Mathematics Student Projects
Below is a list of potential supervisors and suggested research areas for the level 6 project modules MAT-30016 (15 Credit Project) and MAT-30043 (30 Credit Project) as well as for the compulsory level 7 project module MAT-40003 (60 Credit Project).
Each supervisor has indicated whether they can supervise projects at just level 6 or at both levels 6 and 7. Some supervisors have indicated potential titles, while others have chosen to give more details about possible areas of research and have included videos and other media on their research areas. Once you have looked at this content, and prior to selecting potential supervisors, students are encouraged to have an informal discussion with potential supervisors about what possible projects might involve.
While ideally we would like to give every student their first choice of project supervisor, each supervisor has a finite capacity to supervise students and in some cases the number of students wishing to work with a particular supervisor may exceed this capability. Unfortunately, this means that it is unlikely that every student will be able to have their first choice of supervisor.
To assist with the allocation of project supervisors, we kindly ask students to complete a short online form in which they select their first, second and third choice of supervisor. Additionally, we ask students to rank their preferred research area.
Where the number of students selecting a supervisor exceeds their capacity to supervise students, we will first allocate the compulsory MAT-40003 projects, followed by the 30-credit project module MAT-30043 and then the 15-credit project module MAT-30016. Once the capacity of a supervisor has been reached, we will allocate based on students' second and third choices of supervisor. This allocation process will also be done in discussion with supervisors to ensure a good match of students, supervisors and projects.
Once you are ready, please submit your supervisor selection by clicking on this link.
Only one submission per student is allowed. The deadline for submission is 1pm 4th June.
For some examples of recent final year project work please see here.
List of supervisors
Project titles and summary
Project 1: Mathematical models used for modelling COVID-19 data: a literature review
The ongoing COVID-19 pandemic has imposed an unprecedented global health crisis since the 1918 Influenza pandemic. It has already caused over 124 million confirmed cases and 2.7 million deaths worldwide, and 4.3 million confirmed cases with 126,000 deaths in UK. Mathematical models have played an important role in the ongoing crisis as they have been used to inform public policies and planning (such as future projection of resources and health care needs etc.). Mathematical models work under various domains and these domains in the model may differ from disease to disease. These models are useful in hypothesis testing, assessing the best and worst scenarios, evaluating the impact of various policies as well as pharmaceutical and non-pharmaceutical interventions on disease outbreak. Therefore, in this project, we will review mathematical models that were used to support the ongoing planning, understand diseases dynamics, and response efforts.
Project 2: Using meta-analytic approach to estimate average length of hospital stay and incubation period among COVID-19 patients
The COVID-19 pandemic has increased an unprecedented pressure on health systems, with rapidly increasing demand for healthcare in hospitals and intensive care units (ICUs) worldwide. Hospital related care can vary from general ward admissions to high dependency units with oxygen support to intensive care where patients may be intubated for mechanical ventilation. The length of hospital stay may depend on several factors such as the level of care required, and the geographic settings due to different COVID-19 care guidelines. Likewise, reliable estimates of the incubation period (i.e., period from getting infection to first clinical symptom) are important for decision-making to control the infectious diseases in human populations. Knowledge of the incubation period can be used directly to inform decision-making around infectious disease control. Therefore, in this project, we will aim to estimate average length of hospital stay and incubation period among COVID-19 patients from studies reporting these estimates using meta-analytic approach.
Project 3: Estimating sample mean and standard deviation from the sample size, median, range and/or interquartile range
Researchers often pool the results of the sample mean and standard deviation from a set of similar studies in meta-analysis. However, sometimes a number of studies nay report the median, the minimum and maximum values, and/or the first and third quartiles. Hence, in order to combine results, we need to estimate the sample mean and standard deviation for such studies. A number of methods have been suggested in the literature for the estimation of mean and variance. Therefore, in this project, we will explore ways to estimate mean and standard deviation from given median, range and/or interquartile range (IQR), and sample size. We will also compare methods suggested in the literature.
