Several possible projects in mathematics and statistics are offered

The University of Exeter’s Department of Mathematics and Statistics is inviting applications for two PhD studentships fully-funded to commence on 8 January 2025 or as soon as possible thereafter.  The studentship will cover tuition fees plus an annual tax-free stipend of at least £19,237 for 3.5 years full-time, or pro rata for part-time study.  The student would be based in the Department of Mathematics and Statistics in the Faculty of Environment, Science and Economy at the Streatham Campus in Exeter. 

International applicants need to be aware that you will have to cover the cost of your student visa, healthcare surcharge and other costs of moving to the UK to do a PhD

1.      Multi-step look-ahead adaptive design of computer experiments

Supervisor:  Dr Hossein Mohammadi (h.mohammadi@exeter.ac.uk)

Project Description: 

Today, computer simulations play an important role in various scientific fields, ranging from climate change to healthcare. Since these models are typically computationally intensive, we have access to a limited number of model evaluations. To overcome this computational challenge, we usually use a cheap-to-evaluate statistical surrogate to predict the model output. Gaussian process emulators [1, 2] are commonplace surrogates in the field of computer experiments. Building a surrogate requires a set of simulation runs, a.k.a. training data points. These points need to be chosen in a carefully designed manner to achieve maximum possible accuracy. Traditionally, training points are obtained based on a space-filling sampling scheme, which distributes them as uniformly as possible across the space. In recent years, the adaptive design of computer experiments, a.k.a. learning, has gained considerable attention [3, 4]. In an adaptive strategy, information from the emulator and previous simulation runs is used to determine the next model evaluation. This approach offers a smarter way to select future design points, allowing for more sampling in regions of interest compared to space-filling designs, which treat all regions as equally important.

However, most adaptive methods consider only the immediate next step, which may not lead to optimal decisions over the entire sampling process. Multi-step look-ahead strategies, which are novel techniques in active learning, aim to optimise the overall process by planning multiple steps ahead. These strategies evaluate the potential long-term impact of each model evaluation, resulting in a more comprehensive and strategic approach to sampling. This thesis aims to develop multi-step look-ahead active learning approaches for Gaussian processes in the context of computationally expensive simulation models.

2.      A rapidly changing high-latitude environment: a carbon sink or a source?

Supervisor:  Dr Mike O'Sullivan (m.osullivan@exeter.ac.uk)

Project Description: 

The high latitudes comprise of Arctic and Boreal ecosystems. They currently experience the fastest warming, largest variability (extremes), especially in the winter due to cryosphere – albedo feedbacks. As such they are the canary in the coal mine for climate change and climate-carbon cycle feedbacks, given their enormous carbon stocks above-ground in the boreal forests and below-ground in Arctic tundra permafrost soils. IPCC predicts large future changes in snow cover and permafrost, and it is unclear the magnitude impact on the main greenhouse gases (GHGs), CO2, CH4 and N2O. There is a contemporary carbon sink in the growing season, yet future projections in response to climate change are uncertain. Massive stores of organic carbon are stored in permafrost which will likely become unstable under warming, triggering multiple climate feedbacks. There is also large uncertainty around the magnitude of the C sink in boreal forests; atmospheric inversions suggest a larger sink than process-based models. It has been proposed by this team that this may be link to the representation of forest age in process-based models. We therefore urgently need to focus on disturbance and mortality processes in high-latitudes (e.g. fire, permafrost thaw, wind stress) and both in above- and below-ground carbon stocks.

This PhD would form part of Schmidt future’s, Virtual Earth System Research Institute (VESRI). VESRI aims to use big data and AI techniques to improve Earth System Modelling. Under VESRI, the UEXE are funded through the CALIPSO - Carbon Loss In Plants, Soils and Oceans – project, where we have secured 50% funding for a studentship. The goal of CALIPSO is to make a step change in the representation of carbon loss processes in ESMs for three critical knowledge gaps in the global carbon cycle: tree biomass, soil carbon and marine biota.

