# Mathematics & Statistics Available PhD projects

# 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. **

The goal of this project is to explore fundamental structural questions concerning sets of permutations that avoid a single pattern. The study of these combinatorial objects was initiated by Donald Knuth in the 1960s when he investigated which permutations could be sorted by being passed through a stack.

#### Project Details

A *permutation class* is a set of permutations closed downwards under the subpermutation (or containment) order. See [1] for a brief introduction or [5] for a full exposition. A *principal* class is one that can be defined as avoiding a single permutation: Av(τ) = { σ : σ avoids τ }. This project will explore fundamental questions concerning the structure of permutations in principal classes. In particular, two important monotonicity conjectures will be addressed:

Inversion Monotonicity Conjecture [4]: For all *k* and every non-increasing permutation τ, the number of τ-avoiding *n*-permutations with *k* inversions increases as *n *increases.

Block Monotonicity Conjecture [3]: For all *n* and every skew-indecomposable permutation τ of length greater than 2, the number of τ-avoiding *n*-permutations with *m* skew blocks decreases as *m *increases.

For both of these, the smallest case for which the conjecture is open is τ = 1324. Resolving the Inversion Monotonicity Conjecture for 1324-avoiders would yield an improved upper bound on the growth rate of the class. Understanding Av(1324) is one of the primary open problems in permutation classes research [2].

[1] David Bevan, Permutation patterns: basic definitions and notation, 2015. https://arxiv.org/abs/1506.06673

[2] Bevan David, Robert Brignall, Andrew Elvey Price and Jay Pantone, A structural characterisation of Av(1324) and new bounds on its growth rate, *European Journal of Combinatorics*, 88:103115, 2020.

[3] Miklós Bóna, Supercritical sequences, and the nonrationality of most principal permutation classes, *European Journal of Combinatorics*, 83:103020, 2020.

[4] Anders Claesson, Vít Jelínek and Einar Steingrímsson, Upper bounds for the Stanley–Wilf limit of 1324 and other layered patterns,* Journal of Combinatorial Theory, Series A*, 119:1680–1691, 2012.

[5] Vincent Vatter. Permutation classes. In Miklós Bóna, *Handbook of enumerative combinatorics*, 2015. Preprint version: https://arxiv.org/abs/1409.5159

#### Eligibility

You should have (or expect to have) a UK Honours Degree (or equivalent) at 2.1 or above in Mathematics or a closely related subject with a high mathematical content.

The goal of this project is the exact enumeration of classes of permutations that can be plotted on monotone curves in grids. At present, an effective procedure is only known for so-called “skinny” classes. The general problem is challenging because permutations have multiple ways of being plotted on the curves.

#### Project Details

The monotone grid class Grid(M) consists of permutations that can be plotted on monotone curves in a grid. This grid is specified by the matrix M, all of whose entries are in {0, 1, –1}, where 1 and –1 denote monotone increasing and decreasing curves in a cell, and 0 denotes an empty cell. This project concerns the enumeration of these classes.

For a general overview of permutation classes, see Vatter’s recent survey [5]. An introduction to monotone grid classes can be found in [3, Chapter 2]. There is also an interactive demonstration, available from http://demonstrations.wolfram.com/PermutationGridClasses.

Associated with a grid class is a graph whose structure influences its enumerative properties. If the graph is acyclic, then the class has a rational generating function [1]. If the graph is unicyclic, it is conjectured that the generating function is algebraic; otherwise, it is believed that the class is D-finite [3, Chapter 4]. The exponential growth rate of any monotone grid class can be determined from its matrix [2, 4], but an effective procedure for exact enumeration is only known for “skinny” grid classes, those defined by a k × 1 vector [3, Chapter 3]; these are acyclic. The general problem is hard because permutations typically have multiple “griddings” (different ways of being plotted on the curves).

The project is likely to focus on classes defined by a k × 2 matrix, seeking procedures for enumerating acyclic and unicyclic classes of this sort, and in the latter case proving algebraicity.

