Title: Cavitation in nematic elastomer spheres

Code: AM2122A

Project Description:

Nematic liquid crystal elastomers are advanced multifunctional
materials that combine the exibility of polymeric networks with the ne-
matic structure of liquid crystals. Due to their complex molecular architec-
ture, they are capable of exceptional responses, such as large spontaneous
deformations and phase transitions, which are reversible and repeatable
under certain external stimuli (e.g., heat, light, solvents, electric or mag-
netic elds). Their accurate description requires multiphysics modelling
combining elasticity and liquid crystal theories.

In particular, instabilities in liquid crystalline solids can be of potential
interest in a range of applications. This project focuses on the cavitation
instability where a void forms at the centre of a nematic sphere under
radial symmetric tensile load. Assuming an initially unit sphere described
by a simple neoclassical model for ideal nematic elastomers, the aim is
to determine the critical load for the onset of cavitation, and to verify
if the associated bifurcation from the trivial solution where the sphere
remains undeformed is supercritical, i.e., if the cavity radius increases
as the applied load increases. A comparison with similar phenomena in
purely elastic spheres will also be performed.

Project offered as a Double module.

Type: 20 credits

Supervisor: Dr Angela Mihai

Prerequisite 2nd year modules: Real Analysis, Calculus of Several Vari-
ables, Linear Algebra

Prerequisite 3rd year modules for concurrent study: Partial Differential
Equations, Methods of Applied Mathematics, Finite Elasticity

Maximum number of students: 1

Title of project:
Develop Innovative Methods for Teaching Mathematics in Primary School

Code:
FD2122A

Supervisor: Dr F. Dragoni & Dr. M. Pugh

Project description:
1-The project consists in developing an extra-curricular mathematical subject or to be introduced in primary school, at all levels from Reception to Year 6.
The student will need to choose and develop the subject as a topic across all years, and design practical activities that will be used to teach it.
We will contact primary schools to find one willing to participate in the project, and the student will liaise with the school arrange to teach the planned activities to the pupils. It is proposed that these activities would take place during the week around March 14th (Pi-Day).
The student will write a report which discusses the mathematical content chosen, the design of these activities, along with reflection on the success of these activities and recommendations for improving them for future years.
Some possible subjects to develop could be:
– the idea of dimension, from Euclidean space to the fourth dimension (Flatland could be an interesting book to develop activities especially for the youngest children);
– functions, including sets, domain, basic properties of functions, etc;
– limits (extremely challenging);
– any other subject proposed by the student.
2-Alternatively the project can focus on develop innovative methods to teaching numerical skills and/or problem solving in primary schools.
In this direction some possible subjects are developing board games for advance numerical skills and problem solving or explore how to use sport and playing activities to strength the mathematical curriculum in primary schools across Wales.
Also in this case the project will consist in a first part where the student will need to get familiar with the literature and the Welsh math curriculum, a second part where she/he will develop specific activities, a third part where these activities will be delivered to a selected group of primary school pupils and a final part where the student will collect feedback on the activities delivered and reflect on them, implementing when necessary modifications.

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

Prerequisite 2nd year modules:
None

Recommended 3rd year module for concurrent study:
None

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

Teitl y project:   Optimeiddio Systemau Ciwio

Code:  GP2122A

Goruchwyliwr:   Dr Geraint Palmer

Disgrifiad:
Fe ellir modelu systemau ciwio mewn gwahanol ffyrdd, yn analytig trwy ddiffinio cadwynau Markov, neu trwy efelychiad cyfrifiadurol. Mae’r dulliau hyn hefyd yn gallu cael eu hymestyn a’u defnyddio ar gyfer optimeiddio. Gallwn ymestyn cadwynau Markov i brosesau penderfynu Markov (Markov decision processes), a’i ddatrys gyda dulliau rhaglennu deinameg (dynamic programming); a gallwch ychwanegu algorithmau dysgu atgyfnerthol (reinforcement learning) i efelychiadau cyfrifiadurol.

