proper non-commutative generalisation is known, but is yet to be exploited fully.
- Exotic forms of independence. The concept which separates probability from analysis is stochastic independence. Once one moves to the non-commutative world, more than one form of independence exists. Free independence was introduced by Voiculescu in the 1980s, and has important connections to random matrix theory, quantum information theory and representation theory. Connections for other forms of independence remain to be explored.

## Availability

Due to existing commitments, I cannot take new PhD students in the year 2016-17, but would be interested in hearing from suitable applicants wishing to start in 2017-18 or later.

## Current ideas for PhD projects

Here are four possible areas in which I would currently be willing to supervise: each of these is not a specific PhD project, but a setting in which there are various possible research problems that a student could work on. I hope to add more details, or links to more details, in the near future: if you would like to know more then please feel free to get in touch.

- Similarity and perturbation problems for semigroup or Fourier algebras
- The Banach-algebraic version of the "Kähler module of differentials"
- Operator algebras that are locally isomorphic to C* algebras
- Homological algebra for dual Banach algebras and their modules

## Warning notes.

It is possible that you are reading this and have had some exposure to various structural properties of Banach algebras known as "approximate amenability", "character amenability", or "module amenability". **I will not, for the foreseeable future, supervise on any of these three, nor on any hybrid of these.**

("Connes-amenability", on the other hand, is related closely to Theme 4 above, but be warned that this is an area with some nasty traps for the novice, primarily because dual Banach algebras are in some sense still not really understood.)

**Using existing information in multi-arm multi-stage clinical trials**

Multi-arm clinical trials compare several active treatments to a common control and have been proposed as an efficient means of making an informed decision about which of several treatments should be evaluated further in a confirmatory study. Additional efficiency is gained by incorporating interim analyses that allow the study to be stopped early - either because of overwhelming evidence of benefit or lack thereof.

This project will investigate design and analysis of multi-arm multi-stage clinical trials that incorporate existing information (e.g. from previous studies). A Bayesian framework will be use to integrate this information while frequentist properties of the design will be controlled.

Higher structures and noncommutative geometry

Deformation theory of geometric and algebraic structures

Operads and Topological Field Theories

Rational homotopy theory

1.

**Modelling irregularly spaced time series**. Many time series have a natural irregular sampling structure, or feature missingness. For example, this could be due to faulty measurement devices or infrequent event data from environmental processes. Many models do not properly take this structure into account, which can lead to inaccurate modelling and conclusions being drawn. This project would focus on developing new models for such data.

2.

**Long memory**. It has been well established that many time series, from physiological data to climatic series exhibit long memory, i.e. a dependence structure which lasts over long periods. Accurate estimation of measures of persistence can be useful in climate modelling, however, traditional methods often suffer from bias. We would work on new methods of efficient estimation of these dependence measures in a range of time series and image settings.

3.

**Network data**. There has been an explosion of data on networks, from epidemiological processes, to social media. However, not much work has been done linking the dynamics of the network itself together with the process observed on the network. The combine elements from time series and network analysis for computationally efficient inference methods for dynamic processes.

1. Stochastic modelling and object oriented data analysis: This project develops a novel statistical methodology to analyse tree-like data (brain artery trees) based on a topological data representation. Standard methods try to extract high dimensional features from the representation for further analysis. This project considers a stochastic modelling approach similar to queueing models for statistical inference.

2. Prediction models for continuous monitoring data: It is easy to continuously collect and monitor various signals such as physiological or health related information, but is challenging to build a statistical model that takes such information into account. One can view such data as high-dimensional time series but there is more structural information/constraint that can be exploited. This project focuses on developing novel statistical methods using the ideas from functional data analysis and sparsity estimation.

3. Multivariate functional data analysis: Multivariate analysis is well developed for vector-like data, but not well developed for curve-like data such as continuous signals or functional data. Especially capturing (non-linear) dependence in high dimensional setting is challenging due to the inherent geometry of the data. This project develops novel statistical methods that combine the analytical (functional data analysis) and geometrical (shape analysis) approaches to analysing such type of data.

4. Spatial functional data and network regularisation: The spatial data has a natural network structure that is linked to each other through neighbours. When the dimension is high and the information is incomplete, it is difficult to estimate the underlying structure. This project considers to incorporate network regularisation methods in the context of spatial data analysis to tackle statistical inference problems.

Example MCMC areas include: pseudo-marginal, delayed acceptance, HMC, non-reversible, and MCMC and variations for tall data.

My research interstst is insequential monte carlo methods and I am currently inv

estigating their use in the study of motor unit number estimation.