3.      Spatial modelling in complex ecological systems

Supervisor:  Dr Oscar Rodriguez De Rivera Ortega (o.m.rodriguez-de-rivera-ortega@exeter.ac.uk)

Project Description: 

Complex systems, prevalent in ecological, social, economic, and political realms, exhibit several shared characteristics. Firstly, they are thermodynamically open systems, exchanging energy and/or mass with their external environment. Secondly, these systems are typically composed of a multitude of diverse components. Thirdly, the interactions among these components are nonlinear, often characterised by response delays and feedback loops. Fourthly, complex systems demonstrate a high degree of heterogeneity both temporally and spatially. As a consequence of these characteristics, complex systems frequently exhibit emergent properties, multiscale interactions, unexpected behaviours, and self-organisation (Jørgensen, 1995, Prigogine, 1997, Levin, 1999, Wu, 1999).

While the term "complexity" has gained widespread currency across various scientific disciplines, its precise meaning is subject to interpretation. Structural complexity often pertains to the compositional diversity and intricate configuration of a system. Functional complexity, on the other hand, emphasises the heterogeneity and nonlinear dynamics exhibited by a system. Moreover, self-organising complexity is rooted in the emergent properties of systems that co-evolve with their environment, primarily through local interactions and feedback mechanisms operating at diverse spatiotemporal scales. Such self-organising systems are frequently characterised as "complex adaptive systems." (Cowan et al., 1994). According to Levin (1999), a complex adaptive system is a system comprising a diverse assemblage of components, wherein structure and function arise from the interplay between the continuous generation of variation due to various factors and the selective retention of certain variations through local interactions. Many ecological and socioeconomic systems exhibit varying degrees of self-organizing complexity and can thus be categorised as complex adaptive systems. (Cowan et al., 1994, Levin, 1999).

Dealing with such complex, multivariate and multi-scaled problems often invokes the use of quantitative and interdisciplinary approaches. Consequently, this PhD project involves the development and use of complex spatio-temporal statistical models, geographic information systems (GIS), citizen science and earth observation data, and working collaboratively in a multidisciplinary environment.

This PhD project will investigate the spatial and spatiotemporal patterns of ecological processes and their underlying agents. The research will examine how these patterns influence ecosystem variation, how they evolve over time under natural and human-induced pressures, and how this knowledge can be applied to natural resource management. Given that diverse ecosystems coexist within heterogeneous environments that are constantly undergoing disturbance and succession, understanding the relationship between habitat spatial patterns and population dynamics is crucial for predicting and comprehending ecosystem dynamics and natural resources.

Throughout the course of this research, the student will not only gain expertise in advanced statistical techniques but also contribute valuable insights into the field of statistical ecology collaborating with different international organisations. By addressing the challenges posed by different sources of information and developing effective strategies to incorporate them into spatial models, this research will help to advance the integration of diverse data sources in ecological studies. The aim is to enhance the accuracy and reliability of ecological predictions, which can inform better conservation and management decisions.

4.      Improving emulation and calibration of high-dimensional environmental models

Supervisor:  Dr James Salter (j.m.salter@exeter.ac.uk)

Project Description: 

The overarching aims of this PhD project are to improve emulation and calibration of computer models where the model outputs are high-dimensional, with a focus on methodological development but with motivating applications from environmental models.

For computer models to be informative for studying complex real-world systems and making predictions and aiding decisions in the real-world, models must be properly calibrated, and uncertainties in the model understood. Even with increasing computing power, it is not always possible to run large ensembles of high-resolution models that can completely explore the often large input parameter space, with emulators required to predict the model output at unseen versions of the inputs, allowing us to better explore the model and aid tuning/calibration of such models. For large output fields (spatial, spatio-temporal, multiple atmospheric levels), emulation approaches are often based around dimension reduction, with SVD a popular choice (Higdon et al 2008, Salter et al 2019). However, this may struggle with extrapolation; when important real-world patterns are not frequently seen in the model; or when there is large variability in the range of model outputs, leading to trade-offs in the basis vectors that can result in non-stationary behaviour in the projected coefficients.

