Undergraduate research experience

Each year undergraduates can apply to work on a research project during the summer. There are a number of schemes that will provide you with a bursary to spend 4-8 weeks (depending on the scheme) working with one of our academic staff or postgraduate students after the summer exam period. You will be able to get a first taste of what research in Mathematics (Pure or Applied) and Statistics is like, and it is possible that your work will lead to a publication in an academic journal. Even if that doesn’t happen, this research experience can make an excellent addition to your CV.

Our department is devoted to creating an inclusive and welcoming environment to all those who have a passion for mathematics and statistics. In light of this, the Undergraduate Research Internship was set up with the primary goal of offering research experience to students who are typically underrepresented in the mathematical sciences at research level. These underrepresented groups include, but are not limited to: women; people with a disability; people who identify as LGBTQ+; and people in BAME communities.

Undergraduates have the following options for a funded project:

Undergraduate Research Internship

You can apply to work with one of our postgraduate students for 4 weeks in the summer. We usually offer 6 projects: 2 in Pure Maths, 2 in Applied Maths, and 2 in Statistics.

There is a selection process for this scheme which requires that you submit a statement of interest, and, if shortlisted, attend an interview. For Summer 2021, the application is likely to open towards the end of February, which is also when the projects on offer will be advertised. The application is open to students from all years, but students in Levels 2 and Levels 3 will be given priority.

The Heilbronn Institute for Mathematical Research, one of the UK's largest mathematics research institutes, has agreed to fund six UGRI projects as part of their commitment to equality and diversity in higher education. Additional funding is possible through the generous bequests of Harry Burkhill, Barry Jackson, and Chris Cannings, which funded all the UGRI projects in 2020.

Sheffield Undergraduate Research Experience

You apply through the University-run SURE scheme. The application needs the agreement of a member of staff who will supervise the project. If successful, the scheme will provide you with a bursary for 6 weeks, starting on the week after the summer exam period. The application is typically due at some point in March of April.

The SURE scheme is now confirmed to run for summer 2021. Applications are expected to open from Monday 15 February.

To apply for the SURE scheme you need to be in your 2nd year (for a 3-year degree) or 3rd year (for a 4-year degree). It is up to the student to make contact with a potential supervisor and discuss the possibility of a summer project.

Find out more

London Mathematical Society - Undergraduate research bursaries

You can also apply for an undergraduate research bursary from the London Mathematical Society. The application for this is already open (deadline February 1st 2021). The typical duration for these projects is 6 weeks, but it is possible to apply for a longer one.

For this scheme you are required to apply jointly with an academic supervisor, who will submit the application on your behalf. It is your responsibility to initiate contact with a potential supervisor, and they will then need to alert the Tutor for Undergraduate Research (Dr Dimitrios Roxanas) and the Head of School (Prof. Nick Monk) with their intention to submit an application. Only students in their 2nd year (for a 3-year degree) or 3rd year (for a 4-year degree) are eligible to apply.

Find out more

If you have any questions about the various schemes and the application process, please contact the Tutor for Undergraduate Research, Dr Dimitrios Roxanas, who can also help with putting you in touch with potential supervisors.

d.roxanas@sheffield.ac.uk

Student stories

Below you can read about some of the research projects our students have completed.

Jai Schrem

"I'm really happy I was able to complete a short supervised reading project this summer, because it gave me the chance to broaden and deepen my knowledge of group theory and try some challenging problems. Presenting one of the new concepts I learned was also a great opportunity to practice explaining mathematics to an audience and improve my presentation skills."

– Jai Schrem, MMath Mathematics

Read Jai's research presentation (PDF, 224 KB)

Kacper Mytnik

"As a person who is into pure maths and who has considered getting into research in the future, this was a very enlightening experience for me personally because I got a taste of what research mathematics is like. Also it was incredibly fun and interesting learning about big concepts like natural transformations and fundamental groups."