Project 4: Long-term effectiveness of corticosteroid injection versus night splints for carpal tunnel syndrome: secondary analysis from a randomised controlled trial
Carpal tunnel syndrome (CTS) is compression of the median nerve at the wrist. Clinical symptoms of CTS include pain, numbness, tingling, and weakened grip and functional loss in the hand and wrist. CTS is often treated in primary care with 36 patient consultations per 10,000 person years in 2013. Common conservative treatments for CTS are a corticosteroid injection or wearing a wrist splint at night. Currently there is limited evidence on the effectiveness of these treatments over a long period of time.
The INSTINCTS randomised controlled trial aimed to evaluate the clinical effectiveness of corticosteroid injection versus night splints for CTS in the short and long term in primary care patients with mild to moderate CTS. In the short term, injection provided better pain and function and reduced likelihood of insomnia at 6 weeks although no differences between treatment groups were found at 6 months.
The objective of this study is to establish whether the intervention of corticosteroid injection provides superior response to night splints over a two-year period. Outcomes considered are symptoms and function assessed by the Boston Carpal Tunnel Syndrome subscales, sleep disturbance, and healthcare resource.
A repeated measures analysis using data from the 4 follow-up points (6-week, 6-, 12-, and 24-months) will be undertaken using a longitudinal mixed effect model. Random mixed effects models will be used to examine the mean difference (for continuous outcomes) and odds ratio (for binary outcomes) in the two treatment groups over the whole follow-up period. The model will also include an interaction term between treatment and time point to assess mean difference/odds ratio at each time point. All analyses will be performed before and after adjustment for baseline variables, including age, gender, duration of symptoms, and index score and other imbalanced covariates between treatment groups.
Project 5: Factors associated with change in foot pain over time: a 7-year prospective study
Foot pain is common in the general population with prevalence reported between 20-32% and 10% have disabling pain. Foot pain has been shown to be associated with poor health such as mobility limitations and risk of falls. Studies examining foot pain tend to be cross-sectional in nature and little is known how foot pain changes over time and whether certain factors are associated with this change.
The Clinical Assessment Study of the Foot (CASF) is a 7-year population-based prospective study of adults aged ≥50 years. Eligible participants were mailed a Health Survey at baseline, 3- and 7- years on their general health and foot pain. Participants were asked to shade in a foot and ankle manikin comprising of 29 specific pain regions.
The objectives of this study are to (1) model the change in the number of pain sites over 7 years, (2) identify covariates that are associated with the number of foot pain sites over time.
Descriptive statistics would first describe the number of foot pain regions at each time point and explore ways one could reduce the number of pain sites for analysis. Ways to model change in number of foot pain regions over time will be considered. A repeated measures analysis using data from the 3 time points (baseline, 3- and 7 years) will be undertaken using a longitudinal random mixed effects model, modelling the association between time-invariant factors (such as age and sex) and time-varying factors (such as foot pain and function) with the number of pain sites.
1. Time series forecasting
Automatic forecasts of large numbers of univariate time series are often needed in business and other contexts. In this project, two automatic fore-casting algorithms have been introduced and Statistical software R is required to run the algorithm. The first algorithm is based on innovation state space models that underlying exponential smoothing methods. The second one is based on ARIMA models.
The algorithms are applicable to both seasonal and non-seasonal data, and are compared and illustrated using four real time series. The students will be required to apply both forecasting methods to the datasets they are interested in, present the results and discuss the corresponding backgrounds. This project will require some basic time series concepts such as stationarity, autoregressive moving-average models, autocovariance and autocorrelation functions. RStudio is also essential. After the project, you are supposed to know how to use R to run the data analysis.
2. Volatility estimation
Volatility estimation is an important step in the process of many financial decisions and a key input to many financial applications such as option pricing, investment and risk management. Traditionally, ARCH (Engle, 1982) and GARCH (Bollerslev, 1986) models are used to capture time-varying volatility. However, these models have a limitation that they do not allow past shocks to have a different effect on future conditional second moment. This asymmetric property of volatility is called Leverage Effect (LE). The student will be required to extend the existing GARCH models to explore the stylized facts of financial market volatility, fit suitable models to times series data (such as economic indicators, stock prices), investigate the estimated volatility, present and discuss the forecasting results. Statistical software R is required for this project. Of course, you can use other statistical software including Matlab, SPSS or C.
I propose two topics: logic and geometry.
