PhD & Postgraduate Research

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.

1. 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.
2. 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.
3. 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.

View Alexander's profile

I an interested in supervising PhDs in the following topics: Random matrices, high dimensional phenomena, and optimal transportation theory. Specifically, I would be willing to supervise a project `Integrating differential equations in random matrix theory'. This would involve using methods from the theory of linear systems to analyze various operators which arise in random matrix theory. In particular, the aim is to extend ideas of Tracy and Widom to new matrix models. To pursue this project, a student would need a sound background in analysis. While the results have applications to statistical physics, the student would not require much background in physics or probability. This project develops a theme from some previous PhD thesis which I have supervised at Lancaster. Further information is available on request.

View Gordon's profile

Availability

Due to existing commitments, I am unlikely to take any new PhD students in the 2020-21 cycle, although I am always willing to hear from applicants with a strong background in functional analysis. In any case, the list below of possible supervision areas is included for sake of background interest.

Current ideas for PhD projects

Here are three possible areas in which I am 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. If you would like to know more, then please feel free to get in touch.

1. Algebras associated to group actions on L^p spaces
2. Cohomological invariants of Fourier algebras
3. Properties of Banach-Kähler modules

Warning notes.

Themes 1 and 2 are likely to require some prior exposure to the basic theory of C*-algebras. However, some of the necessary background can be learned in the initial stages of the PhD, and depending on the individual strengths of an applicant there may be particular projects that require less functional-analytic machinery; some questions may be suitable for those with solid basic analysis and interests in the representation theory of finite groups.

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.

View Yemon's profile

Interested in supervising doctoral students working on topics in the spectral theory of partial differential operators, particularly those arising in mathematical physics and/or with connections to other areas of mathematics. Potential projects could focus on zero modes of Pauli and Dirac operators, operators with periodic or quasi-periodic coefficients and the stability of embedded eigenvalues.

View Daniel's profile

Students who are interested on any of these topics are more than welcome to contact me for further details.

View Konstantinos's profile

I am currently offering PhD projects in three areas: 1. Quantum groups, including producing novel quantum groups from the double bosonisation construction. 2. Cluster algebras, their quantizations and representation theory. 3. Relationships with mathematical physics, including Verlinde algebras, quantum cohomology and integrable systems and their relationships to quantum cluster algebras.

View Jan's profile

I offer PhD supervision in areas around the mathematics of conformal field theory, operator algebras, and quantum information theory.

View Robin's profile

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.

View Thomas's profile

PhD proposals are welcome in the broad areas of time series analysis, nonstationarity, changepoints, wavelets, streaming data as well as environmental and medical applications.

View Rebecca's profile

I am interested in supervising projects in geometric and combinatorial rigidity (both theory and application areas). I am particularly interested in rigidity properties of periodic structures, rigidity under alternative metrics, and aspects which border analysis, operator theory and Banach space geometry. A good background in either combinatorics or analysis is sufficient to get started. For more information please contact me.

View Derek's profile

Tail asymptotics for perpetuities in continuous time; Heavy-tailed analysis of linear functionals of random walks and Levy processes

View Dmitry's profile

Homological algebra and derived categories
Higher structures and noncommutative geometry
Deformation theory of geometric and algebraic structures
Operads and Topological Field Theories
Rational homotopy theory

View Andrey's profile

I would be interested in taking a PhD student who has some familiarity with algebraic geometry and representation theory. A PhD student of mine would address open questions in the theory of linear algebraic groups over fields. I have potential PhD projects in mind related to birational invariant theory, and also to linear representation theory in positive characteristic.

View Mark's profile

modular representation theory of finite, infinite discrete and profinite groups; (pro-)finite groups; subgroup lattices of finite groups; group cohomology; Burnside rings; (pro-)fusion systems and related topics.

View Nadia's profile

I welcome new PhD students with strong background in statistics and probability to work on projects related to robust estimation in nonlinear time series models and resampling.

View Kanchan's profile

The understanding and control of infectious diseases is of considerable importance to society. How a disease spreads and/or how infectious a disease is, has tremendous implications upon the health and wealth of a community. I am interested in both the probabilistic and statistical analysis of infectious diseases. From a probabilistic perspective, we look to answer questions as: What is the probability that a disease takes hold within a community? How many individuals are ultimately infected by the disease? This involves developing novel probabilistic techniques to answer these questions for realistic population models such as the household and random graph models. Alternatively, having observed an epidemic we can propose a model for the disease spread and estimate the model parameters. However, often the disease data are "incomplete" and novel statistical methods, in particular, Markov Chain Monte Carlo (MCMC) are required to analyse the data. We aim to answer questions concerning the adequacy of the model and the predictive capabilities of the model for the future epidemic outbreaks.

View Peter's profile

I would be happy to supervise a PhD student who is interested in computational methods for Bayesian inference. In particular, the development of new MCMC and SMC algorithms for big data and intractable likelihood problems.

View Christopher's profile

I would be interested in discussing PhD opportunities with a student interested in combinatorics, geometry or both. In combinatorics I am interested in graph theory, matroid theory and combinatorial rigidity. In geometry I am interested in discrete and computational geometry, sphere packing and concrete aspects of differential and algebraic geometry. Unifying these topics is the study of geometric graphs and their configuration spaces. As well as the above purely theoretical topics, I am interested in applications of these topics to biophysical materials and control of robotic formations.

View Anthony's profile

Mendelian randomization, instrumental variables, meta-analysis, structural equation models.

View Tom's profile

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.

View Juhyun's profile

Geometric rigidity, infinite bond-node structures, crystallographic and quasicrystallographic bar-joint frameworks, rigidity operators, rigid unit modes, locally compact graphs and string-node meshes.

View Stephen's profile

Geometric and combinatorial rigidity. Discrete structures. Symmetry.

View Bernd's profile

I am happy to supervise students in topics related to medical statistics. I would particularly welcome applicants interested in developing methodology for the analysis of event history data using multi-state models.

View Andrew's profile

My main research area is in probability with a specific focus on scaling limits of stochastic processes. My work combines ideas from probability theory, real and complex analysis and statistical physics, so I would be interested in discussing possible projects with students who have a strong background in any of these areas.

View Amanda's profile

I am happy to supervise PhD topics in all aspects of extreme value theory. In particular multivariate and spatial extremes are exciting areas with lots of work to be done on relaxing assumptions so that models are more realistic.

View Jennifer's profile

I am happy to supervise students in topics related to medical statistics. I would particularly welcome applicants interested in multiple comparison, simultaneous inference and adaptive design.

View Fang's profile