Student research bursaries

Choose from the list of research projects for 2023 listed below and then submit an Expression of Interest to Dan Rust using the subject heading "STUDENT RESEARCH BURSARY APPLICATION" in capitals by 22 January 2023.

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)
• academic transcript
• a brief statement on if you are considering PhD study and why.

Title: Position Problems: Colouring, Games and Transport

Supervisor: James Tuite

Summary:

In 1900 the famous creator of mathematical puzzles Henry Dudeney posed the following chessboard conundrum: how many pawns can be placed on a n x n chessboard if we require that no straight line in the plane passes through three or more pawns? This problem has been generalised to the setting of graph theory. A graph is a combinatorial object consisting of vertices that are connected by edges. The general position problem asks for the number of vertices in a largest set S such that no shortest path in the graph contains more than two vertices of S. Applications include robotic transport and social network analysis.

Recently there has been a great deal of interest in variations on the general position problem, such as using chordless paths instead of shortest paths, or only requiring the existence of at least one shortest path between any pair of vertices {u,v} in S that contain no vertices of S-{u,v}. The successful candidate would work along with researchers in the OU, Slovenia, Italy and India to extend our knowledge of these variants and explore connections with other areas of mathematics. There are many avenues that could be pursued, depending on the student’s interests. Examples are: colouring problems, connections to robotic transport, extremal questions for graphs with given position numbers, links to domination and Ramsey theory and adversarial games on graphs.

Prerequisite knowledge

Essential:

The candidate should be confident with formulating mathematical proofs and should be familiar with fundamental number theory, linear algebra etc.

Desirable:

Experience with combinatorics and graph theory.

Availability:

July to August

Title: Fluid mechanics of active droplets

Supervisor: Elsen Tjhung

Summary:

Active droplets are internally driven by chemical reactions or microscopic agents, which are present inside the droplet itself. As a consequence, an active droplet may spontaneously move, self-deform, or even split into two smaller droplets. Some examples of active droplets are: a drop of bacterial suspension, a drop of chemical reactants, and a living cell. In this project we will create a mathematical model for these active droplets and solve them numerically or analytically.

Prerequisite knowledge

Essential:

Fluid mechanics, mathematical methods.

Desirable:

Some familiarity with computer programming, e.g. Matlab or Python, and LaTeX.

Availability:

July to September

Title: Working-class mathematicians and 19th-century letters

Supervisor: Brigitte Stenhouse and Andrew Potter

Summary:

In 1831, the President of the British Association for the Advancement of Science George Harvey paid tribute to the working-class mathematicians of Lancashire, and their cultivation of geometrical knowledge. This group of mathematicians was made up of weavers and factory workers, whose primary method of learning mathematics was through question and answer sections of magazines and periodicals. One such Lancashire mathematician was Thomas Turner Wilkinson, who worked on his family’s farm before becoming a mathematics teacher. Along with Thomas Stephens Davies, a mathematics tutor and prolific contributor to mathematical periodicals, Wilkinson chronicled the numerous short-lived mathematical question and answer sections, including the contributions of the Lancashire geometers. This project will involve completing and editing transcripts of letters from Thomas Stephens Davies to Thomas Turner Wilkinson. Using the letters as a jumping off point, we will explore the role of periodicals in building communities for working-class mathematicians in 19th-century Britain, both in cultivating but also in creating racial and gendered barriers to such communities. We will investigate how mathematical knowledge moved from the work of the Lancashire geometers into university curricula.

Research in the History of Mathematics can reveal rich insights into how we think about and do mathematics today. We look at who was doing mathematics, how they learned and communicated with one another, and how new technologies and traditional social hierarchies presented opportunities and barriers respectively for people of different social status. Through looking at history, we start to see mathematics not just as a body of knowledge handed down to us, but a living, breathing community of human interactions and personalities. The successful bursary holder will gain a taster of what it is like to conduct research in the History of Mathematics, and be able to contribute to active research in the area.

Prerequisite knowledge

Essential:

A basic understanding of Euclidean geometry; knowledge of historic and current issues in mathematics related to social class; confidence in typing transcripts and an eye for detail; enthusiasm for reading, especially handwritten letters!

Desirable:

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. Interest in studying geometry outside of the university curriculum.

Availability:

August to September

Title: Modeling of nucleation and growth in macroscopic systems using the Kolmogorov-Avrami-Johnson-Mehl (KAJM) equation

Supervisor: Ivan Sudakow

Summary:

Many macroscopic and microscopic physical systems exhibit so-called nucleation phenomena, the “collective growth” of patterns in the system. Nucleation could be illustrated as infinitesimal seeds of the stable phase from inside the unstable phase. The process of phase transitions, including continuous (second order) or discontinuous (first order), forms the nucleation. Moreover, the fact that the kinetics when the temperature is quenched from above to below the critical temperature is observed in continuous phase transitions. In reality, the formation of clouds, fog, rain, smoke from burning, ice crystals in the refrigerator, bubbles from soda and beer, etc. are all representatives of nucleation phenomena. Thus, nucleation is applicable everywhere from chemistry to climate science.

The objectives of this work are to model nucleation and growth by applying Kolmogorov-Avrami-Johnson-Mehl (KAJM) equation based on the probability equation and to implement a computation algorithm to describe pattern growth.

Prerequisite knowledge

Essential:

Differential equations, probabilities, and programming.

Desirable:

Calculus of variations, dynamical systems, and stochastic processes.

Availability:

July to September

Title: Symbolic dynamics and Walnut

Supervisor: Dan Rust

Summary:

The Thue-Morse sequence is generated by iterating the substitution 0-->01, 1-->10. So, 0-->01-->0110-->01101001-->..., whose 'limit' is the sequence 0110100110010110... . This sequence was introduced over 100 years ago and has some remarkable properties. For instance, it contains no 'cubes', blocks that are repeated three times in a row (like 011011011). Walnut is a new program that is designed to prove theorems about finite state automata (simple theoretical computers) and automatic sequences. So as an example, Walnut can very easily check if an infinite sequences contains any cubes.

Automatic sequences are a special class of sequences that can be generated from a substitution. In this project, we will explore the capabilities of Walnut and its limitations. The goal will be to translate questions from symbolic dynamics into statements that can be understood and evaluated by Walnut, and so fully automate the process of proving some dynamical questions about substitution sequences. This project is ideal for students that have no programming experience but would like to learn how computers can be used to prove theorems in pure mathematics. Students with programming experience are also more than welcome to take part.

Prerequisite knowledge

Essential:

Basic number theory (modular arithmetic etc.), linear algebra, comfort with using computers

Desirable:

Some familiarity with computer programming, some exposure to topology or metric spaces, some understanding of first order logic

Availability:

July to September

Title: Exploring the genetics of our appearance

Supervisor: Kaustubh Adkihari

Summary:

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.

Prerequisite knowledge

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.

Availability:

July to September