See examples of our student's project work in Mathematics
Student work
Some of our students have taken part in the University of Sussex Junior Research Associate (JRA) scheme, which aims to develop future research leaders. They reward academic excellence by supporting highachieving undergraduates to work alongside Sussex's top research faculty during the summer holiday.
Phoebe Valentine  Junior Research Associate (JRA) project in The Isodiametric Inequality
Mathematics BSc student, Phoebe Valentine (pictured above) exhibited her research and won 1st prize in the poster competition with her poster "How to measure a coffee splat".
Phoebe said: "What better way to spend your summer than being paid to delve into some fascinating mathematics?! When I approached supervisors, I did not have a specific project in mind. In fact, I initially said I wanted to do something related to topology. However, when I spoke to Dr Filippo Cagnetti, who was to become my supervisor, he suggested what ultimately became the focus of my JRA: The Isodiametric Inequality.
"This was my first contact with the world of measure theory as it is not part of the syllabus in years one and two. Of course, it is quite possible to hit the library under your own steam, but having the structure and support given by a supervisor is invaluable. Weekly meetings helped to propel the project forwards and resolve any issues that I encountered along the way.
"The final element of doing a JRA is, of course, the poster competition. This is a particular challenge for those doing a JRA in mathematics as we need to make a technical and alienating subject approachable for anyone. After submitting the poster, a handful of people are shortlisted to also give a presentation. This was a daunting but rewarding process which was excellent practice for my subsequent thirdyear project.
George Crowley  Junior Research Associate (JRA) project for numerically simulating curve shortening flow (CSF) with interesting boundary conditions
“I was lucky enough to have participated in the 2021 Junior Research Associate program (JRA), working with Prof Anotida Madzvamuse and Dr James Van Yperen from MPS, in which we worked on a project for numerically simulating curve shortening flow (CSF) with interesting boundary conditions. To put it briefly, CSF is a process that modifies a smooth curve in the Euclidean plane (2D) by moving points on the curve perpendicularly to the curve at a speed proportional to its curvature.
“From my project, I gained a wealth of experience in research and programming. With the help of Anotida and James, I was guided through some useful research papers on CSF and all the relevant material needed to numerically solve the partial differential equation(s) that model CSF. Working with Anotida and James was by far one of the most useful and best experiences I have had since joining Sussex, their wealth of knowledge and generosity with regards to development are unparalleled to any other experience I have had before and were paramount to the success of the project.
“Since the completion of my JRA, I was given the opportunity to continue working with James and Anotida for my BSc dissertation. Thanks to the knowledge gained and relationships made from the research associate program, I was able to work on a further research project created by Anotida within the area of data driven modelling, warranted by the outbreak of the coronavirus. “
Final year projects
The finalyear mathematical research project is undertaken by all Mathematics MMath students, where they have the chance to put their studies into practice and experience academic research.
Ezra Goldsmith  proving the existence of a bound for the Hausdorff dimension of an attractor.
The purpose of Ezra Goldsmith's (pictured above) MMath Project was to prove that a bound exists for the Hausdorff dimension of an attractor, in particular for the subsequent dynamical systems of suitably defined ordinary differential equations. Moreover, though both this result and its area of research have long been established, she wanted to increase the accessibility for those with only an undergraduate level of mathematical understanding and ability, making clear how any involved branches of mathematics are interlinked.
"To achieve this, I had to first establish the groundwork theory of dynamical systems  that is, systems which evolve over time  and their attractors, explaining afterward some different notions of dimension alongside how they're used here to characterise structural complexity. Then I justified why we proceed with the Hausdorff dimension over the rest, and conclude the report by proving the existence of a bound for this dimension of an attractor.
"What I easily enjoyed the most about this project was the opportunity to apply all I have learnt since having begun undertaking my degree, and through a medium that may later help other students like myself overcome the difficulties and challenges this brings.
"While I have had to read many academic articles, papers and other such literature over these past 4 years studying at Sussex, there is something especially fulfilling about conducting your own research, getting to call upon the knowledge gained from your studies, and then see it all brought together in a report of your very own. As it goes, I really could not ask for a better, more meaningful way to wrap up my time as an undergraduate here at Sussex."
Jennifer Ward  numerical solutions of partial differential equations on evolving curves
For her MMath project Jennifer worked on researching how numerical solutions of partial differential equations are found when they have dirichlet boundary conditions, and then expanding to solving in other contexts including periodic conditions and on curves. In this research, she looked at solving using finite difference methods and wrote several different codes in MATLAB to solve equations using these methods. The equations evaluated include Ordinary Differential equations, the Poisson equation and the Heat equation.
She then moved on to looking at Periodic boundary conditions and solving on curves that are both stationary and evolving (moving with time), and writing code to solve in each of these cases.
"My supervisor was Vanessa Styles and she gave me a lot of support, especially when writing the code for the various cases and explaining to me the methods behind solving Partial Differential Equations numerically.
"This project helped me get more experience with MATLAB and increased my knowledge in the area of numerical solutions and why they are useful. I really enjoyed this project, especially as it had uses and applications in a number of my fourth year modules including 'Numerical Solutions of Partial Differential Equations', 'Mathematical Fluid Mechanics' and 'Mathematical Models in Finance and Industry' so it helped my understanding in these lectures."
Cameron Richard  mathematical modelling of infectious diseases
Cameron Richard's project focused on mathematical modelling of infectious diseases, with each of the different diseases' models depending upon their attributes, such as transmission rate, severity, fatality among other attributes.