[1] M. Albert, M.Atkinson, M. Bouvel, N. Ruškuc, V. Vatter. Geometric grid classes of permutations. Trans. Amer. Math. Soc., 2013.

[2] M. Albert, V. Vatter. An elementary proof of Bevan’s theorem on the growth of grid classes of permutations. Proc. Edinburgh Math. Soc., 2019.

[3] D. Bevan. On the growth of permutation classes. PhD thesis, 2015. https://arxiv.org/pdf/1506.06688

[4] D. Bevan. Growth rates of permutation grid classes, tours on graphs, and the spectral radius. Trans. Amer. Math. Soc., 2015.

[5] V. Vatter. Permutation classes. In M. Bóna, Handbook of enumerative combinatorics, 2015. Preprint version: https://arxiv.org/abs/1409.5159

#### Eligibility

You should have (or expect to have) a UK Honours Degree (or equivalent) at 2.1 or above in Mathematics or a closely related subject with a high mathematical content.

**Summary**

Solving well known physical problems modelled by partial differential equations by Machine Learning (ML) techniques remains a difficult task, and a lot of questions of a methodological nature need to be answered. The purpose of this project is to develop new solution methods at the interface of ML and scientific computing for coupled complex models in advanced manufacturing. This is a collaborative project with the Advanced Forming Research Centre (AFRC).

**Details**

The development of ML techniques where sparse data is supplemented with well-established physical and numerical models, has only recently gained attention giving rise to a new paradigm: Scientific Machine Learning (SciML). The synergy between these research fields was made possible thanks to the use of neural networks complemented in part with data and in part with mathematical models and physical laws. However, solving these problems by training neural networks suffers from many drawbacks (slow training, lack of theoretical results). This project will aim at developing faster solvers, provide theoretical insights of their behaviour and apply these to complex models in advanced manufacturing in collaboration with the Advanced Forming Research Centre (AFRC).

This is a very exciting project which will allow the student to work at the interface between computational mathematics and machine learning with a strong potential for industrial application. The student will attend regular research seminars and events within the Mathematics and Statistics department, and so will have many opportunities to interact with a multi-disciplinary team and international collaborators and develop both technically and professionally.

**Eligibility**

The ideal candidate will have a strong background in some of the following areas: linear algebra, numerical analysis, optimisation and/or machine learning. Experience in programming (e.g. MATLAB, Python or Julia) is highly desirable.

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. An MMath, MPhys or MSc degree is desirable.

**Funding**

Fully funded 3-year PhD studentship in the Department of Mathematics and Statistics at the University of Strathclyde, Glasgow, United Kingdom, jointly supervised by Prof Victorita Dolean and Dr Jennifer Pestana. The studentship covers full UK tuition fees and a tax-free stipend for 3 years starting on the **1 ^{st} March 2023**.

**Applications**

Applications can be made here.

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,

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.

The Strathclyde Centre for Doctoral Training (SCDT) in "Data-driven uncertainty-aware multiphysics simulations" (StrathDRUMS) is a new, multi-disciplinary centre of the University of Strathclyde, which will carry out cutting-edge research in data-driven modelling and uncertainty quantification for multiphysics engineering systems. StrathDRUMS will train the next generation of specialists to apply non-deterministic model updating, digital twin techniques, and advanced uncertainty treatments to real-world challenges in civil and aerospace engineering. In our research we aim to study complex systems such as aeroplanes, spacecrafts, buildings and bridges using rigorous mathematical concepts, formulations and computational methods.