1. Logic. This is a very wide field. You might start by looking at the course materials for the third-year Logic module we used to offer: http://www.scm.keele.ac.uk/staff/p_fletcher/logic/ and have a look at the textbooks recommended there. There is a large section of logic books in the library, most of them too advanced for you, but you can find something accessible if you dig. You will begin by studying predicate calculus - this is a simple formal system capable of expressing all mathematical arguments. From there, there are several directions a logic project can take, depending on your interests: (i) a mathematical direction, towards Godel's theorems on completeness and incompleteness of formal logical systems, and questions of decidability or computability; (ii) a philosophical direction, i.e., the foundations of mathematics (how formal logical systems can be used to answer philosophical questions about the nature of mathematics); (iii) set theory (a fundamental theory on which all of mathematics can be based), and transfinite arithmetic; (iv) applications to the physical world (logical systems for reasoning about time, or about parts and wholes, or about necessity and possibility, or about programs running on a computer). You can see my own research interests at https://www.scm.keele.ac.uk/staff/p_fletcher/home/ Please have a look around the subject and try to get a sense of where your interests lie, and then we can meet to discuss your ideas. You should do some serious reading over the summer, so that in September you have a fairly definite idea of the topic you will be concentrating on. Because of the amount of preliminary material that has to be got through to get up to speed, this topic is probably more suitable for a 30- or 60-credit project than a 15-credit project.
2. Geometry. This is a vast area with an ancient history. Possible topics to specialise in include - axiomatic geometry (deriving geometric theorems from axioms), - non-Euclidean geometry (the geometry of the sphere or the projective plane), - applications of non-Euclidean geometry to general relativity (tensor calculus), - projective geometry (an abstract kind of geometry with no distances or angles), - topology (surfaces and so on), - algebraic topology (using group theory to describe surfaces). Look for some accessible books in the Library. Most topology books are too advanced, but you might find the following suitable: S.V. Ault (2018) Understanding Topology: A Practical Introduction. Baltimore: Johns Hopkins University Press. M.A. Armstrong (1979) Basic Topology. London: McGraw-Hill. B. Mendelson (1968) Introduction to Topology. Second Edition. Boston: Allyn & Bacon. D. Roseman (1999) Elementary Topology. Upper Saddle River, NJ: Prentice-Hall. Please have a look around the subject and try to get a sense of where your interests lie, and then we can meet to discuss your ideas. You should do some serious reading over the summer, so that in September you have a fairly definite idea of the topic you will be concentrating on.
(1) Machine learning and its applications
(2) The mathematics of inflation of party balloons
(3) Vibrations, waves, and their surpression
Projects for MAT-30016 or MAT-30043
1. Application of advanced methods to the simple pendulum equation
A swinging pundulum is one of the simplest dynamical systems, but it can be used to introduce methods that are widely used in research, in particular, perturbation methods. A variety of these methods can be applied to the pendulum equation, and their accuracy compared with numerical solutions. The simple context helps us to understand how the methods work, and to see the relative advantages of different approaches. Computer algebra packages, like Mathematica, can be used to continue calculations to high order, and to explore series convergence, and methods to accelerate convergence.
MAT-30002 Nonlinear Differential Equations (strongly recommended)
2. Application of analytical methods to problems in epidemiology
The modelling of epidemics/pandemics often results in sets of nonlinear ordinary differential equations. Large complicated models usually require numerical solution, but in this project we consider relatively simple models and aim to obtain a deeper understanding of their behaviour by using analytical approaches. In particular, we introduce and use perturbation methods. Even these simple models reproduce the main qualitative effects seen, for example, in the covid-19 pandemic (the tricky thing is to make good quantitative predictions, which we do not attempt!) However, even the simplest models may have no exact solution, but perturbation methods can produce approximate formulae for solutions indicating the effects of various parameters on the disease dynamics. In this project we explore how different insights can be obtained by using different approximations.
MAT-30023 Mathematical Biology (strongly recommended, but you may also have encountered similar models in modelling modules)
MAT-30002 Nonlinear Differential Equations (recommended)
3. Bifurcation and chaos in nonlinear equations
Ever since Newton wrote down, and solved, the first differential equations, there has been extraordinary progress in understanding all sorts of systems. Over several centuries a picture of emerged of ‘predictable’ systems modelled by differential equations, and ‘unpredictable’ systems requiring statistical modelling. The discovery in the 1960s that supposedly predictable differential equations could show unpredictable ‘chaotic’ dynamics revolutionized our understanding of all kinds of systems. In this project we explore routes to chaos, where, as one or more parameters are varied in a set of coupled differential equations, solutions evolve from predictable to unpredictable behaviour.
