# How to apply

For more information on studying with this Department go to our** Research Opportunities page.**

To complete an application to study, go to the** University's application page. **

For more information on studying with this Department go to our** Research Opportunities page.**

To complete an application to study, go to the** University's application page. **

**Project Description**

This project concerns the combinatorics of permutations. The goal is to investigate how the structure of a large random permutation evolves as the number of its inversions increases. This has received comparatively little attention in comparison to the analogous theory of the Erdős–Rényi random graph.

An observation shared across mathematical and scientific disciplines is that large random objects satisfying a given set of constraints tend to look alike. The common challenge is then to determine their structure so as to understand their behaviour. One of the most influential and fruitful models of random objects has been the Erdős–Rényi random graph *G*(*n*, *m*), drawn uniformly from all *n*-vertex graphs with exactly *m* edges.

The aim of this PhD is to study an analogous model of the random permutation: σ(*n*, *m*) drawn uniformly from all *n*-permutations with exactly *m* inversions [1]. The permutation model is of a manifestly different nature from the random graph, because σ(*n*, *m*) exhibits a striking local–global dichotomy not encountered in the Erdős–Rényi graph model [2]. The random permutation also lacks the natural probabilistic and evolutionary models available for *G*(*n*, *m*).

A particular focus will be on establishing the thresholds at which certain substructures (“patterns”) first appear asymptotically almost surely. A fundamental question when the number of inversions is sublinear is to establish the threshold for the appearance of any given subpermutation, and determine its asymptotic distribution in the window of its emergence. Also of importance are questions about when larger substructures first appear.

A further emphasis will be on constructing suitable probabilistic and evolutionary models of the random permutation. Another intriguing question concerns the relationship between the number of inversions and the total displacement, both of which are natural measures of how close a permutation is to the identity [3].

References

[1] H. Acan and B. Pittel. On the connected components of a random permutation graph with a given number of edges. J. Combin. Theory Ser. A, 120(8), 2013.

[2] D. Bevan. Permutations with few inversions are locally uniform. https://arxiv.org/pdf/1908.07277

[3] D. Bevan. The curious behaviour of the total displacement. https://tinyurl.com/tdrTalk

**Subject Area**

Mathematics

**Funding Notes**

The studentship covers full tuition fees and a tax-free stipend for four years starting in October 2020. Funding is only available to UK nationals and to EU nationals who have lived in the UK for three years prior to the start of the studentship.

**Application Enquiries**

We are looking for a highly motivated student to join our group. This fully-funded 3 year studentship needs to be filled in as soon as possible, so we recommend applying immediately.

Since Darwin, ecologists have been fascinated by the latitudinal gradients of biodiversity on Earth. Why are there so many phytoplankton species in the subtropical ocean gyre, which is believed as an “ocean desert” (Barton et al. 2010; Righetti et al. 2019)? This is particularly puzzling if you think about the principle of “Competitive exclusion” that states that the number of coexisting species should not exceed the number of different resources (Hardin 1960)? You will have the opportunity to attack this problem using a variety of tools such as analyzing partial differential equations, numerical simulation, and statistical analysis.

It is anticipated that you will receive substantial trainings in mathematical and statistical modelling including but not limited to analyses of ordinary and partial differential equations and Bayesian inference. You will also have the opportunity to use the high-performance computing system in Strathclyde (https://www.archie-west.ac.uk/). Your mathematical, statistical, and programming skills are expected to be substantially enhanced during the PhD training. These skills will be very useful for securing some of the most popular jobs in this Big Data era.

You will mainly work within the Marine Population Modelling group, Department of Mathematics and Statistics, University of Strathclyde (https://www.strath.ac.uk/science/mathematicsstatistics/smart/marineresourcemodelling/). You will also have the opportunity to collaborate with the group of Prof Hongbin Liu in Hong Kong University of Science and Technology.

Applicants should have or expect to obtain a good honours degree (1, 2.1, or equivalent) in applied mathematics, statistics, theoretical ecology, oceanography, or a highly quantitative science. Experience of numerical modelling and programming in Fortran, Matlab or R would be highly beneficial, but not essential.

To apply, send 1) a complete CV, 2) a 1 page personal statement explaining your interests and skills for this project, and 3) names and contact information of three references to the lead supervisor, Dr Bingzhang Chen, Department of Mathematics and Statistics, University of Strathclyde, Glasgow at bingzhang.chen@strath.ac.uk.

We value diversity and welcome applications from all sections of the community.

The University currently holds a Bronze Athena SWAN award, recognising our commitment to advancing women’s careers in science, technology, engineering, maths and medicine (STEMM) employment in academia.

The studentship is co-funded by both the University of Strathclyde and Hong Kong Branch of Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou) and are open to all nationalities. However, it is expected that non-EU/UK students should bring their own funding to match up with the extra international fee.

Armstrong, R. A., & McGehee, R. (1980). Competitive exclusion. The American Naturalist, *115*, 151-170.

Barton, A.D., Dutkiewicz, S., Flierl, G., Bragg, J. and Follows, M.J. (2010). Patterns of diversity in marine phytoplankton. Science, 327, 1509-1511.

Chesson, P. (2000). Mechanisms of maintenance of species diversity. Annual review of Ecology and Systematics, 31, 343-366.

Hardin, G. (1960). The competitive exclusion principle. Science, 131, 1292-1297.

Huisman, J., & Weissing, F. J. (1999). Biodiversity of plankton by species oscillations and chaos. Nature, 402, 407-410.

Hutchinson, G.E., 1961. The paradox of the plankton. The American Naturalist, 95, 137-145.

