Mathematics

Dynamical systems and computation: Opening the black box

Supervisor

Dr Claire Postlethwaite, Dr Matthew Egbert
Faculty of Science
Project code: SCI001

Artificial neural networks can be trained to perform very complex tasks such as voice recognition or image classification. However, it is often very unclear how such networks work, or how they will behave when given a new type of input. This project involves using a neural network of a special type to solve similar problems, where the network is designed in such a way that we are able to “open the black box” after training the network parameters.

Prerequisites: Knowledge of differential equations, and some programming and/or computer science experience.

Post-quantum cryptography

Supervisor

Prof. Steven Galbraith
Faculty of Science
Project code: SCI002

Post-quantum cryptography is a hot topic at the boundary of mathematics and computer science. The project will consider post-quantum cryptosystems based on isogenies of elliptic curves and/or lattices. Depending on the skills and interests of the student, the project may include computer programming and computer experiments. As well as having good computer skills, the ideal candidate will have taken courses in linear algebra, abstract algebra and number theory.

Numerical methods for finding homeostasis in biological models

Supervisor

Dr Graham Donovan
Faculty of Science
Project code: SCI003

Homeostasis is an important biological phenomenon, in which the output of a biological system is relatively insensitive to changes in input over a certain range; e.g. body temperature (output) as a function of air temperature (input). One key question is: how can we find homeostasis within (often large) parameter spaces?

This project will explore numerical methods for finding homeostasis, using as an example system models of gene regulatory networks (GRN). In GRNs, the relatively low dimensionality allows one to use a geometric approach in which homeostasis is expected to emerge generically only via certain geometric structures. These geometric insights allow direct methods of finding such structures via construction of augmented systems in which the desired structures are attracting equilibria.

Interested students should have at least some background in (ordinary) differential equations and numerical methods, and an interest in biological applications.

Predicting lung function from incomplete data: Graph dynamical systems under uncertainty

Supervisor

Dr Graham Donovan
Faculty of Science
Project code: SCI004

Models predicting lung function from structural measurements of the airways are potentially useful for designing improved, patient-specific treatments for disease. However, this is only true of the uncertainty of those structural measurements is sufficiently low to allow the uncertainty of the resulting predictions to lie within an acceptable range. Because some such models can be thought of as graph dynamical systems, this amounts to the study of graph dynamical systems under uncertainty.

Understanding the origin of streaming driven by the ER network in plant cells

Supervisor

Dr Graham Donovan
Faculty of Science
Project code: SCI005

Many types of plant cells exhibit a streaming behaviour in the cell cytoplasm which is thought to be driven by the endoplasmic reticulum (ER) network. The dynamic origins and stability of these streams are not well understood; this project will study the interactions between cytoplasmic flow and ER network dynamics which may lead to, and modulate, these streaming dynamics.

Students' systematic mistakes as a window into their mathematical thinking

Supervisor

Dr Igor' Kontorovich
Faculty of Science
Project code: SCI027

In the daily teaching-and-learning reality, we treat mistakes as something to be avoided. If you think about assessment, for example, it often acts as an institutionalized punishment for those who “don’t get it right”. From the perspective of mathematics education research, the instances of “not getting it right” provide a window into silent mechanisms of students’ thinking. Hence, the enhanced interest in the systematic mistakes that students make.

The scholarship student will engage with final exams in Stage-I mathematics courses and dissect the ‘logic’ behind students’ reasoning and common answers. This work might lead to insights about the sources of these mistakes and pedagogies to overcome them.

This project is intended for all students with a solid mathematical background and a genuine interest in educational issues.

Unlocking the potential of non-traditional mathematics teaching

Supervisor

Dr Igor' Kontorovich
Faculty of Science
Project code: SCI028

This project is driven by the recent calls to transform the traditional university mathematics teaching and learning to become more engaging and student-centered. The scholarship student will analyze data collected from a variety of mathematics courses where innovative pedadogies were implemented. In some courses, these were creative tasks asking students to script mathematical dialogues between fictional characters. In other courses, the students constructed proofs in small groups and presented them to their peers as a means to im-prove their work.

This project is aimed at characterizing the potential of selected pedagogies in mathematics teaching and learning based on students' experiences.

The project is intended for all students with a solid mathematical background and a genuine interest in educational issues.

Hamiltonian cycles in vertex-transitive graphs

Supervisor

Dr Gabriel Verret
Faculty of Science
Project code: SCI030

A vertex-transitive graph is, informally, a graph in which all the vertices are "identical" with respect to the structure of the graph. A Hamiltonian cycle in a graph is a cycle going through every vertex exactly once. There are only five known connected vertex-transitive graphs without Hamiltonian cycles and some conjecture that there are no others. The general problem is probably hard, but various cases can be investigated, including graphs of small order, where a computer can be very helpful.