– Kacper Mytnik, BSc Mathematics

Read Kacper's research presentation (PDF, 452 KB)

Louisa McKenna

"It is an excellent insight into what further study would be like. Working hours are flexible and I was able to focus my project on an area of research I am interested in. I really enjoyed myself and learnt a lot from the very best researchers at the University of Sheffield."

- Louisa McKenna, BSc Mathematics

Read Louisa's research presentation (PDF, 1.06 MB)

Elisabetta Dixon

"This project was the perfect chance to explore what postgraduate research could be like. We got to explore the fun, independence and frustrations this sort of stuff has to offer, whilst knowing we were in the reassuring hands of friendly supervisors ready to help at any moment. I am so glad I seized the opportunity."

- Elisabetta Dixon, BSc Mathematics with study in Europe

Read Elisabetta's research presentation (PDF, 1.60 MB)

Previous research projects

Below you can find out about research projects from the past academic year, 2019-2020.

Chameleon mechanism in cosmology

Student: Cameron Heather

Supervisor: Richard Daniel (PGR)

One of the biggest remaining mysteries in physics is dark energy, which we know is the cause of the expansion of the universe, but still eludes our understanding. In this project, we looked at a very general model with a scalar field which mediates a fifth force. Via the chameleon mechanism this fifth force is dependent on the density of interacting bodies. As such, Newtonian mechanics can be recovered despite this ‘hidden force’.

Read Cameron's research presentation (PDF)

Cameron Heather

"I really enjoyed my time on the internship. It was the first research project we have had the chance to do and it is a great taster of what's to come in fourth year maths and beyond! I definitely look forward to more research led aspects of maths thanks to this opportunity."

- Cameron Heather

Rarity of nonconvergence in preferential attachment graphs with three types.

Student: Benjamin Andrews

Supervisor: Dr Jonathan Jordan

Preferential attachment graphs are a model of random network, where nodes (which have one of several types) are added to a network and connected to others already in it at random. This is a good model for a lot of situations in the real world, such as people’s brand preferences (where the types are the brands). Preferential attachment means that nodes are more likely to be connected to others that already have more connections, and choose their type based on them. This can be likened to people being more likely to choose their brand preference based on that of influential people, for example. It was conjectured that with three different types of nodes, the proportions of each type in the network would always converge. However, this is not the case in at least one setup, where new types are selected using a "rock-paper-scissors" style process. We explore similar processes to these, which do end up converging, and discuss the idea that cases where convergence does not happen are probably rare and special.

Read Benjamin's research report (PDF)

Benjamin Andrews

"The UGRI is a really good way to get a taste of what PhD research is like, as well as look further into a bit of maths you enjoy!"

-Benjamin Andrews

Representation Theory of Finite Groups and Burnside's Theorem

Student: Jae Irvine

Supervisor: Joseph Martin (PGR)

In some sense, simple groups are to groups what prime numbers are to natural numbers. Burnside’s theorem was a large step in the classification of all the simple groups, stating that a group G for which |G|=paqb cannot be simple if p and q are primes and a,b are non-negative integers whose sum is greater than two. It was first proved using representation theory, rather than directly through group theory. A purely group theoretical proof was not completed for another 66 years, which is enough to display representation theory’s effectiveness.

An Exploration into the Effects of Various Modelling in Chronology Construction in Archaeological Dating


Student: Elisabetta Dixon

Supervisors: Prof Caitlin Buck, Bryony Moody (PGR)

Accuracy in Archaeological dating is important. There are different models we can use that gives us a calendar date range from a radiocarbon date to estimate the age of an archaeological find. Differing models can produce contradictary date ranges, and with multiple phases or multiple dates within a phase, this 'difference' can multiply significantly. We wanted to explore and compare these differences to see just how significant model choice might be to an archaeologist wanting to construct a chronology from a particular site.

Watch Elisabetta's research presentation


Elisabetta Dixon

"This project was the perfect chance to explore what postgraduate research could be like. We got to explore the fun, independence and frustrations this sort of stuff has to offer, whilst knowing we were in the reassuring hands of friendly supervisors ready to help at any moment. I am so glad I seized the opportunity."