Righetti, D., Vogt, M., Gruber, N., Psomas, A. and Zimmermann, N.E., 2019. Global pattern of phytoplankton diversity driven by temperature and environmental variability. Science advances, 5, p.eaau6253.

Classification of teas (types, quality grades, region of origin etc) has been examined by many researchers, using both chemical composition and more recently digital image analysis techniques to extract features from the image that are useful for classification.

Factors affecting the success of classification include the choice of features, the classifier and the imaging modality. Building on previous work at Strathclyde in collaboration with the EEE department, this project will allow the student to examine the choice of any of these to achieve optimal results. Intending students should have a strong statistical background and excellent computer skills, and be competent/be able to quickly become competent in the use of both R and Matlab.

**Second supervisors: George Gettinby, Magnus Peterson**

Worldwide losses of honey bee colonies have attracted considerable media attention in recent years and a huge amount of research. Researchers at Strathclyde have experience since 2006 of carrying out a series of surveys of beekeepers in Scotland (http://personal.strath.ac.uk/a.j.gray/) and now have 5 years of data arising from these surveys. These data have been used to estimate colony loss rates in Scotland and to provide a picture of beekeepers’ experience and management practices.

This project will examine the data in more detail than has been done so far, and is likely to involve data modelling and multivariate methods to identify risk factors. Part of the project will involve establishing the spatial distribution of various bee diseases.

This project will build on links with the Scottish Beekeepers’ Association and membership of COLOSS, a network linking honey bee researchers in Europe and beyond. Intending students should have a strong statistical background and excellent computer skills, and be competent/be able to quickly become competent in the use of R.

**Introduction**

In this project we shall look at how stochastic models can be used to describe how infectious diseases spread. We will start off by looking at one of the simplest epidemic models, the SIS (susceptible-infected-susceptible) model. In this model a typical individual starts off susceptible, at some stage catches the disease and after a short infectious period becomes susceptible again.

These models are used for diseases such as pneumococcus amongst children and sexually transmitted diseases such as gonorrhea amongst adults (Bailey, 1975). Previous work has already looked at introducing stochastic noise into this model via the disease transmission term (Gray et al. 2011). This is called environmental stochasticity which means introducing the random effects of the environment into how the disease spreads. This results in a stochastic differential equation (SDE) model which we have analysed. We have derived an expression for a key epidemiological parameter, the basic reproduction number.

In the deterministic model this is defined as the expected number of secondary cases caused by a single newly-infected individual entering the disease-free population at equilibrium. The basic reproduction number is different in the stochastic model than the deterministic one, but in both cases it determines whether the disease dies out or persists. In the stochastic SIS SDE model we have shown the existence of a stationary distribution and that the disease will persist if the basic reproduction number exceeds one and die out if it is less than one.

**SDE Models with Environmental Stochasticity**

We have also looked at other SDE models for environmental stochasticity. One of these took a simple deterministic model for the effect of condom use on the spread of HIV amongst a homosexual population and introduced environmental stochasticity into the disease transmission term. Again we found that a key parameter was the basic reproduction number which determined the behaviour of the system.

As before this was different in the deterministic model than the stochastic one. Indeed it was possible for stochastic noise to stabilise the system and cause an epidemic which would have taken off in the deterministic model to die out in the stochastic model (Dalal et al., 2007). Similar effects were observed in a model for the internal viral dynamics of HIV within an HIV-infected individual (Dalal et al., 2008).

**Demographic Stochasticity**

The real world is stochastic, not deterministic, and it is difficult to predict with certainty what will happen. Another way to introduce stochasticity into epidemic models is demographic stochasticity. If we take the simple homogeneously mixing SIS epidemic model with births and deaths in the population we can derive a stochastic model to describe this situation by defining p(i, j, t) to be the probability that at time t there are exactly i susceptible and j infected individuals and deriving the differential equations satisfied by these probabilities.

Then we shall look at how stochastic differential equations can be used to approximate the above set of equations for p(i, j, t). This is called demographic stochasticity and arises from the fact that we are trying to approximate a deterministic process by a stochastic one (Allen, 2007). Although the reasons for demographic and environmental stochasticity are quite different the SDEs which describe the progress of the disease are similar. The first project which we shall look at is analysis of the SIS epidemic model with demographic stochasticity along the lines of our analysis of the SIS epidemic model with environmental stochasticity. **Further Work**

After this we intend to look at other classical epidemiological models, in particular the SIR (susceptible-infected-removed) model in which an individual starts off susceptible, at some stage he or she catches the disease and after a short infectious period he or she becomes permanently immune. These models are used for common childhood diseases such as measles, mumps and rubella (Anderson and May, 1991). We would look at introducing both environmental and demographic stochasticity into this model.

Other epidemiological models which could be analysed include the SIRS (susceptible-infected-removed-susceptible) epidemic model, which is similar to the SIR epidemic model, except that immunity is not permanent, the SEIS (susceptible-exposed-infected-susceptible) model which is similar to the SIS model, but includes an exposed or latent class, and the SEIR (susceptible-exposedinfected- removed) model, which similarly extends the SIR model.

We would also aim to look at other population dynamic models such as the Lotka-Volterra predator-prey model. There is also the possibility of developing methods for parameter estimation in all of these epidemiological and population dynamic models, and we have started work on this with another Ph. D. student (J. Pan).

**References**

1. E. Allen, Modelling with Itˆo Stochastic Differential Equations, Springer-Verlag, 2007.

2. R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.

3. N.T.J. Bailey, The Mathematical Theory of Infectious Disease and its Applications, Second Edition, Griffin, 1975.

4. A.J. Gray, D. Greenhalgh, L. Hu, X.

5. N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for AIDS and condom-use. J. Math. Anal. Appl. 325, 36-53, 2007.

6. N. Dalal, D. Greenhalgh and X. Mao, A stochastic model for internal HIV dynamics. J. Math. Anal. Appl. 341, 1084-1101, 2008.

