Applications are invited from Open University undergraduate and master's students in mathematical sciences for research bursaries in the School of Mathematics and Statistics at the Open University in summer 2022.

**Application deadline: 30 April 2022
Four bursaries available**

The scheme offers research experience for those considering future PhD degrees. Each successful bursary holder will be supervised by a member of academic staff at the School of Mathematics and Statistics to work on a research project. They will be expected to write a report of their findings to be submitted at the end of the project.

The work can be carried out on campus in Milton Keynes with face-to-face meetings, or it can be carried out remotely from home with online meetings (or some combination of the two). The four bursary holders will interact with each other as well as with their supervisors and others in the School, including academics, postdocs and PhD students.

The placements will take place between July and September 2022, inclusive. Each project is expected to require around 140 hours of study, which can be spread over 4 to 8 weeks, depending on the successful applicant’s and the project supervisor’s availability. The weeks of study need not be consecutive.

The value of the bursary is £900 to support you with the cost of your studies. This rate corresponds to that of the London Mathematical Society Undergraduate Research Bursaries scheme. The bursary will be paid in two instalments of £300 and £600. The first instalment will be paid after one week of the project start; the second instalment will be paid one week before the project finishes. No further expenses or allowances are available in conjunction with this bursary.

Bursaries will be awarded to four student applicants based on the following criteria.

- Current Open University student, studying a qualification with substantial mathematical content
- Completed (or expected completion of) Levels 1 and 2 by July 2022
- Grade 1 or Grade 2 passes at most Open University modules studied so far, or evidence of similar levels of achievement at another Higher Education institution
- Evidence of enthusiasm for one of the research projects listed below, and evidence of meeting the essential prerequisites of that project
- Ability to work independently to agreed timescales
- Ability to keep in regular contact with your supervisor, by email, phone, face-to-face or video conferencing
- Excellent written communication skills

The Open University is committed to supporting the rights, responsibilities, dignity, health and wellbeing of staff and students through our commitment to equality, diversity and inclusion. We value diversity and we recognise that different people bring different perspectives, ideas, knowledge, and culture, and that this difference brings great strength. We encourage and welcome applications from all sections of the community, irrespective of background, belief or identity, recognising the benefits that a diverse organisation can bring.

Choose from the list of research projects for 2022 listed below and then submit an Expression of Interest to the Postgraduate Research Tutor using the subject heading "STUDENT RESEARCH BURSARY APPLICATION" in capitals by 30 April 2022.

Your Expression of Interest should be no longer than 500 words and should contain:

- your name, Personal Identifier and preferred email address
- your choice of projects, in order of preference (at most two)
- a summary of how you meet the eligibility criteria above and why you are suitable for your project choice(s).
- Additions for 2023: academic transcript and question on considering PhD study.

Successful and unsuccessful applicants will be informed by 6 May 2022. Sorry but we will not provide feedback on applications. There are no interviews in the application process.

Our appearance is a major component of our social and personal identity. Human appearance is multi-faceted, with many different aspects such as skin colour, facial shape, body height, and so on. Unfortunately, some aspects of our appearance have also been the subject of societal misunderstandings, including discrimination and racism.

Modern genetics is using various methods to understand the biological basis of several aspects of our appearance. Many genes have been identified that explain the huge variability observed between ethnicities and also within any group of people. Scientists have used such knowledge to increase public understanding around such issues and push back against discriminatory notions and misunderstandings.

Underlying all this are mammoth efforts in generating data and analysing it with various statistical techniques. The student will be working as part of a research team to work with a small dataset to investigate a specific aspect of the diversity of our appearance and its connection with diverse ethnicities and genetics. The immediate goal is to produce a research output, but the broader goal is to have a better understanding of the delicate issues around this area, and to support the student in their own journey in public engagement on human diversity and fighting against discrimination. It will be a useful experience for someone planning to do further study in biomedical or bioethics fields.

*Essential*: Basic understanding of probability and statistics. Some familiarity with programming or data analysis software (such as MS Excel) would be helpful.

*Desirable*: Some basic knowledge of biology or genetics would be helpful to grasp the context of the problem easily, but is not essential.

July to September

A group can be extended to a ring by defning an extra binary action on it which satisfies certain conditions. In this project you will learn how we can extend a group to a new object, that is called an "ordered group", by adding an ordering to it satisfying certain conditions. The notions of ideals and simplicity will be defined analogously using the ordering. Ordered groups are important in classifying symbolic dynamical systems, which are systems where time and space is discrete.

The research will consist of learning basic properties of the ordered groups and then a scheme to produce such groups; that is called "direct limit systems".

*Essential*: Basic algebra, linear algebra.

*Desirable*: Dynamical systems.

July to September

This is a hands-on project to work closely with the supervisor to build a suite of routines in the Julia programming language to implement rigorous numerical functional analysis for area-preserving twist maps of the plane via the generating function approach. No prior knowledge is assumed (in particular you are not expected to understand very much of the last part of the previous sentence). However good computer-programming skills are essential. Subject to quality control, it is intended that the routines will be made available to the academic community via an online repository with joint authorship with the supervisor.