This project will explore and develop alternative methods for emulating high dimensional output fields, with an emphasis on developing methods that can be used efficiently in calibration problems, and for studying model discrepancies (differences between the real world and model-world). Potential directions include: machine learning-based approaches to dimension reduction (convolutional neural networks, variational autoencoders), if the uncertainty in predictions can be properly characterised; and using models of the same system at different resolutions and learning patterns from the faster, lower resolution models and using these to aid emulation of input-dependent patterns at higher resolutions.

In this project, there will be the opportunity to apply methods to real environmental modelling problems, including relating to land surface modelling and atmospheric dispersion/volcanic ash modelling.

5.      From dynamics to number theory

Supervisor:  Dr Tony Samuel (a.samuel@exeter.ac.uk)

Project Description: 

This is an open call for applications to pursue a PhD in a topic at the intersection of dynamical systems and number theory, or in another area related to Dr. Tony Samuel’s research interests, please see https://experts.exeter.ac.uk/43201-tony-samuel/ for further details. Therefore, a strong MSc-level understanding of at least three of the following topics would be highly advantageous: number theory, ergodic theory, measure theory, fractal geometry, or stochastic processes.

The PhD project offers flexibility and may take various directions depending on the applicant's specific interests. Indeed, potential research themes include, but are not limited to, the complexity of mathematical quasicrystals, the ergodic properties of continued fraction maps, multi-fractal analysis of non-integer-based expansions, or random number generators.

6.      Limit theorems and mixing properties for dynamical systems with an infinite invariant measure and a non-integrable observable

Supervisor:  Dr Tanja Schindler (t.schindler@exeter.ac.uk)

Project Description:

We consider an ergodic system (X, μ, T) and an observable f:X→ℝ. It is one of the most classical problems in ergodic theory to study the long term behaviour of f,...,f∘︎T^n; in particular of the Birkhoff sum f+...+f∘︎T^n. Such systems are often better suited as models than iid random variables as short term dependencies can be taken into account.
Besides the easiest case - to study limit theorems for μ a probability measure and f an integrable (often even very nicely behaved, e.g. Hölder continuous) observable - in the last 10 to 20 years a number of limit theorems have been proven for the following settings:
a) μ is probability measure and f is a non-integrable observable - in this setting often clustering, i.e. the successive occurrence of large events, made those dynamical systems behave qualitatively different from iid random variables with the same distribution function.
b) μ is an infinite measure space and f is either integrable or at least bounded - this setting has e.g. been used to model anomalous diffusion.

In this project the general aim is to combine the above two settings, namely we assume that μ is an infinite measure and additionally f is non-integrable - even non-integrable over a finite measure set. In this setting, both the infinite measure as well as the non-integrable observable influence the qualitative behaviour and it is the aim of the project to study under which conditions each of the two has the leading influence on the system.
The main limit theorems to be proven for such settings are stable laws, extreme value laws, and almost sure limit theorems - possibly two sided and possibly under truncation.
Moreover, a suitable notion of mixing is supposed to be developed. While there are notions like ψ- or α-mixing in the case of random variables over a finite measure space and Krickeberg-mixing for dynamical systems with a infinite invariant measure, so far there is no notion of mixing that helps describing the systems in our setting.

Some of the results are expected to be obtained mostly by a applying a combination of the methods from settings a) and b), e.g. in the case of stable laws or two-sided almost sure limit theorems. These parts of the project can be considered as a lower risk and in an optimal case give already early publishable results.

Depending on the student's progress and interest there are different possibilities in which the project can continue: For instance, there are a number of number theoretic examples which exhibit this behaviour, e.g. different non-standard continued fraction expansions which could give nice applications. On the other hand, there are also a number of zero-entropy dynamical systems (like suspension flows or adding machines) and together with a non-integrable observable they fall into the above described setting - however for them even in setting a) and b) a lot less is known.