Differential equation models are commonly used to model infectious diseases. The population is divided up into compartments and the flow of individuals through the various classes such as susceptible, infected and removed is modelled using a set of ordinary differential equations (Anderson and May, 1991, Bailey, 1975). A basic epidemiological parameter is the basic reproduction number. This is defined as the expected number of secondary cases produced by a single newly infected case entering a disease-free population at equilibrium (Diekman and Heesterbeek). Typically the disease takes off if R0 > 1 and dies out if R0≤1.

However media awareness campaigns are often used to influence behaviour and if successful can alter the behaviour of the population. This is an area which has not been studied much until recently. The student would survey the existing literature on media awareness models in the literature and with the supervisor formulate mathematical models using differential equations for the effect of behavioural change on disease incidence. These would be examined using both analytical methods and computer simulation with parameters drawn from real data where appropriate. The mathematical techniques used would be differential equations, equilibrium and stability analyses and computer simulation.

**References**

1. R.M. Anderson and R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.

2. N.T.J. Bailey, The Mathematical Theory of Infectious Disease and its Applications, Second Edition, Griffin, 1975.

3. O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and computation of the basic reproduction number R0 for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365-382.

4. A. K. Misra, A. Sharma and J. B. Shukla, Modelling and analysis of effects of awareness programs by media on the spread of infectious diseases. Math. Comp. Modelling 53, 1221- 1228.

5. A. K. Misra, A. Sharma, V. Singh, Effect of awareness programs in controlling the prevalence of an epidemic with time delay, J. Biol. Systems, 19(2), 389-402,

Summary of PhD project: The project deals with VISIBILITY GRAPHS. It will involve numerical work (kinetic Monte Carlo simulations) as well as methods of network and graph theory, primarily spectral analysis of adjacency matrices and Laplacians of the visibility graphs, and centrality and communicability indices of these graphs, Voronoi cell decompositions, and at a later stage Minkowski functionals and persistent homology. Thus this is a project that applies a variety of analytical, topological, and computational tools to an important industrial problem which we will describe below.

Visibility graphs will be used to allow us better to differentiate between mechanisms of realistic multidimensional extended island Submonolayer Deposition (SD) processes as the first step to a better control of the resulting morphology. For different processes, different morphologies of the resulting surface (substrate covered in islands) are important. For quantum dots, for example, one would like to have islands of particular size and shape periodically spaced on the surface; in other contexts uniform layers of deposition are needed as basis for ``sandwich''-like arrangements of materials. The overall aim is to extend the machinery created by Allen, Grinfeld and Mulheran Physica A 532 (2019), 121872) for SD in a one-dimensional setting and point islands to a realistic multidimensional context involving extended islands.

Supervisors: M. Grinfeld (Mathematics and Statistics) and P. A. Mulheran (Chemical and Process Enginnering)

Funding: this project is fully funded by EPSRC for 48 months.

In recent years there has been much work on reaction-diffusion equations in which the diffusion mechanism is not the usual Fickian one. Examples are integro-differential equations, porous media type equations, pseudodifferential equations, p-Laplacian type equations and prescribed curvature type (saturating flux) equations.

The motivation for this work comes from material science and mathematical ecology. However, there are applied contexts where these diffusion mechanisms have never been considered. One is in the area of combustion and the other is in the area of regularised conservation laws and shock propagation. This project, which would build on the work I did through the years on integrodifferential models and recently with M. Burns on the prescribed curvature equations, will use PDE, asymptotic, and topological methods to explore the dynamics of blowup and of shock propagation in canonical examples of reaction equations and nonlinear scalar conservation laws regularised by non-Fickian diffusion terms.

References:

[1] M. Burns and M. Grinfeld, Steady state solutions of a bistable quasilinear equation with saturating flux, European J. Appl. Math. 22 (2011), 317-331.

[2] M. Burns and M. Grinfeld, Steady state solutions of a bistable quasilinear equation with saturating flux, European J. Appl. Math. 22 (2011), 317-331.

Recently, a new class of model has been developed to describe, for example, phase separation in materials such as binary alloys. These take the form of integrodifferential equations. Coarsening, that is, creation of large scale patterns in such models is poorly understood.

There are partial results [1, 2] that use the maximum principle, while for most interesting problems such a tool is not available. This will be a mixture of analytic and numerical work and will need tools of functional analysis and semigroup theory.

References:

[1] D. B. Duncan, M. Grinfeld, and I. Stoleriu, Coarsening in an integro-differential model of phase transitions, Euro. J. Appl. Math. 11 (2000), 561-572.

[2] V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. J. Appl. Math. 17 (2006), 221-232.

**Project Description**

Marine sonar records have long documented the existence of a layer of acoustic scattering at around 400m depth throughout the world’s oceans, which can undertake diel (night-time) vertical migrations to the near-surface waters. For many years the origins this phenomenon were uncertain, but now we know that it is due to concentrations of small fish and invertebrates which make up what we call the “mesopelagic community” of marine life. We also know that this community plays key role in the ocean food web, providing food for top predators such as tuna, seabirds and whales. Mesopelagic organisms also affect the vertical flux of carbon into the interior of the ocean as a result of eating at the surface and respiring at depth, and so are involved in global climate regulation [1,2].

