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# Student research internships

## Applications for internships in 2022

Applications are invited from undergraduate and master's students in mathematical sciences for research internships in the School of Mathematics and Statistics at the Open University in summer 2022. Both full-time (35 hours per week) and part-time (17.5–35 hours per week) applications are welcomed, and flexible working hours will be supported.

The scheme offers research experience for those considering future PhD degrees.

Four internships are available. The four interns will interact with each other as well as with their supervisors and others in the School, including academics, postdocs and PhD students.

### Dates, duration and stipend

The internships will take place between July and September 2022, inclusive.

Full-time internships will last 4 weeks and part-time internships will last 4 to 8 weeks (to be negotiated with the project supervisor). Those weeks may not necessarily be consecutive.

Each student will receive a stipend in line with the real living wage.

### Eligibility

To be eligible for the scheme you must be studying a degree in mathematical sciences or a related discipline in the UK at a UK higher-education institution. By July 2022, you must have completed at least the first two full years of an undergraduate degree and you cannot have completed a master's degree. We normally require students to have achieved first-class or high 2.1 grades in most of the modules they have taken.

We aim to award at least two of the four internships to students from under-represented groups in mathematical sciences, including (but not limited to) women, ethnic minority students, disabled students, carers, and students from low-income groups.

## Application procedure

[Section to be amended.] Choose from the list of Research projects for 2022 listed below and then submit the following (and nothing else) as pdf documents to the Postgraduate Research Tutor.

2. A single A4 page which should include
• your name and contact details
• your choice of projects, in order (at most two)
• a summary of why you are suitable for your project choice(s).

Successful applicants will be informed by 30 April 2022. There are no interviews in the application process.

## Frieze patterns and Farey complexes

Ian Short

### Project summary

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.

### Prerequisite knowledge

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.

### Availability

July to September

## Generating matrices whose leading eigenvalues are Pisot or Salem numbers

Reem Yassawi

### Project summary

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 $$P(x)$$.

### Prerequisite knowledge

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