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.

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.

Homotopy theory, homotopical algebra, higher structures, higher category theory, string topology, topological quantum field theories, rational homotopy theory

## 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.)

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.

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.

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

**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.

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

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.

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

I am happy to supervise PhD students who wish to work on a project related to my research expertise in Operator Theory and Non-commutative Analysis. Possible topics include: ideals of the algebra of bounded operators on a Banach space; the existence of approximate identities in such ideals; commutators and traces.

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

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.

modular representation theory of finite groups (endotrivial, endo-p-permutation modules, relative syzygies), invertible modules (for discrete groups), finite (p-)groups theory (soft subgroups, subgroup lattices of finite groups), and related topics to these.

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.

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.

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 the above topics to biophysical materials and control of robotic formations.

I am interested in supervising PhD students in the broad areas of time series and image analysis, as well as multiscale (wavelet) methods in statistics and analysis of network data. Example research questions include:

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.

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

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.

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

Geometric and combinatorial rigidity. Discrete structures. Symmetry.

Projects are available in combinations of (1) MCMC theory: efficiency as a function of tuning parameters, and convergence; (2) MCMC methodology: developing new algorithms or variations on existing algorithms; (3) MCMC applications in e.g. the environment or veterinary epidemiology; (4) particle filters and bridges for stochastic processes.

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

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.

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.

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.

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.