The fish component of the mesopelagic community (e.g. lantern fishes) is attracting particular attention. Recent global assessments suggest that these fishes represent a huge unexploited resource that could be used to meet human feed demands [1]. However the technological and economic challenges of harvesting, and the biodiversity and climate change consequences, raise doubts about the viability of such a venture. These issues will be addressed in a newly funded EU Horizon2020 project “Sustainable management of mesopelagic resources (SUMMER)” due to start in September 2019 [3]. This studentship will form part of the UK contribution to SUMMER.

The studentship will be part of a Work-package on “Food-web structure and resilience”, specifically on the development and application of mathematical models of ocean food webs incorporating mesopelagic fish and invertebrates, and top predators, to investigate the consequences of mesopelagic exploitation. The student will be based in the Marine Population Modelling group, Department of Mathematics and Statistics, University of Strathclyde [4] and focus on extending an existing food web model of shelf seas to represent ocean food webs [5,6]. The work will involve close collaboration with other modelling teams in Denmark, Portugal and Turkey, and also with sea-going researchers who will be gathering new data to support the models. In the first instance the model developments will focus on legacy field data archives from the Irminger Sea region east of Greenland, and extend to the Mediterranean and equatorial regions of the Atlantic later in the project.

The project will require the student to become an expert in ocean ecology and the application of mathematics and computer programming skills to ecological problems. Realistically, the applicant should have, or expect to obtain, a good honours degree (1, 2.1, or equivalent) in applied mathematics, statistics, earth science, ecology, or a highly quantitative science, and be able to demonstrate experience of programming in C and/or R. It is anticipated that you will receive trainings on mathematical and statistical modelling including but not limited to numerical implementation and solving of differential equations and optimisation methods. Your mathematical, statistical, and programming skills are expected to be substantially enhanced during the PhD training. These skills will be very useful for securing some of the most popular jobs in this Big Data era.

To apply, send 1) a complete CV, 2) a 1 page personal statement explaining your interests and skills for this project, and 3) names and contact information of three references to the lead supervisor, Prof. Michael Heath, Department of Mathematics and Statistics, University of Strathclyde, Glasgow at m.heath@strath.ac.uk

There is no specific closing date for this project – we will appoint when we find the most appropriate candidate, but the earliest starting date is 30 September 2019.

We value diversity and welcome applications from all sections of the community.

The University currently holds a Bronze Athena SWAN award, recognising our commitment to advancing women’s careers in science, technology, engineering, maths and medicine (STEMM) employment in academia.

**Funding Notes**

This studentship is funded by the EU Horizon 2020 Programme and is open to all nationalities. However, it is expected that non-EU/UK students should bring their own funding to match up with the extra international fee. The participation of the University of Strathclyde in the H2020 SUMMER project is not expected to be dependent on the outcome of Brexit.

**References**

[1] Irigoien, X., et al. (2014). Large mesopelagic fishes biomass and trophic efficiency in the open ocean. *Nat. Communicat.* 5, 3271. doi: 10.1038/ncomms4271

[2] St John et al. (2016). A Dark Hole in Our Understanding of Marine Ecosystems and Their Services: Perspectives from the Mesopelagic Community. Front. Mar. Sci. 17 March 2016 | https://doi.org/10.3389/fmars.2016.00031

[3] EU CORDIS website – Horizon 2020 Sustainable management of mesopelagic resources. project. https://cordis.europa.eu/project/rcn/223251/factsheet/en?WT.mc_id=RSS-Feed&WT.rss_f=project&WT.rss_a=223251&WT.rss_ev=a

[4] Strathclyde Marine Population Modelling Group: https://www.strath.ac.uk/science/mathematicsstatistics/smart/marineresourcemodelling/

[5] Heath, M.R. ( 2012). Ecosystem limits to food web fluxes and fisheries yields in the North Sea simulated with an end‐to‐end food web model. *Prog. Oceanogr.* 102, 42– 66.

[6] Heath M.R. et al. (2014). Understanding patterns and processes in models of trophic cascades. *Ecology Letters* 17, 101-114.

The dispersal of plant diseases is among the most important side effects of a closely integrated global economy. There are many examples of plant diseases that were introduced to the UK and other countries by trade, including Phytophthora ramorum, Dutch elm disease and ash dieback. The number of new plant diseases appearing in Europe each year more than quadrupled during the 20th and the early 21st century.

The plant trading network of the UK is likely to undergo significant modifications due to policy changes, both Brexit-related and due to potential reorientation of the British agriculture towards more emphasis on conserving the environment. This might result in an increase in long-range trade with more diverse partners, resulting potentially in the rise of the risk of importing pests, or an increase in domestic production where non-compliant businesses create larger risks. Thus, introducing stricter regulation might have an unintended consequence of a rearrangement of trading networks which may lead to a shift in the risk pathways rather than the overall reduction. The diversification and increased short-term dynamics of the trading networks is also associated with the rise of e-commerce. The rise of internet trading affects both the traditional supplier (e.g. plant nurseries) and the individual customer, resulting in a potential loss of traceability and quality.

By combining data analysis with bioeconomic modelling, this project aims to develop a framework to analyse the risks associated with plant trade, to highlight the pathways associated with high risk, and to assess strategies that can be used to minimise the risks at different border points (international and domestic “borders”). Firstly, the research will look to assess the gains from combining different data sources as well as improved data. This might include studying public data such as that held for England & Wales at FERA, as well as a consideration of statutory plant movement data that is currently privately held and unlinked. Secondly, we will use the novel bioeconomic framework to study influence of changes in disease risk on the change of trade decisions on the way buyers and sellers are connected. Finally, we will use the cutting-edge mathematical models of dynamic and adaptive networks to capture the traders behaviour whereby the individuals alter the pattern of with whom to trade and how much to trade in response to the epidemic.