MAT-30002 Nonlinear Differential Equations (strongly recommended)
4. Hydrodynamic instabilities
When fluid flows, e.g. an ocean circulation current, like the gulf stream, or around an aerofoil, like a wind turbine, then it might, or might not, become turbulent. This can have a huge impact on heat transfer affecting weather and climate, or on the performance of a device. However, whether the flow is on a planetary scale, or in, or around, a piece of technology, there are common fundamental issues concerning whether a given flow is stable or unstable. Although the governing equations (NavierStokes equations) are notoriously challenging, it turns out that, remarkably, for some flows, they can be reduced to low-order, constant coefficient, linear ordinary differential equations, that can be solved with first year calculus methods! In this project we use these methods to explore the stability of some simple flows.
MAT-30004 Fluid Mechanics (strongly recommended)
MAT-30011 Waves (recommended)
Project for MAT-40003
5. Hydrodynamic instabilities
When fluid flows, e.g. an ocean circulation current, like the gulf stream, or around an aerofoil, like a wind turbine, then it might, or might not, become turbulent. This can have a huge impact on heat transfer affecting weather and climate, or on the performance of a device. However, whether the flow is on a planetary scale, or in, or around, a piece of technology, there are common fundamental issues concerning whether a given flow is stable or unstable. Although the governing equations (Navier-Stokes equations) are notoriously challenging, it turns out that, remarkably, for some flows, they can be reduced to low-order, constant coefficient, linear ordinary differential equations, that can be solved with first year calculus methods! In this project we use these, and more advanced, methods to explore the stability of some simple flows.
MAT-30004 Fluid Mechanics (strongly recommended)
MAT-40004 Hydrodynamic Stability Theory (strongly recommended)
- Parametric analysis of a viscoelastic oscillator.
- Harmonic vibrations of composite rods.
My research is in the area of computational mathematics/computational mechanics. I develop accurate computational methods for the accurate solution of partial differential equations with application to practical problems in engineering and physics including electromagnetism, fluid flow and mechanics. I have particular expertise in computational schemes such finite elements and finite differences and often develop MATLAB and Python codes as well as using existing libraries.
I’m also interested in the computational solution of problems involving locating and identifying hidden targets (eg landmines, security screening at transport hubs and public events and medical imaging). The solution of such problems often combines applied mathematical modelling and the application of modern data science techniques such as classification using machine learning.
To get an overview of one of my current research projects please see a recent video available here.
I can supervise projects related to the above areas and these will typically involve computational mathematics and its interface to data science. Projects can range from practical applications and coding through to a deeper mathematical understanding of the numerical schemes and/or some combination of these. Please get in touch for an informal discussion.
Available to Supervise projects in MAT-30016, MAT-30043 and MAT-40003
Chaos in 1D Maps (level 6 only)
Chaotic dynamics pose a significant barrier to making predictions and accurately modelling many physical systems. Coming up with simple systems to investigate complicated phenomena is a key theme in applied mathematics. To this end this project will study the remarkable case of chaos in very simple 1D recurrence relations, like the logistic map, and consider typical ways of characterising chaos in these equations. Such chaos also results in the unexpected formation of fractals; self-repeating patterns which famously appear in nature (snowflakes, ferns, clouds). A project on this topic could link with an investigation of fractals.
The Navier-Stokes equations for an incompressible, viscous, Newtonian fluid are used for a multitude of engineering and environmental applications, however no proof exists that the equations will exhibit well-defined solutions from a suitably smooth initial condition. This question forms one of the famous “Millennium Prize” million dollar problems in mathematics. The way in which such loss of regularity is seen is through a “finite-time blow-up”, i.e. within a finite amount of time the solution becomes infinitely valued. One way to direct analysis of the equations is with well-designed numerical solutions and careful analysis of the results; however, this is difficult as singularities are extreme conditions to resolve and track. It has been shown that novel transformations of the governing equations can improve the accuracy and validity of such experiments. This project will study simplified PDEs and determine effective ways to diagnose singularities in numerical experiments.