We are pleased to announce an available funded 3.5 year PhD studentship project within the centre on “High-dimensional computations with applications to uncertainty quantification for multiphysics engineering systems”, supervised by Dr Yoshihito Kazashi in the Department of Mathematics and Statistics and co-supervised by Dr Sifeng Bi (Mechanical & Aerospace Engineering), Dr Marco de Angelis (Civil & Environmental Engineering) and Dr Michele Ruggeri (Mathematics and Statistics). In engineering applications, various types of uncertainty are modelled with a large number of parameters. However, working with such models yields high-dimensional problems, such as high-dimensional integration to compute expectations and high-dimensional approximation to construct surrogate models, that are computationally very challenging. The successful candidate will develop and analyse numerical techniques in computational uncertainty quantification. They will focus specifically on cutting-edge numerical analysis of high-dimensional integrals (that outperform Monte Carlo methods), function approximation, probability density function estimation as well as optimal solvers of partial differential equations that take uncertainty into account.

Although the student will be based in the Department of Mathematics and Statistics at the University of Strathclyde, they will be benefit from supervision by the wider SCDT team. According to the student interest and background, the research will be aligned and supported by one of StrathDRUMS partners, which include National Manufacturing Institute Scotland (NMIS), National Physical Laboratory (NPL), and UK Atomic Energy Authority (UKAEA). The student will thus be integrated within a vibrant and active multi-disciplinary research community with in-house training opportunities available across multiple faculties. In addition to undertaking cutting-edge research, the student will be registered for the Postgraduate Certificate in Researcher Development (PGCert), which is a supplementary qualification to develop core skills, networks, and career prospects.

The successful candidate will be expected to conduct high-quality research in the areas of computational uncertainty quantification, participate in relevant training activities and events provided by StrathDRUMS, disseminate research findings through publications and presentations, contribute to the wider research community through engagement and collaboration with other researchers.

We encourage applications from UK-based students for this position. However, we also welcome strong international students to apply, provided they are able to cover the difference between the home and international tuition fees on their own.

The studentship should start on 1 October 2023. The studentship will fund the annual Home tuition fees and a tax-free stipend for 3.5 years. The stipend rates are announced annually by UKRI. To give an idea, for the 2022/23 academic year, the annual UKRI stipend is £17,668.

Applicants should have, or be expecting to obtain soon, a first class or good 2.1 honours degree (or equivalent) in mathematics or in a closely related discipline with a high mathematical content. Excellent written and verbal communication skills, analytical and problem-solving skills, ability to work independently and as part of a team are essential. Programming skills and some knowledge of analytical and numerical methods in uncertainty quantification are desirable.

To apply, candidates should send their CV, transcript, and cover letter to y.kazashi@strath.ac.uk. Review of applications starts on 15 June 2023. Applications will be accepted until the position is filled.

The focus of this PhD would be on adapting existing graph alignment methods and developing new approaches so that they can handle large-scale (~1 million nodes), directed (i.e. nonsymmetric) networks and represents a potential breakthrough for many real-world applications. Furthermore, it would benefit from an expertise built upon a successful previous collaboration between Dstl and the University of Strathclyde.

Networks (or graphs) are a powerful tool to represent systems of interacting or related data. They can be used to understand a broad range of real-world applications: Internet of Things, biological systems such as protein-protein interactions, neuronal functions, food webs; social systems such as online social networks, phone calls, bank transfers, collaborations, physical contacts or genealogical relationships, to name just a few. Complex network theory thus provides a versatile and adaptive framework to model and analyse various systems from very different fields, making discoveries in this theory potential breakthroughs impacting many areas of application.

Graph alignment is concerned with the assessment of the similarity between different networks in order to identify common actors. Alternatively, differences can indicate anomalous behaviour or alternative pathways.

The project is a partnership between academia and industry and attracts generous funding and an enhanced stipend. As part of the project, the successful applicant will spend at least 3 months working at one of Dstl’s sites in the south of England. There will be opportunities to present the results of the research both nationally and internationally.

**Eligibility**

Students will require at least an upper second-class honours degree in Mathematics (or a very closely related area) by the time the project starts. Strength in linear algebra/graph theory is a plus, as is a good background in programming mathematical software.