The studentship is jointly funded by Defra and the University of Strathclyde for a 3.5 years study period. Applicants should have a first-class or a 2.1 honours degree in a relevant subject; Masters degree will be an advantage. A more detailed plan of the studentship is available to candidates upon application. Funding is available for UK and European applications.

**Project Description**

A fully-funded four year EPSRC Industrial Case PhD studentship is available, starting on or after 1st October 2020. The research will be undertaken within the Numerical Analysis group in the Department of Mathematics and Statistics at the University of Strathclyde, supervised by Professor John Mackenzie and Dr Alison Ramage. Please see http://www.mathstat.strath.ac.uk/research/groups/nasc for details of the group's research. The iCase partner is the National Physical Laboratory (NPL), the UK's National Measurement Institute, and the industrial supervisor will be Professor Alastair Forbes.

The student will be part of a project which addresses the challenge of modelling one of the most important environmental problems currently affecting people's health, urban air pollution. Although it is well established that urban air quality can be modelled mathematically using partial differential equations, the inclusion of uncertainty propagation for this class of models requires multiple model evaluations with many different inputs, leading to excessive computational demands if only a high-fidelity model is used. There is therefore a pressing need for models which combine such techniques with reduced order approaches and parameter estimation informed by observational data sets.

The aim of this project is to develop such multifidelity methods to accelerate the solution of uncertainty propagation by combining techniques from mathematical modelling, statistics, linear algebra and data science. The project will place the student at the forefront of research in numerical methods, and provide an excellent opportunity to develop skills working at the interface between applied mathematics, engineering and industry.

Applicants should have, or be expecting to obtain in the near future, a first class or good 2.1 honours degree (or equivalent) in mathematics, engineering or a mathematical science. The studentship is available for UK/EU candidates subject to specific eligibility criteria (see, e.g., https://www.strath.ac.uk/studywithus/scholarships/strathclyderesearchstudentshipscheme-studentexcellenceawardsepsrc/). In addition to the payment of fees (UK/EU) for the duration of the project (4 years), the award includes provision for a student maintenance grant above the standard EPSRC rate (£15,285 GBP for the first year of study, the level will be reviewed annually). The student will spend some time working at NPL in Teddington, Middlesex during the period of the

Details of how to apply can be found at https://www.strath.ac.uk/courses/research/mathematicsstatistics/#apply.

**The closing date for applications is 31st May 2020.**

Informal enquiries should be addressed to Professor John Mackenzie [+44 (0)141 5483668, j.a.mackenzie@strath.ac.uk].

**Project Description**

A fully-funded three year PhD studentship is available, starting on or after 1st October 2020. The research will be undertaken within the Numerical Analysis group in the Department of Mathematics and Statistics at the University of Strathclyde, supervised by Professor John Mackenzie and Dr Alison Ramage. Please see http://www.mathstat.strath.ac.uk/research/groups/nasc for details of the group's research.

The student will be part of a project which addresses the challenge of modelling one of the most important environmental problems currently affecting people's health, urban air pollution. Although it is well established that urban air quality can be modelled mathematically using partial differential equations, the inclusion of uncertainty propagation for this class of models requires multiple model evaluations with many different inputs, leading to excessive computational demands if only a high-fidelity model is used. There is therefore a pressing need for models which combine such techniques with reduced order approaches and parameter estimation informed by observational data sets.

The aim of this project is to develop such multifidelity methods to accelerate the solution of uncertainty propagation by combining techniques from mathematical modelling, statistics, linear algebra and data science. The project will place the student at the forefront of research in numerical methods, and provide an excellent opportunity to develop skills working at the interface between applied mathematics, engineering and industry.

Applicants should have, or be expecting to obtain in the near future, a first class or good 2.1 honours degree (or equivalent) in mathematics, engineering or a mathematical science.

The studentship is available for UK/EU candidates subject to specific eligibility criteria (see, e.g., https://www.strath.ac.uk/studywithus/scholarships/strathclyderesearchstudentshipscheme-studentexcellenceawardsepsrc/). In addition to the payment of fees (UK/EU) for the duration of the project (3 years), the award includes provision for a student maintenance grant at the standard rate (£15,285 GBP for the first year of study, the level will be reviewed annually).

Details of how to apply can be found at https://www.strath.ac.uk/courses/research/mathematicsstatistics/#apply.

**The closing date for applications is 31st May 2020**.

Informal enquiries should be addressed to Professor John Mackenzie [+44 (0)141 5483668, j.a.mackenzie@strath.ac.uk].

Up to 2002, most of the existing strong convergence theory for numerical methods requires the coefficients of the SDEs to be globally Lipschitz continuous [1]. However, most SDE models in real life do not obey the global Lipschitz condition. It was in this spirit that Higham, Mao and Stuart in 2002 published a very influential paper [2] (Google citation 319) which opened a new chapter in the study of numerical solutions of SDEs---to study the strong convergence question for numerical approximations under the local Lipschitz condition.

Since the classical explicit Euler-Maruyama (EM) method has its simple algebraic structure, cheap computational cost and acceptable convergence rate under the global Lipschitz condition, it has been attracting lots of attention.

Although it was showed that the strong divergence in finite time of the EM method for SDEs under the local Lipschitz condition, some modified EM methods have recently been developed these SDEs. For example, the tamed EM method was developed in 2012 to approximate SDEs with one-sided Lipschitz drift coefficient and the linear growth diffusion coefficient. The stopped EM method was developed in 2013. Recently, Mao [3] initiated a significantly new method, called the truncated EM method, for the nonlinear SDEs. The aim of this PhD is to develop the truncated EM method. The detailed objectives are:

(1) To study the strong convergence of the truncated EM method in finite-time for SDEs under the generalised Khasminskii condition and its convergence rate.