- Elisabetta Dixon

The Impact of Local Interactions on an Epidemic

Student: Lydia Wren

Supervisor: Dr Alex Best

Mathematical modelling has long been used to analyse the dynamics of infectious disease. Since the early twentieth century, the SIR model has been used to track individuals within a population who are either susceptible to, infected with or have recovered from a disease but this model has its shortcomings. One issue with the SIR model is that it assumes every individual has the same likelihood of being infected, whereas, in reality, an individual is much more likely to become infected when they are in close contact with an infected individual, either physically or socially. In my research I have used a pair-approximation model (based on the SIR), which allows for global interactions to be limited, and my results show the impact that local interactions can have on an epidemic.

ABC methods and their use in ‘step and turn’ animal movement models


Student: Sam Felton-Dobbs

Supervisor: Dr Alison Poulston

When performing Bayesian inference, often the likelihood cannot be evaluated. In this case, methods have been developed to approximate the target posterior distribution. This presentation will introduce rejection ABC as such a method and show how it can be used in a toy example. I will also present an example animal movement model to demonstrate the real world impact of these likelihood-free inference methods

Read Sam's research presentation (PDF)

Sam Felton-Dobbs

Computational Euclidean Geometry

Student: Jamie Wright

Supervisor: Dr James Cranch

In this project me and Dr James Cranch have began development on software which can use polynomial rings and ideals to help deduce certain Euclidean geometrical axioms. Euclidean geometry is a rich source of mathematical problems and is a cornerstone of the school syllabus. Such problems have appeared in competitions for school-aged students for more than sixty years now. For most of these problems, the range of concepts used is quite limited: nearly everything can be broken down to discussion of points, straight lines, circles, lengths and angles. As a result, it is not difficult to translate these problems into algebra, where we assign variables to the coordinates of all points involved. Such algebraic problems, despite being near impossible for a human to work with, can be used by computers extremely efficiently to do computations on these equations. The main goal of this project was to create a library of Python code which helps convert geometric problems into algebra, enabling that algebra to be solved by existing methods.

Loops and covers


Student: Kacper Mytnik

Supervisor: Prof Sarah Whitehouse

The project is focused on a certain object in algebraic topology; the fundamental group of a space X with base point x. It is denoted π1(X,x) which is a set of equivalence classes of loops based at the point x, endowed with the group structure under the operation of concatenation of loops. This object is very useful in classifying connected spaces up to homotopy equivalence however it can be notoriously hard to compute. A very useful tool for computing the fundamental group of a space is the Van Kampen theorem. Briefly, it says that for sufficiently nice spaces that can be broken up into two pieces U and V, such that the intersection is non-empty and the union of the two pieces is the whole space, we have π1(X,x)=π1(U,x)*π1(V,x), which is the amalgamated free product of the fundamental groups of the two pieces. In order to prove this theorem, one can proceed by directly examining the loops or by employing an equivalence of categories amongst other methods. In this project I did the latter and here I will discuss what went into establishing the equivalence of the category of covers and the category of π-sets.

Read Kacper's research presentation (PDF)


Kacper Mytnik

"As a person who is into pure maths and who has considered getting into research in the future, this was a very enlightening experience for me personally because I got a taste of what research mathematics is like. Also it was incredibly fun and interesting learning about big concepts like natural transformations and fundamental groups."

– Kacper Mytnik

Monte Carlo Methods

Student: Louisa McKenna

Supervisor: Ines Krissaane (PGR)

The aim of the project was to familiarise myself with Bayesian statistics, as I did not take Bayesian Statistics as a third year module, and to use this field of statistics to solve problems, such as the Birthday problem and the Monty Hall problem, in R. I then adapted this to explore a range of sampling methods including Monte Carlo, Gibbs and Metropolis-Hastings, meaning that when we cannot infer exactly, we approximate the true distribution by taking samples. I was able to use these sampling methods alongside the Ising model, and link this further to images created by MRI scans. Finally, I was able to use these imaging techniques to show how to make an image less blurry through the R Shiny App.