**Funding**

Fully funded 4-year PhD studentship in the Department of Mathematics and Statistics at the University of Strathclyde, Glasgow, supervised by Dr Philip Knight, Dr Francesca Arrigo (both Strathclyde) and Dr Robert Keens (Dstl) The studentship covers full UK tuition fees and a tax-free enhanced stipend for 4 years starting on the **1 ^{st} October 2023**.

**Further informtion**

Further information can be obtained by contacting Dr Philip Knight (p.a.knight@strath.ac.uk).

**Applications**

Applications can be made here.

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.

Cholesteric liquid crystals are chiral systems which possess a spontaneously formed helical structure with the pitch in micron range which is important for various applications in optics and nanophotonics. In recent years the interest has shifted in the direction of lyotropic cholesterics which are the solutions of various chiral macromolecules, viruses or chiral nanocrystals. These systems are important for biology (for example, cholesteric states of DNA) and also provide some very useful natural anisotropic chiral materials.

Among the most interesting resources to explore are cellulose and chitin, key biopolymers in the plant and animal world, respectively. Both have excellent mechanical properties and can be extracted as nanorods with high degree of crystallinity.Both are also chiral. Molecular chirality of such nanorods is amplified into a helically modulated long-range ordered cholesteric liquid crystal phase when they are suspended in water.

The aim of this project is to develop a molecular-statistical theory of chirality transfer in cholecteric nanorod phase, determined by steric and electrostatic chiral interactions, and quantitatively describe the variation of helix pitch as a function of rod length, concentration, dispersity and temperature. The theory will be built upon the previous results, obtained for different cholesteric liquid crystals ((see some references to our work below).

The project will include a collaboration with two experimental group at the University of Luxembourg and the University of Stuttgart. These groups have an enormous expertise in the field of lyotropic liquid crystals.

[1] Honorato-Rios, C., Lehr, C., Sch¨utz, C., Sanctuary, R., Osipov, M. A., Baller, J. and Lagerwall, J. P.F.., Fractionation of cellulose nanocrystals: enhancing liquid crystal ordering with- out promoting gelation, *Asia Materials, *10, 455–465 (2018).

[2]. Dawin, Ute C., Osipov, Mikhail A. and Giesselmann, F. Electrolyte effects on the chiral induction and on its temperature dependence in a chiral nematic lyotropic liquid crystal J *of Phys. Chem. B*, 114 (32). 10327-10336 (2010)

[3] A. V. Emelyanenko, M. A. Osipov and D. A. Dunmur, ] Molecular theory of helical sense inversions in chiral nematic liquid crystals *Phys. Rev. E, *62, 2340 (2000)

Elastic constants of nematic liquid crystals, which describe the energy associated with orientational deformation of such anisotropic fluids, are among the most important parameters for various applications of liquid crystal materials. The elastic constants of nematic liquid crystals have been well investigated both experimentally and theoretically in the past. During the past decade a number of novel liquid crystals materials with unconventional molecular structure have been investigated and it has been found that these systems are characterised by the anomalous values and behaviour of the elastic constants. In particular, it has been shown that in the nematic phase exhibited by the so-called V-shaped bent-core liquid crystals two of the three elastic constants decrease nearly to zero with the decreasing temperature. This behaviour is still very poorly understood.

It should be noted that bent-core liquid crystals attract a very significant attention at present because they also exhibit a number of unusual novel phases with a nanoscale helical structure. It is now generally accepted that a transition into these unusual phases may be driven by the dramatic reduction of the elastic constants.

The aim of this project is the generalise the existing molecular-statistical theory of elasticity of nematic liquid crystals to the case of bent-core nematics, composed of biaxial and polar molecules, using the preliminary results obtained in recent years (see, for example, the references to some of our recent papers given below). Another aim is to explain the existing experimental data on the temperature variation of the elastic constants of bent-core nematic liquid crystals.

The project will include a collaboration with the experimental group at the University of Leeds and the theoretical group from Russian Academy of Sciences. These collaborations are very important for the success of the project.