(2) To use the truncated EM method to investigate the stability of the nonlinear SDEs. Namely to study if the numerical method is stochastically stable when the underlying SDE is stochastically stable and to study if we can infer that the underlying SDE is stochastically stable when the numerical method is stochastically stable for small stepsize.

A PhD studentship might be available for the project.

References:

[1] Mao X., Stochastic Differential Equations and Applications, 2nd Edtion, Elsevier, 2007.

[2] Higham D., Mao X., Stuart A., Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40(3) (2003), 1041--1063.

[3] Mao X., The truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 290 (2015), 370--384.

Income needs seem to be a top priority at the moment and with low savings rates and top UK equity income funds yielding less than 4\%, it is perhaps easy to understand why. With savings rates continuing at their record lows, some savers are turning to alternatives. It is in this spirit that many financial institutions are offering stock market linked savings plans to those looking to combine a high yield opportunity with some protection against a falling stock market. That is why the stock market linked savings accounts have recently become very popular. The returns of these accounts are random so the returns, even the initial capital, are not guaranteed. They are very much different from the familiar fixed-term-fixed-rate savings accounts.

The aim of this PhD project is to perform the stochastic and numerical analysis on the stock market linked savings accounts in order to establish the theory on the mean percentage of return (MPR). The MPR depends mainly on two factors:

(1) The structure of the saving accounts, namely the terms of the portfolios (plans);

(2) The behaviour of the stock market linked.

There are various portfolios in the market. This PhD project will analyze a number of typical plans. On the stochastic modelling of the stock market, there are various stochastic differential equation (SDE) models. In this project, some of these SDE models will be used. The key objective here is to establish the explicit formulas for the MPRs of the underlying stock marker linked saving accounts if possible; otherwise develop some new techniques to establish better approximation schemes.

The project will then develop the techniques of the stochastic and numerical analysis to deal with other more complicated financial derivatives including bond, fund.

A PhD studentship might be available for the project.

The area of active fluids is currently a “hot topic” in biological, physical and mathematical research circles. Such fluids contain active organisms which can be influenced by the flow of fluid around them but, crucially, also influence the flow themselves, i.e. by swimming. When the organisms are anisotropic (as is often the case) a model of such a system must include these inherent symmetries. Models of bacteria and even larger organisms such as fish have started to be developed over the last few years in order to examine the order, self-organisation and pattern formation within these systems, although direct correlation and comparison to real-world situations has been limited.

This project will use the theories and modelling techniques of liquid crystal systems and apply such modelling techniques to the area of anisotropy and self-organisation derived from active agents. The research will involve a continuum description of the fluid, using equations similar to the classical Navier-Stokes equations, as well as both the analytical and numerical solution of ordinary and partial differential equations.

Flow of groundwater in soils with vegetation and variable surface influx Groundwater is the water underneath the surface of the earth which fills the small spaces in the soil and rock and is extremely important as a water supply in many areas of the world. In the UK, groundwater sources, or aquifers, make up over 30% of the water used, and a single borehole can provide up to 10 million litres of water every day (enough for 70,000 people).

The flow of water into and out of these aquifers is clearly an important issue, more so since current extraction rates are using up this groundwater at a faster rate than it is being replenished. In any specific location the fluxes of water occur from precipitation infiltrating from the surface, evaporation from the surface, influx from surrounding areas under the surface, the flow of surface water (e.g. rivers) into the area, and the transpiration of water from underground directly into the atmosphere by the action of rooted plants.

This complicated system can be modelled using various models and combined into a single system of differential equations. This project will consider single site depth-only models where, even for systems which include complicated rooting profiles, analytical solutions are possible, but also two- and three-dimensional models in which the relatively shallow depth compared to the plan area of the aquifer can be utilised to make certain "thin-film" approximations to the governing equations.

We wish to develop innovative methods for modelling high-dimensional time series. Practical time series data, including both continuous-valued and discrete-valued data such as climate record data, medical data, and financial and economic data, are used for empirical analysis.

Models for forecasting multivariate conditional mean and multivariate conditional variance (volatility) are concerned. Techniques for dimension reduction, such as dynamic factor analysis, are used.

The estimation of models for panel data analysis and the option valuation with co-integrated asset prices is discussed.

Nowadays people often meet problems in forecasting a functional. A functional may be a curve, a spatial process, or a graph/image. In contrast to conventional time series analysis, in which observations are scalars or vectors, we observe a functional at each time point; for example, daily mean-variance efficient frontiers of portfolios, yield curves, annual production charts and annual weather record charts.

Our goal is to develop new models, methodology and associated theory under a general framework of Functional Time Series Analysis for modelling complex dynamic phenomena. We intend to build functional time series models and to do forecasting.

When the true economic system consists of many equations, or our economic observations have a very high dimension, one may meet the ``curse of dimensionality" problem. We try to impose a common factor structure to reduce dimension for the parametric and nonparametric stability analysis of a large system. Replacing unobservable common factors by principle components in parametric and nonparametric estimation will be justified.

In contrast to conventional factor models which focuses on reducing dimensions and modeling conditional first moment, the proposed project devotes attention to dimensional reduction and statistical inference for conditional second moments (covariance matrices). The direct motivation lies in the increasing need to model and explain risk and uncertainty of a large economic system.

The other distinctive point is that the proposed project considers factor models for high frequency data. A key application is the analysis of high dimensional and high-frequency financial time series, although the potential uses are much wider.