7.      Mathematical modelling of perception and action dynamics

Supervisor:  Dr Piotr Slowinski (p.m.slowinski@exeter.ac.uk)

Project Description: 

Hand and eye coordination is a basic skill crucial to perform countless daily activities. Hand-eye coordination is shaped by perception and action dynamics, the way we are reacting to external stimuli (e.g., spotting notification and reaching for a mobile phone). The perception and action dynamics depends on sensory-motor and neural latencies, i.e., time it takes to notice stimulus (perception), time to process it and decide on appropriate action (processing) and time it takes to complete the movement (action).

Deficits in perception and action dynamics are frequently observed in neurological and mental disorders. Similarly, it is commonly observed that outcomes of the neuropsychological tests of people experiencing mental health issues differ from the outcomes of typical population. However, understanding of how perception and action dynamics affects outcomes of neuropsychological tests is lacking.

To address this knowledge gap, the project seeks to develop and analyse mathematical models of perception and action dynamics. The models will allow to investigate both sources and effects of (1) slower and more variable reaction times observed in people at risk of or experiencing mental illness (represented in the models as a temporal delay), and (2) variability in sensory processing or inaccuracies in motor execution (represented in the models as a different sources of noise). The models will use a system of non-homogeneous (driven by external signal) delay differential equations (with scalar, distributed or state dependent delays). The equations will describe eyes and hand movement driven by an input signal (e.g., object observed on the screen). The models will include neurologically motivated coupling between hands and eyes. Model analysis will combine dynamical systems theory, computational non-linear dynamics approaches and simulations.

Mathematical modelling will allow to understand complex interactions between parts of the nervous system (eyes, brain, motor neurons, muscles) involved in on eye-hand coordination and perception and action dynamics more generally. The causal/ mechanistic relations encoded in the model parameters (eye-brain-hand coupling, sources of noise, sources of sensory-motor latencies) will allow to precisely define and compare behavioural mechanisms (cognitive/ compensatory strategies) employed by people to complete neuropsychological tests. Crucially, the models will be validated in behavioural experiments using digital versions of neuropsychological tests.

Project outcomes will directly help in development of devices and technologies fostering better mental health (faster and personalised diagnosis), individually adaptive, minimally intrusive monitoring technologies (cheap, non-invasive, portable data collection set-up, suitable for clinic and home) and new methods of recognising abnormal data patterns (based on individual cognitive strategies and objective, model-driven metrics).

In a longer term, understanding relation between perception and action dynamics and cognitive function might revolutionise the way we define symptoms used to diagnose mental health problems e.g., by facilitating development mathematical nosology and neurosymptomatics.

Entry requirements

Applicants for this studentship must have obtained, or be about to obtain, a First or Upper Second Class UK Honours degree, or the equivalent qualifications gained outside the UK, in an appropriate area of science or technology. 

If English is not your first language you will need to meet the required level as per our guidance at https://www.exeter.ac.uk/pg-research/apply/english/

How to apply

Apply now

·       Contact the supervisor of the project you wish to apply for to discuss the suitability of your application.  The supervisor will provide an evaluation to the admissions panel directly.

PLEASE ENSURE YOU STATE THE PROJECT AND SUPERVISOR ON YOUR APPLICATION FORM.

In the application process you will be asked to upload several documents. 

·       CV

•         Letter of application (outlining your academic interests, prior research experience, the specific project you wish to undertake, the name of the supervisor of the specific project and reasons for wishing to undertake the specific project).

•         Transcript(s) giving full details of subjects studied and grades/marks obtained (this should be an interim transcript if you are still studying)

•         Two references from referees familiar with your academic work. If your referees prefer, they can email the reference direct to PGRApplicants@exeter.ac.uk quoting the studentship reference number.

•         If you are not a national of a majority English-speaking country you will need to submit evidence of your proficiency in English.

The closing date for applications is midnight on 5 November 2024.  Interviews will be held virtually in the week commencing 18 November 2024.

If you have any general enquiries about the application process please email PGRApplicants@exeter.ac.uk or phone +44 (0)1392 722730 or +44 (0)1392 72515.  Project-specific queries should be directed to the relevant supervisor (contact details from https://mathematics.exeter.ac.uk/people/academicstaff/).

Summary