Read Louisa's research report (PDF)


Louisa McKenna

"It is an excellent insight into what further study would be like. Working hours are flexible and I was able to focus my project on an area of research I am interested in. I really enjoyed myself and learnt a lot from the very best researchers at the University of Sheffield."

- Louisa McKenna

Simplicial Sets

Student: Calum Hughes

Supervisor: James Brotherson (PGR)

Simplicial sets are an extension of the notion of the simplicial complex, which is (roughly) a way to split up a topological shape into n-dimensional triangles in order to learn information about the number of n-dimensional holes and connectedness of the shape. It has uses in algebraic topology, and especially in homotopy theory, but my project favoured building geometric intuition for what a simplicial set is. The terminology for a lot of algebraic topology is written in the language of category theory, which provides shortcuts to understanding and proving related ideas in different mathematical areas. A few of the main ideas of category theory were first introduced before moving onto a relatively intuitive introduction to simplicial sets.

Read Calum's research presentation (PDF)

Calum Hughes

"This project really helped me decide that I'd like to do a PhD in maths and it was a great way to get paid to learn some amazing stuff!"

-Calum Hughes

Machine Learning - An Introduction to Gaussian Processes

Student: Callum Murton

Supervisor: Valentin Breaz (PGR)

Machine Learning is an extensive field that lies at the intersection between Statistics and Computer Science, with several exciting applications. One key algorithm of Machine Learning is Gaussian Processes, which is a very popular method of predicting data. Gaussian Processes come with many great advantages which makes predicting data very flexible. Especially when compared to other methods such as Linear Regression, which has its limitations. From the research, we wanted to motivate the use of Gaussian Processes and investigate why they are so commonly used. Also, we wanted to understand how they work and the theory behind them; enabling us to apply them for ourselves. This led us to the use of Bayesian Inference and Covariance Functions (Kernels), which are instrumental aspects of Gaussian Processes. Research methods included, reading ‘Gaussian Processes for Machine Learning’, and attending an online Gaussian Processes Summer School. Finally, we applied our research to a practical problem. Where given some training data, we fitted a Gaussian Process such that it gave us accurate predictions with uncertainty.

Read Callum's research report (PDF)


Callum Murton

Chaos and fractal boundaries


Student: Murat Delibas

Supervisor: Dr Ashley Willis

To trigger chaotic dynamics, sometimes we need to apply a small ‘kick’ to get things started. We tend to be interested in equilibrium solutions to a dynamical system, when in practice, even if a solution is stable, a small perturbation from it could lead to chaos. The onset of turbulence in flow through a pipe (or between flat plates) is a good example – simple flow straight through the pipe (‘laminar’ flow) is possible, but large enough perturbations trigger a chaotic turbulent flow. A nice system that mimics this with only 9 coefficients was derived by Moehlis, Faisst and Eckhardt (2004). A perturbation of special interest is the smallest perturbation that will trigger chaos, but how can we find what this perturbation is? Generically the basin of attraction of a chaotic attractor is fractal, but is that the case in the vicinity of this ’optimal’ perturbation?

Read Murat's research presentation (PDF)

Murat Delibas

"Internships, either in research or in industry, are perfect for refining your skills. And this particular one gave me invaluable experience as an applied mathematician, whose work consists of a sudden shifts from understanding problem mathematically to translating it to the efficient code compilable by computer in a reasonable amount of time"

-Murat Delibas

Finding a strong knot invariant

Student: Shengzhi Luo

Supervisors: Dr James Cranch, Dr Fionntan Roukema

With the knowledge that there are two knots in different shapes with the same invariant the approach of this project is to try to use different invariants to tell knots apart with various crossings and strings and compare each invariant to get a relatively reasonable invariant. To learn about invariants, the project started with braids theory in two dimensions which was a relation to knots in three dimensions. After done with colorings, Python was used to count all braids in the list by 3-coloring to see if they had the same braid closure and thus there would be 1, 9, 27 three-colorings to sort knots. Therefore, on the basis of coloring, quandles were introduced to involve more colorings and get programming digitally. For the second goal, entropies were needed to measure how good a knot invariant was and by running Python through the list of quandles which was up to 14 numbers entropy showed some regularity and similarity. There were symmetries on knots and quandles for the same entropies with different invariants and those usually wouldn’t be strong invariants. Moreover, quandles with more elements normally would have lower entropy among all knots.