Marine ecosystems are generally composed of large numbers of species of widely varying sizes, ranging from unicellular species, through zooplankton and up to large fish and whales. The distribution of the total biomass of all species by size is known as the biomass spectrum.

In addition to the size variation due to differing characteristic sizes of the different species, and unlike the case in terrestrial systems, individuals themselves often undergo increases in body size of several orders of magnitude, from small eggs and larvae at a few millimetres in length up to large adults at the metre length scale.

At different parts of its life cycle an individual will be present at different parts of the biomass spectrum. Despite this apparent complexity it has long been known from field observations that biomass spectra show many regularities. In particular the logarithm of biomass density is approximately linearly related to the logarithm of body length with negative slope. Moreover, the slope of the spectrum is potentially sensitive to environmental and anthropogenic perturbations, for example the removal of large fish due to commercial fishing.

For these reasons biomass spectra have gained currency in recent years as a tool for studying the integrated ecosystem impacts of climate change and human exploitation of the seas.

The relative simplicity of representing the entire ecosystem as a size-structured spectrum has also permitted the development of mathematical representations in terms of partial differential equations, and these can be used to make predictions about ecosystem level responses to fishing and environmental change.

To date most modelling efforts have focused on steady state analyses of biomass spectra, which may be compared to annually averaged size spectra from field observations. In temperate shelf seas such as the North Sea, however, the biomass spectrum is not static but subject to seasonal impulses caused by increased primary production from phytoplankton in the spring bloom, and by seasonal reproduction by zooplankton and fish. This is known from both observational and mathematical models to induce annual ripples in the biomass distribution that propagate up the spectrum, gradually attenuating at larger sizes.

Fish larvae can exploit this by growing in size at a rate that allows them to feed near the peak of the wave, a phenomenon that has been dubbed by John Pope and his co-workers “surf-riding the biomass spectrum”. This project will focus on developing existing models to better represent these processes. In order to test the models the parameters will be estimated using Bayesian inference methods by fitting to a variety of data sets, including a long term (multi-annual) high temporal resolution (weekly) data from the North Sea.

Many fish populations worldwide have been heavily exploited and there is accumulating evidence from both observational and theoretical studies that this harvesting can induce evolutionary changes. Such responses can affect the stock sustainability and catch quality, and so there is a recognized need for new management strategies that minimise these risks. Most results suggest that high mortality on larger fish favours early maturation.

However, recent theoretical work has shown that trade-offs between growth and maturation can lead to more complex evolutionary responses. Surprisingly, harvesting large fish can select for either late or early maturation depending on the effect of maturation on growth rate. To date most theoretical studies have used evolutionary invasion analyses on simple age-based discrete-time models or on continuous-time coupled ODE representations of size structure.

In common with many generic models of fish population dynamics, population control occurs by unspecified density-dependence at settlement. While these simplifications carry the advantage of analytical tractability, the analysis assumes steady state populations. This, together with the stylised life-histories precludes comparing model results with field data on secular changes in size-distributions an sexual maturity.

The work in this project will develop a new generation of testable model for fisheries-induced adaptive changes with the potential to inform future management decisions. This will involve developing a consumer-resource model which a length-structured fish population feeding on a dynamic biomass spectrum.

Differently-sized fish will compete for food by exploiting overlapping parts of the food size spectrum. The population will be partitioned by length at maturity, and this will be the heritable trait under selection. The model will be used to explore how changes in mortality and food abundance affect the evolutionarily stable distribution of maturation lengths.

Comparisons with survey data on North Sea demersal fish will be used to assess whether the historical harvest rates are sufficient to explain growth rate changes as an evolutionary response. Finally the evolutionarily stable optimal harvesting strategies will be identified.

**Project Description: **

A fully-funded four year PhD studentship is available, starting on or after 1st October 2020. The intra-disciplinary research will be supervised by Dr Katherine Tant and undertaken primarily within the Continuum Mechanics and Industrial Mathematics group in the Department of Mathematics and Statistics at the University of Strathclyde. Please see https://www.strath.ac.uk/research/subjects/mathematicsstatistics/continuummechanicsindustrialmathematics/ for details of the group’s research. The student will also have the opportunity to interact with the Stochastic Analysis group and the Numerical Analysis and Scientific Computing group, facilitated by their second supervisors, Dr Mohammud Foondun and Professor Victorita Dolean Maini.

Ultrasonic non-destructive evaluation (NDE) is critical for the structural assessment of the UK's aging industrial infrastructure, as well as for the monitoring and quality control of modern additive manufacturing methods. It concerns the practice of transmitting mechanical waves through a solid object and subsequently using the reflected wave data collected on its surface to ‘see the unseen’; that is to create an image of the object’s interior which highlights any embedded defects. Mathematically, this is known as an inverse problem. This project will focus on developing new ultrasonic tomography techniques for NDE based on the principles of Bayesian inference, which provide a convenient mathematical framework to estimate the joint conditional probability distribution of the spatially varying material properties of an object given some observed boundary measurement data (this is the *posterior* distribution). Typically, Markov chain Monte Carlo sampling methods are used to obtain numerical approximations of the true posterior distribution, but these approaches are often computationally intractable for high-dimensional parameter spaces (such as those present in ultrasonic tomography problems). This project will examine Variational Bayesian (VB) approaches instead, where the Bayesian inverse problem is reformulated as a more computationally efficient deterministic optimisation problem. Although the project will focus primarily on NDE applications, the techniques developed will be applicable across many sectors, including medical imaging and seismology.

