Department of Mathematics MSc Courses in Mathematical Sciences

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### Three Sample MSc Projects

The following projects should give a flavour of what we can offer for MSc projects. We have a wide range of interests in mathematics and probability & statistics.

### Integro-differential equations in Neural Field Models

A mathematical description of synaptically coupled neural tissue involves the use of integro-differential equations. Similar to PDEs we can observe pattern formations and waves.

This project would consider the analysis and numerical simulation of waves and patterns in one and two-dimensional neural field theories of thalamic and cortical tissue.

Project offered by : Dr Gabriel Lord

### Vortex dynamics in the Landau-Ginzburg model

The Landau-Ginzburg model is a non-linear Schroedinger equation which is used in various physical contexts such as superfluidity and superconductivity. It admits vortex solutions of quantized vorticity. The goal of this project is to study the slow dynamics of several vortices using an adiabatic approximation. Earlier work has revealed the existence of solutions which consist of two vortices orbiting each other, with the angular velocity depending on the separation. Specific questions to be addressed in this project are

• 1. Are there similar solutions involving three or more vortices?
• 2. How accurate is the adiabatic approximation? In particular, do radiative effects lead to significant corrections?
• The project will involve a combination of analytical and numerical work.

Project offered by : Dr Bernd Schroers

### Modelling "Vasculogenesis" - formation of the blood flow network

The arteries, capillaries and veins that transport blood around the body are collectively referred to as the vascular network. The formation of this system during embryonic development is referred to as vasculogenesis, and has been subject to great attention due to its importance in pathological processes such as cancers.

A lot of attention has focussed on the chemotactic nature of the cells that form the capillary walls. Chemotaxis is a mechanism by which cells move in response to gradients in an environmental chemical factor. Mathematical models based on systems of partial differential equations have been developed to describe the chemotactic movement of cells, and have been applied to a wide variety of biological systems of pattern formation, including bacterial movement, immune response and cancer growth.

In this project, a mathematical model will be developed to understand whether chemotaxis is capable of generating a vascular network.

Project offered by : Dr K Painter

For more details of any aspect of the MSc/Diploma Programme, please contact us.