Bydd y prosiect hwn yn edrych ar ryw system ciwio benodol a ellir ei optimeiddio, a defnyddio’r dulliau hyn i archwilio i mewn i’r strategaethau gorau ar ei chyfer. Er enghraifft, pryd yw’r amser gorau i ychwanegu gweinyddion ychwanegol? Pryd yw’r amser gorau i ddargyfeirio cwsmer i weinydd cyflymach, ond fwy costus? Pa mor wael oes angen i’r system fod cyn gwrthod cwsmeriaid newydd?

Rhagofynion modiwlau’r ail flwyddyn:  Bydd MA2601 yn ddefnyddiol iawn, ond nid yw’n hanfodol. Bydd angen fod yn gyfforddus yn defnyddio Python. 

Nifer o fyfyrwyr y gellid eu goruwchwylio am y project hwn: 2

Code: JS2122A

Title of project:
Arithmetic Functions and the Riemann Zeta Function
Code:
MCL2122A
Supervisor: 
Dr. M. C. Lettington
Project description:

Project offered as double module, single module, or both:
Double
Prerequisite and Recommended 2nd year modules:
MA2011 Introduction to Number Theory I
Recommended 3rd year modules for concurrent study:
MA3011 Introduction to Number Theory II
Number of students who could be supervised for this project:
1

Title of project:
Representation Theory Of Finite Groups

Code:
MP2122A

Supervisor: 
Dr. M. Pugh

Project description:
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces. In particular, group elements can be represented as matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra.
The main task of this project will be to classify (real and complex) representations of the dihedral and cyclic groups.
Project offered as double module, single module, or both:
Double

Prerequisite 2nd year modules:

MA2013 Groups

Recommended 3rd year module for concurrent study:

MA3013 Rings and Fields

Number of students who can be supervised on this project:

1

Title of project:
Qualitative Theory of Partial Differential Equations

Code:
ND2122B

Supervisor: Prof. N. Dirr

Project description:

Certain partial differential equations of the form

∂tu(x,t)=∆u(x,t)+f(x,u(x,t))

can be transformed in an integral equation and solved by a Peano-Iteration in a similar way as it is done for ODEs. The difference is however, that we have to work in infinite dimensional vector spaces instead of R. Combining ideas from ODEs and Functional Analysis, a lot can be said about the qualitative behaviour of such equations (stability, long-time behaviour etc.) similar to the ODE case.
These equations, called reaction-diffusion equations, have applications in chemistry, biology and physics.
A project could, depending on the interest of the student,  focus on numerics and/or on analysis.
Background Reading:
L.C. Evans, Partial Differential Equations, AMS Grad. Studies in Math. 19

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

Prerequisite 2nd year modules:
Real Analysis, Series and Transforms. Not prerequisite but recommended is Modelling with ODEs or a Year 2 Numerical Analysis Module.

Recommended 3rd year module for concurrent study:
Ordinary Differential Equations, Fourier and Functional Analysis

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

Title of Project: Topics in Combinatorics

Code: RB2122A

Description: Combinatorics is the branch of mathematics concerned with the theory of arranging or selecting finitely-many objects according to specified rules. The objects can be material (such as people in a group or cards from a pack) or abstract (such as numbers, symbols or shapes). A primary aim of combinatorics, when applied to particular cases, is to determine the number of arrangements or selections, but without actually listing them. Accordingly, combinatorics primarily involves the theory of counting or enumeration. This project will first study the enumeration of various fundamental discrete mathematical structures, including permutations of finite sets and multisets, combinations of finite sets and multisets, partitions of finite sets, and partitions and compositions of integers. It will then study the enumeration of a range of further mathematical objects, such as plane partitions, tableaux, graphs and lattice paths. The methods for enumeration will involve bijective arguments, recurrence relations and generating functions. The project will include guided reading of relevant texts, completion of combinatorial exercises, and computer calculations and simulations. There will be some flexibility, based on the preferences of the student, in the choice of the combinatorial topics and methods which are studied.

Type: Single 10 credit module, Autumn semester

Supervisor: Prof. R. Behrend

Prerequisite 2nd year modules:

MA2008/2058 Linear Algebra II

MA2011 Introduction to Number Theory I

MA2013 Algebra 1: Groups

Prerequisite 3rd year modules for concurrent study:

MA3013 Algebra 2: Rings

Maximum number of students: 2