Turbulence in a fluid is often defined as some form of “spatiotemporal chaos” meaning it exhibits complexity in space as well as time. This project will consider the difficulties in modelling systems which have this two-fold complexity. This could involve a number of avenues; how do we numerically solve the equations accurately? How do we describe or characterise spatiotemporal dynamics? How does turbulence arise and sustain itself? Does any coherence remain buried within the disorder? Note: all projects will involve some computational element, usually in Python. Other topics related to chaos, nonlinear dynamics, numerical solutions & fluid mechanics can be offered, please get in touch if there is something specific of interest.
Mathematical models of real-world problems
(Level 6 15-credit project only; not available to students registered for MAT-30051 – Mathematical Modelling)
Mathematical models are useful to understand real-world behaviour relevant to a variety of problems relevant to industry, commerce and the environment. Students are encouraged to discuss with me real-world problems that you may be interested in developing mathematical models for. This will typically involve using analytical and computational techniques for the solution of ordinary and partial differential equations. Previous projects include models for infectious diseases, pattern formation in biological systems (tiger stripes and leopard spots), economic and business cycles, options pricing using the Black-Scholes model and capital asset pricing model.
Fluid dynamical instabilities in thin-film flows
(Level 7 project only)
Fluid dynamical instabilities in thin-film fluid flows are ubiquitous in a wide variety of problems, e.g., finger-like patterns are observed when coating a wall with paint (see figure on the left), advancing lava flows develop toe-like protrusions (see figure on the right), to name a few.
These spatial instabilities are unwanted, e.g., a fingering instability can prevent uniform coating of a wall, branching of lava flows enhances its spreading, which can lead to death and can cause damage to infrastructure and buildings. It is vital that one better understands the mechanisms driving fluid instabilities and develop strategies on how to control or eliminate them. Motivated in part by the above applications, the project will investigate model thin-film flow problems incorporating physical effects such as gravity, electric and magnetic fields, chemicals, etc. The main focus will be on the theoretical modelling including asymptotic and analytical techniques, and stability analysis combined with numerical simulations to elucidate the mechanisms underlying fluid dynamical instabilities in thin-film flows.
Adaptive Numerical Methods for Partial Differential Equations
(Level 7 Project Only)
Many evolutionary partial differential equations (PDEs) develop structure as the solution evolves. For example, a travelling front is observed in PDEs modelling gravity-driven flows, such as lava and glacier flows, and avalanches. Numerical methods to solve such problems often have difficulties in accurately resolving such structures due to their localised nature. Amethod to overcome this is to use some sort of adaptive approach in which either the number of mesh points used to solve the PDE is increased or the local density of themesh points is increased, particularly, near suchstructures.Moreover, this adaptive mesh would also be required to move along with the structure as it evolves in time. The project will develop the R- adaptive process in which mesh points are moved to areas of the solution where greater resolution is required. This method has shown great promise, but a systematic analysis of its performance and a comparison with other adaptive Methods (such as the h-adaptive or h-p adaptive method) remains to be done. This will be then applied to canonical evolutionary PDEs such as the Burgers and Korteweg-De Vries (KdV) equation.
See here for details.
The proposed projects are in the general area of mathematical elastodynamics.
1. Elastic surface waves Rayleigh and Rayleigh-type waves form an important class of waves propagating in elastic solids, having various engineering applications, including seismic protection, non-destructive testing and many others, In our research we study the equations describing propagation of such waves and deduce associated physical interpretations.
2. Moving load problems A class of problems in which the applied loading is moving along the boundary of an elastic solid is considered, having natural applications to transport. An interesting feature worth studying is associated with critical (resonant) speeds of the moving load as well as near-critical ones.
