Mariia Sinkevich - Junior Research Associate (JRA) project on Mathematical Queuing Models and their Applications in Everyday Problems

3rd year Mathematics BSc student, Mariia Sinkevich (pictured above) exhibited her research at the 2023 JRA poster exhibition. Her research focused on queuing problems. Using mathematical modelling and probability theory, she modelled queuing systems and analysed behaviour.

“After developing core theories and mathematical models, I implemented them in Python programmes to perform predictive analysis, analysed long-term behaviour, and created animations of queues and traffic based on initial parameters.”

“It was a great opportunity to see all the amazing projects that undergraduate students had developed over the summer. The exhibition showcased a variety of posters on different topics, which led to many engaging discussions throughout the day.

“The fact that we spend around 5 years of our lives waiting in queues made people especially engaged with my research. Visitors were curious how we could decrease this number and how we could relate traffic jams and probability.

“I enjoyed explaining the power of maths and probability and the amazing things we can achieve with them. Visitors also enjoyed playing with coding programmess and animations, attempting to model queue behaviour in real time. I also had some in-depth scientific conversations about the research and how we can utilise the results.”

“I intend to delve deeper into mathematical modelling and system analysis. It amazes me how we can utilise mathematical theories to model systems and anticipate their behaviour. My research this summer has been instrumental in determining the career path that I want to pursue, and I’m looking forward to beginning my postgraduate journey.”

"How to measure a coffee splat" by Phoebe Valentine

Phoebe Valentine - Junior Research Associate (JRA) project in The Isodiametric Inequality

Mathematics BSc student, Phoebe Valentine 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 third-year 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. “

Displaying the iterative complexity of the Koch curve

The Lorenz attractor

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 person-to-person 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

Figure 1

"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).

Figure 2

"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 two-phase 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 problem-solving skills by having to think in different ways to solve problems encountered in the project."

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.

"Post-graduation, 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!"