*Essential*: Good computer-programming skills in any language, an interest in learning the Julia programming language.

*Desirable*: Knowledge of the Julia programming language, knowledge of online repositories such as Github.

July to September

The project is to build on the ideas contained in the paper

B. D. Mestel. Optimal battery charge/discharge strategies for prosumers and suppliers. Energy Syst (2016). doi:10.1007/s12667-016-0211-y

which uses elementary analytical techniques from the calculus of variations to study optimal battery charging for a simple battery model. Further progress with more realistic battery models will require numerical solution of the equations on a computer. The project will implement the technique in a suitable computer language such as Matlab, Python or Julia. No prior knowledge of battery modelling is required.

*Essential*: Calculus of variations, differential equations. Programming experience in a major language such as Matlab, Python or Julia including graphics.

*Desirable*: Experience of solving differential equations numerically.

July to September

Andrew Potter and June Barrow-Green

In 1884, Edwin A. Abbott (1838—1926) anonymously published the novella *Flatland: A Romance of Many Dimensions*. Set in a two-dimensional planar universe, the story follows the adventures of the protagonist, A. Square, as he discovers the third spatial dimension. Over the years since its publication, it has been extremely popular in educating non-mathematical audiences in the geometric ideas of higher spatial dimensions.

*Flatland* is a product of its time, and its treatment of gender, race and class can be quite jarring for a modern reader. On the other hand, however, the story contains strongly subversive themes, as A. Square transcends the petty restrictions of *Flatland*’s established social order. In this project, we will explore to what extent *Flatland* reinforces or challenges social injustices still pertinent today in the world of mathematics. We will look at the text itself, along with critical and literary responses to *Flatland* from the past 140 years, to see how *Flatland* has related to – and still relates to – issues of equality, diversity and inclusion in mathematics.

*Essential*: A secure understanding of how spatial geometry can be generalised to four or more dimensions; knowledge of historic and current issues in mathematics related to gender, race and class; enthusiasm for reading!

*Desirable*: Some understanding of non-Euclidean geometry; experience of study in English literature and/or history at A-Level / AS-Level (England, Wales, Northern Ireland), Higher (Scotland), or at an equivalent level or above.

July to September

This is a companion project to my PhD project with the same title.

For the first couple of weeks of the project you will learn about frieze patterns, hyperbolic geometry and the universal Farey complex. After this we will investigate Farey complexes associated to finite rings. This is likely to involve computational work in visualising the Farey complexes. We will consider the combinatorial properties of the relationship between these properties and properties of the associated frieze patterns.

*Essential*: Elementary linear algebra and number theory.

*Desirable*: Some experience of ring theory and hyperbolic geometry would be handy but not necessary. Programming experience would also be helpful.

July to September

Rare events describe events affected by small fluctuations that occur with extremely low frequency even in a long period of time. Some examples of rare events include: nucleation phenomena, stock market crash, heatwave, violent conflicts etc. In this project, we are going calculate the probability that a particular rare event will happen. And if such event occurs, we are also going to calculate its typical trajectory. To do this, we are going to use tools from statistical physics, such as path integrals and instantons. For example, the probability that a Brownian particle moves from position A to position B in space can be written as a sum over all possible trajectories starting at A and ending at B with suitable statistical weight (path integral).

*Essential*: Differential equations, probabilities, and elementary linear algebra.

*Desirable*: Calculus of variations, dynamical systems, and stochastic processes.

July to August

Dudeney posed the following puzzle in 1900: what is the largest number of pawns that can be placed on an n by n chessboard such that no horizontal or vertical line contains more than two pawns? This problem has been generalised to the more general setting of graphs as follows: given a graph G, what is the size of the largest set S of vertices such that any shortest path in G contains at most two points of S? In this project, you will investigate new variants of this problem, for example replacing ‘shortest path’ by ‘induced path’ or considering visibility from a single vertex. Depending on your interests the project can go in several directions, including extremal problems for the size of graphs of graphs with given position numbers, packing problems for general position sets, complexity results, or extending previous work to the setting of directed graphs or hypergraphs. The selected candidate would become part of a collaboration with researchers from the Open University, University of Kerala, India and University of L'Aquila, Italy.

*Essential*: Basic algebra and number theory.

*Desirable*: Knowledge of elementary graph theory/combinatorics. Programming skills would also be helpful.

July to September

When is a polynomial \(P(x)\) the characteristic polynomial of a matrix with nonnegative integer entries? It turns out that \(P(x)\) is such a characteristic polynomial precisely when its largest root is what is known as a *Perron number*. Special families of Perron numbers are *Pisot numbers*, *Salem numbers*, and *Mahler measures*.

In this project, given a suitable polynomial \(P(x)\), you will generate nonnegative integer matrices whose characteristic polynomial is \(P(x)\).

*Essential*: A solid familiarity with elementary linear algebra, including eigenvalues of a matrix.

*Desirable*: Field extensions. Familiarity with some software for numerical computation.

July to August