Generally I am interested in studying waves and vibrations in solids. In particular, I could offer the following projects:
1. Edge waves in structures (localized waves propagating in thin elastic structures)
2. Dynamics of high contrast composites (structures composed of parts with highly different material properties, having applications in e.g. laminated glass, insulation panels, etc)
The titles of four possible projects are:
1. 'Black holes' for inertia-gravity waves in the uniformly stratified fluid at the critical latitude
2. Generation of inertial waves in the subsurface layer of the ocean by a drop of wind
3. Nonlinear evolution of edge waves
4. Nonlinear seiches
For more details of the projects see Projects 2021 Victor Shrira.pdf
- Inertia-gravity waves beyond the inertial latitude.pdf
- Long nonlinear surface and nternal gravity waves.pdf
- Near-inertial waves in the ocean.pdf
- Near-inertial waves.pdf
- Nonl Vorticity Waves.pdf
- Observations of meteotsunami on the Louisiana shelf.pdf
- On the Nature of Near-Inertial Oscillations.pdf
I will be able to supervise projects in algebra and/or number theory. Many of the projects I have supervised in the past have given students the opportunity to explore a topic that extends the material they have studied at level 6, or to assemble the machinery required to prove a result that was too difficult to prove at level 6. This typically requires students to learn new material from textbooks and journal articles, and to synthesise this into a free-standing account that is accessible to their peers. Students can demonstrate their understanding by providing their own descriptions, proofs, or examples. Many of the projects listed below have quite steep prerequisites; I urge potential students to discuss them with me before completing any forms!
Examples of titles might include:
Proofs of the Law of Quadratic Reciprocity: This famous and beautiful result appeared in level 6 Number Theory without proof. There are over 200 different proofs in the literature, drawing on a wide range of different mathematical techniques. It would be interesting to compare and contrast a selection of these.
The structure of the Zn*: In level 6 Number Theory we saw that an understanding of the group Zn* (for a natural number n) is important for solving problems in number theory and cryptography. We derived a formula for the order of this group, and stated the values of n for which it is cyclic. This project might begin with a proof of that result, and move on to study how the structure of this group is connected with the prime factorisation of the natural number n.
Module theory: A module is a generalisation of a vector space in which coefficients are drawn from a more general ring rather than a field. In general they are less "well behaved" than vector spaces, but they are important because they appear in a wider variety of contexts. This project might explore this "bad" behaviour, survey some examples, and prove some of the structural results that hold in some cases.
Error correcting codes: An error correcting code is a system for increasing the reliability of communication over a channel in which there is a risk of noise distorting the message. Examples include QR codes, CDs and DVDs, and satellite communication. The mathematics of these systems in a mixture of linear algebra and modular arithmetic. This project might start with the basics of these systems and lead on to the construction of some famous and powerful examples.
Quaternion Algebras: The quaternions are a noncommutative number system that extends the complex numbers. They form a 4-dimensional vector space over the real numbers with basis 1,i,j,k, where i^2=j^2=k^2=ijk=-1. They have applications in geometry and physics, and appear in many classification theorems in algebra. More generally, a quaternion algebra over a field F is a four dimensional vector space over F with basis 1,i,j,k, where i^2=a, j^2=b, k^2=-ab, for some a,b in F. It is interesting and important to understand how properties of these systems depend upon the field F and the elements a,b.
Modular Forms: A modular form is a special type of function on the complex numbers that exhibits a large amount of symmetry. Properties of modular forms are very important in number theory; for example, they played a crucial role in the proof of Fermat's Last Theorem. This project would blend complex analysis, group theory, and number theory to explore some of these properties.
Dirichlet's theorem on primes in arithmetic progressions: this famous and beautiful theorem states that if a,b are coprime natural numbers then there are infinitely many prime numbers of the form an+b (with n in the natural numbers). The proof is an intricate mixture of group theory and complex analysis.
Additive number theory: This topic includes results such as Lagrange's Four Square Theorem (every nonnegative integer is the sum of at most 4 square numbers) and conjectures such as the Goldbach conjecture (every natural number greater than 5 can be expressed as the sum of 3 prime numbers). One direction for the project would be to survey some of these results and conjectures, and the techniques used to attack them. These techniques are often relatively elementary, but the arguments can be complicated and intricate.
Recently a significant amount of interest in research and industry has been in finding ways to redirect and control waves using composite 'meta-materials'. Projects in this area would involve studying the mathematical properties of ordinary materials and the treatment of junction conditions to find a way to produce desired wave behaviours. This could involve the redirection of waves in a particular direction or the suppression of certain wave frequencies.