"One of the most interesting parts of the project that could be looked at in further detail would be that of external factors contributing to disease spread. An example of this is Cholera, as mentioned in the project, where keeping drinking water filtered appropriately effectively removes the disease from circulation where the filtration is occurring, but as soon as that filtration stops the disease takes hold once again.
"This is shown in the graphs where the filtration starts at time 2000 but ends at time 3000, ω here is the period of immunity from the disease in time after recovery. This idea can be extended to other external factors such as general cleanliness of surfaces in public places.
"Another way in which the models could be improved upon is by taking a more realistic approach to modelling transmission with people going to work, to restaurants, etc. and contact at these places either via persontoperson transmission or contact with infected surfaces.
"Working on this project has given me a greater insight into how exactly the different attributes of a disease affect its spread, along with ways to mitigate this spread in more precise detail than before, using mathematics to model diseases with different attributes.”
Jonathan Robinson  The Hammersley Model
"My MMath project concerns the discrete Hammersley model, which focuses on the positive integer lattice with each point marked as 1 (present) or 0 (occupied) according to i.i.d. Bernoulli random variables with parameter p. The project explores the problem of the 'longest strictly increasing path' in a grid, which is the path with positive slope that passes through as many occupied points as possible, but the path may not be fully horizontal or vertical. Figure 1 gives an example of a maximal path in a grid of Bernoulli points (points with value 1 are marked).
"For a square grid, the value of the length of the longest increasing path divided by the size of the grid is of interest. The relative simplicity of the discrete Hammersley model enables an explicit formula for this value to be found. My project explores the application of this formula to a grid where the points above a straight line are more likely to be occupied than the points below. The shape of the maximal path changes and using the formula allows the optimal path shape to be found through a maximisation problem. Figure 2 illustrates the maximal path in a twophase Bernoulli grid. The predicted points of where the dividing line is entered and exited are shown and match the behaviour of the maximal path.
"The project has taught me a lot about academic research and how the concepts learned in my degree can be used to explore and solve problems. I have also developed my problemsolving skills by having to think in different ways to solve problems encountered in the project."
More student work
 Jordan Boote  developing a numerical method to construct a contraction metric
 "There exists a Theorem which proves the existence, uniqueness and exponential stability of a periodic orbit for a specific type of nonsmooth differential equations. This Theorem is equivalent to the existence of a contraction metric. My MMath Research Project was about developing a numerical method to construct this contraction metric and apply the method to several examples.
Essentially, my project was about writing a MATLAB algorithm to construct a valid function which can be used in conjunction with this existence and uniqueness Theorem, to prove results about solutions to this specific type of differential equation. Some of the constructed functions the algorithm generated are seen in the two figures.
"During the project, I gained valuable insight into the processes and skills associated with mathematical research. It taught me how to bring different aspects of my degree together in order to understand extremely complicated results and how to present them in a simplified way which can be easily followed. In addition to this, it has developed my programming skills as the MATLAB program I wrote is something I never would have believed I could do. The MMath Research Project has been my favourite module by far throughout University as it really captures your individuality and gives you the freedom to write your own work.
"Postgraduation, I am currently editing the paper with my supervisor with the goal of publishing it. It is an amazing feeling, knowing that my own work could be published as a new research paper within the field of Mathematics. I would recommend every undergraduate student to get involved in a research project as despite the difficulty, it’s something you’ll gain invaluable experience from, and there is always the exciting possibility of getting your research published!"
 Emma Muijen  mathematical modelling of infectious diseases

"My project was about the mathematical modelling of infectious diseases. I found this fascinating as the models revolve around how the disease behaves in the future and what might be the best preventative measures to prevent the spread. The mathematics allows you to become a kind of fortune teller!
"I also really enjoyed working alongside my supervisor. She was so helpful and always willing to teach me new concepts so that I could get the most out of my project. Anything I got stuck on, I always ended up managing to understand it with her help."
 Jessica Abbott  impact of vaccinations on mathematical modelling of diseases

"My MMath project was about disease and how this is modelled mathematically, specifically looking at the impact of vaccinations on the models. This can be explained as seeing how effective vaccinations are and how their effect on those susceptible to the disease, those infected and those that have recovered. The way I showed the impact of vaccination was through mathematical manipulation of the model allowing me to describe the stability of the disease and when this was the case, which was displayed in a figure produced through coding on a programme called MATLAB.
"The model I looked at is the SIR model, representing how those susceptible, infected, and recovered change over time particularly with the assumption that those that have recovered are vaccinated. Through the mathematical manipulation I managed to fulfil the aim of the project which was the effectiveness of vaccinations and I confidently proved that vaccination programmes have massive impacts; mainly good; on the SIR model. Throughout the project I studied how previous side effects of other vaccines impact the uptake of vaccinations later on and could see a strong correlation between these side effects and fewer people deciding to get vaccinated. All my mathematical techniques supported my arguments in this project.
"I enjoyed the MMath project massively especially since I hope to research disease in my future. The support and feedback I received from my supervisor enabled me to produce work to my best standard but also correct errors if they did arise. The research and Mathematics used in this project gave me the opportunity to take previous knowledge developed throughout my degree and gain skills further with time management and research, both essential for my future."
You might also be interested in:
 topics available for final year projects
 enhancing your study and research opportunities
 the Junior Research Associate scheme
 our research
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Mathematics
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