Read Shengzhi's research report (PDF)

Shengzhi Luo

Investigating properties of the Solar Corona with Fourier Transforms


Student: Philip Kmentt

Supervisor: Farhad Allian (PGR)

The project revolved around Fourier transforms and their importance in signal processing and solar physics. Briefly, a Fourier transform is a mathematical transformation that transforms a signal in time into its dominant frequencies and portrays these in a frequency domain. Fourier transforms have vast applications in data analysis and especially physics, as they help to describe the behaviour of waves, which I investigated in my summer project, both theoretically and by use of code. The aim of my project was to enhance my coding skills, to develop a better understanding of the Fourier transform and also to appreciate how the Fourier transform can be used to model the sun’s corona, an area in which my project supervisor is actively engaged in. I achieved this successfully by dedicating my time towards creating and improving code in Python to model explicit data from NASA I was sent.

Read Philip's research presentation (PDF)


Philip Kmentt

"Thanks to the UGRI summer project scheme, I really enjoyed working on Fourier Transforms and investigating the Solar Corona together with my project supervisor Farhad Allian. Also, I am really grateful for the invaluable experience of discovering what scientific and mathematical research is like and for the opportunity to enhance my coding skills."

-Philip Kmentt

Lagrangian points and spacecrafts

Student: Barry Zhang

Supervisor: Dr Rekha Jain

Lagrangian points L1 to L5, discovered by French mathematician Joseph Louis Lagrange, are locations in space where gravitational forces and the orbital motion of a body balance each other. Therefore, these points are used by spacecrafts to float. In 1994, NASA launched the Wind spacecraft, and it was placed, with nudge, in the L1 Lagrangian point of the Sun-Earth system. The Solar and Heliospheric Observatory (SOHO) also slowly orbited around the L1 Lagrangian point. WMAP, Planck and the James Webb Space Telescope are all using L2. The L3 point is hidden behind the Sun at all times and so not used but is a popular topic for science fiction movies. The L4 and L5 points could provide stable orbits under certain conditions.

Read Barry's research presentation (PDF)


Barry Zhang

"Together with Dr Rekha Jain, we spent this summer for the project. I learnt a lot of new things from celestial mechanics, which I never have a chance in previous study. In general, it was a very meaningful project to me."

-Barry Zhang

Direct Systems of Groups: How can the concept of a limit be applied in group theory?


Student: Jai Schrem

Supervisor: Dr Moty Katzman

We first construct a direct system of groups and its direct limit in abstract terms, beginning with some basic definitions and notations that are required, such as a partially ordered set and some new notation for homomorphisms. Next, we looked at an example of a direct system of prime power order cyclic groups and construct the Prüfer group of type p∞ as its direct limit.

Read Jai's research presentation (PDF)


Jai Schrem

"I'm really happy I was able to complete a short supervised reading project this summer, because it gave me the chance to broaden and deepen my knowledge of group theory and try some challenging problems. Presenting one of the new concepts I learned was also a great opportunity to practice explaining mathematics to an audience and improve my presentation skills."

– Jai Schrem

Using Bayesian thinking to find the law of economic development between nations

Student: Jialiang Sun

Supervisor: Dr Miguel Juarez

Economic development is a hot topic in recent years, so the research on the commonness of economic development among developing countries has become a statistical problem worth discussing. This project combined the statistical thinking of Bayes and applied it to the topic of economic development.

Read Jialiang's research presentation (PDF)