Title of Project: Dimension Reduction and Classification Algorithms

Code: AA2122A

Supervisor: Dr A. Artemiou

Project description:
Data mining and dimension reduction are two topics which receive great attention in the big data era we are in. There are a number of algorithms which were proposed to discuss data mining algorithms (like Support Vector Machines) and dimension reduction algorithms (like Sufficient Dimension Reduction). Recently, the two areas were combined to create new algorithms for dimension reduction in regression using ideas from the data mining literature. This created a class of new algorithms that perform very well and are able to do linear and nonlinear dimension reduction in a unified framework.
The project objective is to create first new algorithms for data mining by combining existing algorithms. The student will review the given literature and then derive results for new methodology for classification. The performance of the new algorithm will be evaluated using real datasets. Then, the algorithms will be used to perform dimension reduction to regression problems based on existing literature. Good programming skills are essential for this project. (In 2017-2018 a student work on this project and developed a modified version of SVM)

Prerequisite 3rd year module:
MA2500: Foundations of Probability and Statistics;
MA 2501: Programming and Statistics
and MA3505: Multivariate Statistics is helpful but not essential.

Number of students who can be supervised on this project:
1

Title of Project:
Viscosity Solutions and Nonlinear Partial Differential Equations

Code: FD2122B

Supervisor: Dr. F. Dragoni

Project description:
Nonlinear partial differential equations (PDEs) describe many phenomena
in sciences and economics, e.g. the value of an optimal control problem,
minimal surfaces, porous media, conservation laws, etc. Solutions to these
equations are often not sufficiently regular in order to write all the derivatives
appearing in the equation (e.g. they can be just continuous and not
differentiable). So the solutions have to be understood in a suitable generalized
sense. We consider the so-called viscosity solutions which consist
in using sufficiently smooth test functions to approximate from above and
below a unique continuous solution for a very large class of first-order and
second-order PDEs.
A project would consist of an introduction to the general theory of viscosity
solutions and applications to a specific nonlinear PDE (e.g. Hamilton-Jacobi
eq.s, infinite-Laplace eq., p-Laplace eq., etc.). One possibility is to investigate
the relation between viscosity solutions and control theory or games
theory. Another possibility is to focus on degenerate elliptic/parabolic equations
(e.g. the evolution by mean curvature flow) or the relations with convexity.
If required, it will be possible to deal with more geometric equations
(space-dependent) and metric formulas. One possibility is to investigate the
relation between viscosity solutions and control theory or games theory.

Project offered a double module, single module, or both:

Prerequisite 3rdd year modules:
The following classes may be helpful:
MA3303 Theoretical and Computational PDEs

Number of students who can be supervised on this project:
1

Title of project: Unitary Braidings

Code: GL2122A

Supervisor: Dr G. Lechner

Project description:

Project offered as double module, single module, or both: Double

Prerequisite modules: Linear Algebra I+II, Groups, Functional and Fourier Analysis

Number of students who could be supervised for this project: 1

Year: 4

Title of project: Reproducing kernel Hilbert C∗ -modules and applications to machine learning

Code: GL2122B

Supervisors: Dr. B. Gauthier and Dr. G. Lechner

Project Description:

Project offered as double module, single module, or both: Double

Prerequisite modules: Functional and Fourier Analysis

Number of students who could be supervised for this project: 1–2

Year: 4

Title of project:
Modelling of Calcium Waves in In-Vitro Fertilization

Code:
KK2122A

Supervisors: 
Dr Katerina Kaouri & Dr Thomas Woolley

Project description:
Calcium (Ca2+ is a life and death signal, the most important second messenger in the body, carrying important information across all our cells. It also plays a very important role in embryogenesis, beginning at fertilization when fast Ca2+ waves sweep through the egg after sperm enters. These Ca2+ waves and their characteristics are a predictor of the embryo viability and pregnancy.

In the project we will look at mathematical models of calcium signalling appropriate for fertilization, which are systems of nonlinear differential equations. The models will be analysed computationally with MATLAB and COMSOL Multiphysics and, when possible, they will be studied analytically through asymptotic analysis.

This project falls within the booming, interdisciplinary area of mathematical/quantitative biology and more specifically in the area of In-Vitro Fertilization (IVF), where an egg is fertilized outside the woman and inserted later with the hope of leading to a pregnancy. More than 6 million babies have been born through IVF to date and this number is increasing rapidly in many countries, including the UK. The project is, thus, of great interest to experimentalists and IVF clinics. We have an ongoing collaboration with Prof. Karl Swann, Chair of Developmental Biology at Cardiff Biosciences, and, if time allows, the models will be validated with data from the Swann lab. We also have a collaboration with the London Women’s Clinic (Cardiff branch).

The required mathematical background is differential equations and some acquaintance with programming, preferably in Matlab. The modelling and simulation skills that will be developed can be used in many other real-life problems. No biological background is needed, as any necessary knowledge can be acquired during the project.

Useful references:
• Dupont et al. “Models of calcium signalling” (2016). (In the library.)

Project offered as double module, single module, or both:
Double

Prerequisite modules:
MA0232: Modelling with Differential Equations
MA3304: Methods of Applied Mathematics
MA3303: Theoretical and Computational Partial Differential Equations

Number of students who could be supervised for this project:
1

Title of project:
Models of Cancer and Calcium Signalling

Code:
KK2122B

Supervisor: 
Dr Katerina Kaouri

Project description:
Cancer cells exhibit increased motility and proliferation, which are instrumental in the formation of tumours and metastases. Calcium (Ca2+), the most important second messenger in our body, also plays a prominent role in the evolution of cancer. We will look at a model of cancer cell movement that will account for cancer cell diffusion, advection and proliferation. We will couple this cell movement model with established models of calcium signalling which reproduce experimentally observed calcium oscillations in the cells. Such insights could provide a step forward in the design of new cancer treatments that may rely on controlling the dynamics of cellular calcium.

The models will be analysed computationally with MATLAB and COMSOL Multiphysics and, when possible, they will be studied analytically.

This project falls within the booming, interdisciplinary area of mathematical/quantitative biology. The project is, thus, of interest to experimentalists and clinicians. There is an ongoing collaboration with experimentalists and the models could be validated with experimental data, if time allows.

The required mathematical background is differential equations and some acquaintance with programming, preferably in Matlab. The modelling and simulation skills that will be developed can be used in many other real-life problems. No biological background is needed, as any necessary knowledge can be acquired during the project.

Useful references:
• Dupont et al. “Models of calcium signalling” (2016). (In the library.)
• Kaouri K et al, https://arxiv.org/abs/2003.00612/

Project offered as double module, single module, or both:
Double

Prerequisite Modules:
MA0232: Modelling with Differential Equations
MA3304: Methods of Applied Mathematics
MA3303: Theoretical and Computational Partial Differential Equations

Number of students who could be supervised for this project:
1-2