PhD opportunities

Diseases and pests of trees lead to large economic costs to the forestry industry, limit natural spaces for recreation and impact climate change by reducing trees’ ability to capture carbon. A key element of infection specific to plants is spatial structure: since plants are usually fixed in location, infection tends to be localised. In this project you would develop advanced, spatially-explicit mathematical models using nonlinear ordinary differential equations and computational individual-based models. You will fit your models to real data of disease spread and then test different management and mitigation strategies.  

Antagonistic coevolution between parasites and the hosts they infect is a major driver of disease dynamics. Mathematical models have been used for some time to understand these co-evolutionary dynamics, however, they tend to assume constant environments and no external interactions with other species. In this project you will develop coevolutionary models of hosts and parasites within the adaptive dynamics framework, primarily using nonlinear ordinary differential equations and some stochastic numerical models. You will incorporate complex environments either through fluctuating parameters (for example representing seasonality) or external interactions (such as competiiton with other species).

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Random graph models are probabilistic models for the development of networks. They are often inspired by networks found in areas such as computing, social networks and biology, but the  emphasis is on studying mathematical properties of the graphs. One family of random graph models which has been studied extensively for the last 25 years or so is the preferential attachment model and its variations. Some of these variations feature an interesting phenomenon often referred to as condensation, where a small proportion of vertices become unusually important in the network in that a large proportion of edges connect to them. The aim of this project will be to study this phenomenon further and its occurrence in new graph models.

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Index trackers (ETFs, index funds, index mutual funds) aim to replicate the return of a reference equity index while keeping implementation frictions---notably turnover and trading costs---under control. Index tracking is a passive investment approach that frequently beats their actively managed counterparts over the long term. In practice, full index-replication can be costly or operationally infeasible; consequently, pure tracking with a sparse basket is common for segregated mandates, often subject to constraints such as a full-investment budget and, in some mandates, long-only weights. A large body of work formulates index tracking as an optimisation problem with convex losses and penalties, or as mixed-integer programs that directly control support size and transactions. These approaches typically deliver a single allocation weight vector and tune regularisation hyperparameters by heuristics or cross-validation. Such tuning requires manual interventions and is often subject to uncertainty.  

This project develops novel Bayesian methodology with the aim to return a posterior over weights, tuning parameters, and the residual variance that drives tracking error. The aim is decision-grade uncertainty quantification (UQ) and principled data-driven parameter selection. Hence, the project will develop automatic portfolio rebalancing driven by UQ analysis, based on the posterior distribution of the allocation weights.

We will model the tracking residual returns—index minus portfolio—as a noisy outcome with unknown variance, regressing the index returns on the returns of the individual assets. Portfolio weights live in a constrained space reflecting business rules: non-negativity or signed positions, leverage or budget limits, optional sector exposures, and explicit cardinality. Sparsity comes from imposing hierarchical priors that prefer few active names: either spike-and-slab families (selection indicators plus a continuous “slab” for active weights) or continuous shrinkage alternatives such as horseshoe or Dirichlet-Laplace. Tuning parameters are not fixed numbers in this approach: global and local shrinkage scales, inclusion probabilities, tracking-error and turnover multipliers, and the residual variance all carry their own priors. The project is expected to result in a coherent hierarchy that links portfolio structure, transaction costs, prior beliefs, and a data-driven narrative for model selection.

Inference combines two ingredients. First, state-of-the-art MCMC samplers, which are designed to handle non-smooth penalties and hard constraints. These samplers perform gradient-informed random walks while enforcing feasibility at every step. These take advantage of recent advances and contribute to convex nonsmooth optimisation and numerical analysis of stochastic differential equations. Second, Gibbs or Metropolis updates for the hyperparameters: residual variance, global shrinkage, local scales, and selection indicators. 

Alongside expert-set priors, the project proposes to also use machine learning techniques to learn parts of the prior and penalty structure from data in an interpretable way. Examples include mapping liquidity and volatility features to a prior preference for sparsity, learning sector-level shrinkage from historical co-movement, or learning a turnover multiplier that depends on spread and market depth. The plan is to start with simple parametric maps and progress to neural networks with monotonicity or convexity constraints so that learned priors remain stable and defensible. Expert ranges can be encoded as hyperpriors, allowing the data to refine but not overrule domain knowledge.

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The cost and duration of traditional clinical trials imply a high societal and economic cost, hampering the release of new treatments to public. One possible avenue to improve on this situation is the use of mathematical models to produce synthetic data, which can then be shared with contemporary clinical trials to reduce the number of patients and the duration of the trial. The approach can take advantage of well established mathematical models of diseases, or use current information to generate statistical forecasting models for less well understood systems. A Bayesian approach allows combining disparate sources of information, along with coherent uncertainty propagation, fundamental to feed into validation, verification and uncertainty qualification needed for clinical approval of these novel medical devices. 

Panel (longitudinal) data enables learning the dynamics and relations of (groups of) units, strengthening the inference on both cross-sectional and dynamic parameters. The dominant approach to modelling unbounded continuous quantities of interest is to assume symmetric error terms (eg Gaussian or Student), but this assumption is often not met in practice. The idea is to explore, from a Bayesian perspective, the use of skew error distributions from well-known skewing mechanisms (eg Azzalini, 1985; Ley 2010), while allowing the parameter(s) controlling the skewness to vary over time. This implies the design of a stochastic path to control the process in time and exploring conditions for convergence. 

The Brownian web is a dense system of coalescing Brownian motions in 1+1 dimensional space time. It is known to be the universal scaling limit of a wide variety of models, ranging from population genetics and interface growth to theoretically important processes such as orientated percolation and directed spanning forests. The project will develop scaling limit theory for dense interacting systems of particles in cases where the limiting particle motions are not simply Brownian motions e.g. they might feature spatial inhomogeneity or heavy tailed behaviour.

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Large random trees have well-known discrete and continuous limits which have been studied since the 90's. The aim of this project is to investigate how much changes if we force the trees to be in unusual regimes.