This is a very exciting project which will allow the student to work at the interface between mathematics and industry. The student will attend regular research seminars and events within the Mathematics and Statistics department and the Centre of Ultrasonic Engineering at Strathclyde, and so will have many opportunities to interact with a multi-disciplinary team and develop both technically and professionally.

Applicants should have, or be expecting to obtain in the near future, a first class or good 2.1 honours degree (or equivalent) in mathematics or a mathematical science.

**Funding notes**: The studentship covers full tuition fees and a tax-free stipend for four years starting in October 2020. Funding is only available to UK nationals and to EU nationals who have lived in the UK for three years prior to the start of the studentship.

**Deadline **Sunday 28^{th} June

The behaviour of sessile droplets is an area of very active international research, with new publications appearing on an almost daily basis and entire conferences now dedicated to the topic. Over the last decade, Professor Stephen Wilson has collaborated very successfully with Professor Khellil Sefiane from the School of Engineering at the University of Edinburgh (and a Visiting Professor at the University of Strathclyde) of a variety of practically important fluid-dynamical problems, including evaporating droplets, bubble dynamics, self-rewetting fluids, and anti-surfactants.

The aim of the present project is to build on the proposed supervisors’ previous work on evaporating sessile droplets to explore two exciting new aspects of this scientifically and practically important problem.

Very recent work by Sefiane and his collaborators have provided the first comprehensive experimental investigation of vapour absorption by sessile droplets of a desiccant liquid (i.e. one which draws moisture from the air). The mathematical modelling and analysis of such systems is an exciting and challenging open problem.

In practice, droplets almost never occur singly, but, due to the inherent complexity of the multiple-droplet problem, so far very little work has been done on the interactions between evaporating droplets. The aim of the proposed work is to investigate the fascinating but virtually unexplored subject of the collective behaviour of large arrays of small sessile droplets. Interactions between the droplets (in particular, so-called “shielding” effects) are expected to lead to very different collective behaviour compared to that of isolated droplets.

The student will join a lively and mutually supportive cohort of fellow PhD students within the Continuum Mechanics and Industrial Mathematics (CMIM) research group.

You should have (or expect to have) a UK Honours Degree (or equivalent) at 2.1 or above in Mathematics, Mathematics and Physics, Physics or a closely related discipline with a high mathematical content. Knowledge of Continuum Mechanics and mathematical methods (such as asymptotic methods) for the solution of partial differential equations is desirable (but not essential, particularly for overseas applicants).

Informal enquiries can be made to the lead supervisor, Professor Stephen K. Wilson, Department of Mathematics and Statistics, University of Strathclyde, Glasgow at s.k.wilson@strath.ac.uk and/or +44(0)141 548 3820. However, formal applications must be made via the online application procedure which can be found at

https://www.strath.ac.uk/courses/research/mathematicsstatistics/

remembering to list the title of the project as “Mathematical Modelling and Analysis of Sessile Droplets” and Professor Stephen K. Wilson as the first supervisor. There is no need to include a detailed research plan, but a brief outline of your relevant experience (if any) and your motivation for choosing this project would assist with the selection procedure.

This project will be supported by a Research Excellence Award (REA) Studentship funded by the University of Strathclyde with the support of the University of Edinburgh.

Unfortunately, funding rules mean that this studentship is only open to UK and EU students, and not to EEA or International students. Moreover, EU students are only eligible for the stipend element of the studentship if they have been resident in the UK for 3 years, including for study purposes, immediately prior to starting their PhD.

In recent years there has been an explosive growth of interest in the behaviour and control of fluids at small (typically sub-millimetre) scales motivated by a range of novel applications including ink-jet printing and lab-on-a-chip technologies.

Much of the most exciting current research concerns the interaction between fluids and both rigid and flexible structures at small scales, and so the aim of the present project is to use a judicious combination of asymptotic methods and judiciously chosen numerical calculations to bring new insight into the behaviour of a variety of novel fluid-structure interaction problems in microfluidics.

In the last decade or so there has been an explosion of interest in droplet evaporation, driven by new technological applications as diverse as crop spraying, printing, cooling technologies such as heat pipes, and DNA micro-array analysis.

One particularly interesting aspect of this problem which has thus far received relatively little attention is that of fluids whose surface tension exhibits a local minimum with temperature, known as self-rewetting fluids, a property that can have a profound effect on the dynamics of droplets on heated substrates.

The aim of the project is to build on the existing literature on conventional surface-tension-gradient driven spreading and droplet drying (see, for example, the references to some of our recent work on these problems given below) to bring new physical insight into this challenging scientific problem, and hence to harness the novel properties of self-rewetting droplets in a range of applications.

The project will be a collaboration colleagues at the University of Edinburgh who will be undertaking a parallel series of experimental investigations on this problem which will be key to the successful outcome of the project.

Dunn, G.J., Wilson, S.K., Duffy, B.R., David, S., Sefiane, K. “The strong influence of substrate conductivity on droplet evaporation” J. Fluid Mech. 623 329-351 (2009)

Dunn, G.J., Duffy, B.R., Wilson, S.K., Holland, D. “Quasi-steady spreading of a thin ridge of fluid with temperature-dependent surface tension on a heated or cooled substrate” Q. Jl. Mech. appl. Math. 62 (4) 365-402 (2009)

The MSP group engages in research at the frontier of Mathematics, Physics and Theoretical Computer Science with particular strengths in category theory, type theory, logic, functional programming and quantum computation. Fundamentally, we want to change the world around us and believe we have the mathematical ideas to do so. If you want to help us achieve this, look us up at http://msp.cis.strath.ac.uk.

If you would like to discuss opportunities to do graduate work with the MSP group then please contact Dr. Anders Claesson at anders.claesson@strath.ac.uk