[1] M. A. Osipov and G. Pajak, Effect of polar intermolecular interactions on the elastic constants of bent-core nematics and the origin of the twist-bend phase, The European Physical Journal E 39, 45 (2016).

[2] M. A. Osipov and G. Pajak, Polar interactions between bent-core molecules as a stabilising factor for inhomogeneous nematic phases with spontaneous bend deformations, Liquid Crystals 44, 58 (2016).

[3] S. Srigengan, M. Nagaraj, A. Ferrarini, R. Mandle, S.J. Cowling, M.A. Osipov, G. Pająk, J.W. Goodby and H.F. Gleeson, J. Mater. Chem. C, 2013, 6, 980

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.

A fully funded three-year PhD studentship under the supervision of Dr Michele Ruggeri, Dr Yue Wu and Prof. John Mackenzie (starting on 1 October 2023) is available in the Department of Mathematics and Statistics of the University of Strathclyde (Glasgow, United Kingdom). A recent technological innovation to increase the capacity of hard disk drives is heat-assisted magnetic recording (HAMR). Exploiting the fact that many magnetic properties of materials are temperature-dependent (for example, anisotropy, that is proportional to the strength of the magnetic field needed to write information on a disk, can be reduced by increasing the temperature), the basic idea of HAMR is to temporarily heat the disk via thermal pulses of a laser diode during writing. This reduces the anisotropy, which makes the material much more receptive to magnetic effects and allows to write to much smaller regions, increasing the overall capacity of the disk. The aim of this project is to advance the numerical analysis of the (stochastic) partial differential equations modelling HAMR. These include the (stochastic) Landau-Lifshitz-Gilbert and Landau-Lifshitz-Bloch equations, coupled with the Maxwell equations (modelling the magnetic field) and the heat equation (modelling the evolution of the temperature). We aim to develop structure-preserving finite element methods and study their rigorous convergence. This is a very exciting project which will allow the student to work at the interface between mathematical modelling, stochastic analysis and computational mathematics with applications firmly in sight.

The student will be a member of the Analysis group (Themes: Numerical Analysis and Stochastic Analysis) of the Department of Mathematics and Statistics. The project will provide training in mathematical modelling, partial differential equations, numerical analysis, scientific computing and stochastic analysis, thus equipping the student with highly desirable skills for working in either academia or industry. Further training will be provided for giving research talks, writing scientific papers and using computational software. Participation in seminars, workshops and conferences as well as research visits to the international collaborators will be strongly encouraged.

The studentship will fund the annual Home tuition fees and a tax-free stipend for three years. The stipend rates are announced annually by UKRI. For the 2022/23 academic year the annual UKRI stipend is £17,668.

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 in a closely related discipline with a high mathematical content. Programming skills and some knowledge of analytical and numerical methods for the solution of (stochastic) partial differential equations are desirable.

Informal enquiries should be sent to Dr Michele Ruggeri (michele.ruggeri@strath.ac.uk). Applications, including a motivation letter (max 1 page A4) and a curriculum vitae, should be sent should be sent to Dr Michele Ruggeri (michele.ruggeri@strath.ac.uk). The position will remain open until filled (a first screening of all applications will take place on 31 May 2023), so we recommend applying as soon as possible.

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.

This EngD will work collaboratively between the Centre for Ultrasonic Engineering (Prof. C. MacLeod), the Department of Mathematics and Statistics (Dr. K. Tant) and Rolls Royce. This project promises to be an exciting, fun and industrially relevant project, working alongside skilled engineers and scientists with state-of-the-art robotic equipment to deliver meaningful industrial change.

**Research Project**

Ultrasonic non-destructive evaluation (NDE) describes the practice of transmitting sound waves through solid objects and analysing the resulting scattered waves to construct images of the object’s interior, facilitating defect detection and materials characterisation.

The ultimate aim of this project is to investigate enhanced ultrasonic NDE of metal components directly at the point of manufacture, to deliver high-quality components right, first time. Importantly, this in-process approach requires the development of real-time data processing algorithms. Machine learning algorithms used in conjunction with models and simulations of wave propagation will be explored to facilitate this.

Specifically, this project will examine how to compensate for the effects that thermal gradients, complex build geometries and heterogeneous microstructures have on the probing ultrasonic waves. Three research questions will be addressed:

- Can complex and dynamic build geometries be accurately mapped out in near real-time using in-process ultrasonic inspection data, where extreme thermal gradients (generated by the manufacturing process) cause distortion of the expected wave paths?
- Can this knowledge of the component geometry, coupled with models of the thermal gradient be used to better constrain and drive the microstructure characterisation problem?
- Can the knowledge from steps i) an ii) be used to reliably image components as they are built?

The final deliverable of the project will be a fully automated in loop capability, to be demonstrated on test structures manufactured for aerospace and energy applications (with other industrial members applications supported as appropriate).

**Funding **

For UK students, Tuition Fees and a generous enhanced stipend of £26,168 tax-free per annum for up to 4 years. The student will also have access to substantial international travel and project funds.

**Research Environment**

The student will be based in the newly opened £2.1M Sensor Enabled Automation & Control Hub (SEARCH) Laboratory at the University of Strathclyde, working alongside a research team of over 35 researchers and PhD Students, while also having access to state of the art sensor, robotic and welding equipment. The student will work in collaboration and spend time on site working with the lead industry partner (Rolls Royce) to gain a greater appreciation of the specific industrial challenges and opportunity for automated inspection during fusion welding.

This EngD will be aligned to the EPSRC Centre for Doctoral Training in Future Innovation in Non-Destructive evaluation (FIND). The student will undertake specific industrial technical training courses (Ultrasonics, Welding and KUKA Advanced Robotic Programming) along with the University Research Development Program (RDP) to deliver training and development on traditional PhD activities such as presentations, conferences and technical writing.

In this project, we propose to harness the potential of artificial intelligence (AI) for efficient ocean climate forecast. We will examine physics informed machine learning (PIML) as a means of opening the ‘black box’ behind the AI system.

Conventionally, numerical modelling is widely used to provide ocean forecasts. These models involve simulating the ocean climate using mathematical equations and physical models, and exploit complex input data arising from various sources, such as satellites, in-situ measurements, and re-analysis models. However, these models are computationally expensive and time consuming to run, especially for long-term forecasts. In this studentship, we propose to harness the potential of artificial intelligence (AI) for efficient ocean climate forecast. We will examine physics informed machine learning (PIML) as a means of opening the ‘black box’ behind the AI system. This will be achieved by encoding the differential equations which govern the ocean climate into the learning algorithms, amplifying the value of the training data and optimising the exploration of the model space for improved efficiency. Furthermore the fusion of multi-modal training data from both high and low fidelity sources will be explored, facilitating predictions even in the scenarios where data is sparse.

The studentship will fund the annual Home tuition fees and a tax-free stipend for 42 months. The stipend rates are announced annually by UKRI. For the 2022/23 academic year the annual UKRI stipend is £17,668.

**Eligibility:**

We encourage applications from UK-based students for this position. However, we also welcome international students to apply, provided they are able to cover the difference between the home and international tuition fees on their own.

#### Qualifications:

Essential:

- Applicants should hold (or expect to get) a minimum of an upper second class honours degree or an MSc in mathematics, engineering, physical sciences, computer science or a related field.
- Excellent written and verbal communication skills
- Strong analytical and problem-solving skills
- Ability to work independently and as part of a team

Desirable:

- Prior knowledge or experience in mathematical modelling and/or Artificial Intelligence (AI)
- Programming experience in analytical languages such as MATLAB, Python or R
- A background in HPC-based computer modelling
- Experience in research, preferably in the form of a research project or dissertation

**Further informtion**

Further information can be obtained by contacting Dr Katy Tant (katy.tant@strath.ac.uk).

**Applications**

Applications can be made here.

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)