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

Title of project:
The Use of Quizzing in the Learning of Mathematics

Code:
RHW2122A

Supervisor: Dr. R. Wilson & Dr. M. Pugh
Project description:

Quizzing is an increasingly common tool being utilised in education, especially in online education. Research suggests that this approach can help with concentration, identify gaps in knowledge, build confidence and help learners retain information more effectively.
This project will investigate the effectiveness of quizzing in the context of mathematics. Potential avenues for the student undertaking the project to explore will include (but are not limited to):
• The cognitive science behind the concept of quizzing, and how the process of quizzing aligns to well known effective learning strategies.
• Exploring the advantages and dis-advantages of quizzing in mathematics.
• The impact of using quizzes in formative and summative assessments.
NOTE: Those interested in selecting this project are strongly encouraged to arrange to see the supervisors in order to discuss the project in further detail.
Project offered as double module, single module, or both: 
Double
Prerequisite 2nd year modules:
None

Recommended 3rd year module for concurrent study:

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

Title of Project:
Approaches to Learning in Mathematics
Code: RHW2122B
Supervisor: Dr. R. Wilson & Dr M. Pugh
Project description: 

Approaches to learning refers to the differences in intentions and motives a student has when facing a learning situation, and the corresponding strategies they utilise. One such distinction is between deep, surface and strategic approaches. Deep approaches are characterised by learning strategies that focus on meaning, directed towards understanding by critically relating new ideas to previous knowledge and experience. Conversely, surface approaches focus on memorising without reflecting on the task or thinking about its implications in relation to other knowledge. On the other hand, a strategic approach to learning uses a deliberate combination of both approaches. This project will explore the effect different approaches to learning have on students’ learning of mathematics.
It is widely accepted that assessment drives what students learn. One potential aspect to explore is the relationship between approaches to learning and types of assessment questions. Assessment questions in mathematics can be categorised in terms of the skills required to complete them – from routine use of procedures to problem solving. Some questions related to assessment that this project could consider are the impact that different types of assessment questions have on students’ approaches to learning, or the effectiveness of different approaches to learning for answering different types of assessment questions.
NOTE: Those interested in selecting this project are strongly encouraged to arrange to see the supervisors in order to discuss the project in further detail.
Project offered a double module, single module, or both:
Double
Prerequisite year 2 and 3 modules for study:
None
Number of students who can be supervised on this project:
1

Title of Project:
Engaging Students in their Mathematics Learning: What Works?

Code: RHW2122C

Supervisor: Dr. R. Wilson & Dr M. Pugh

Project description: Research has shown that engaging students in the learning process increases attention, focus, and motivates them to practice higher-level critical thinking skills and promotes meaningful learning experiences. This leads to a key question: “what does it mean to engage with mathematics?”. The project will look to investigate this question in further detail, and potential avenues for the student undertaking the project to explore will include (but are not limited to):
• Considering examples of effective teaching practices that are commonly used in mathematics to encourage engaged student learning,
• Exploring why current students engage (or not) with existing learning opportunities.
• The impact of assessment on meaningful engagement with mathematics learning.
NOTE: Those interested in selecting this project are strongly encouraged to arrange to see the supervisors in order to discuss the project in further detail.

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

Prerequisite year 2 and 3 modules for study:
None

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

Title of project:
Category Theory and Homological Algebra

Code:
TL2122A

Supervisor: 
Dr. T. Logvinenko

Project description:
Category theory enjoys an unfortunate reputation of a somewhat dry and abstract subject. Yet it is firmly established as the main language of modern pure mathematics, just as homological algebra is established as its main toolset.

Most new developments in algebraic and differential geometry, number theory and theoretical physics are stated in the language of category theory, and it is essential for anyone interested in these subjects to learn this language.

The project would consist of a general introduction to category theory and homological algebra, focusing on the elements which enjoy broad applications mentioned above. The students would learn the following notions: categories, functors and natural transformations, examples of commonly occuring categories, adjunctions, representable functors and the Yoneda Lemma, additive and abelian categories. The students would study in detail the category of modules over a ring and prove it to be abelian. They would then study complexes of objects in an abelian category, their cohomology groups, short and long exact sequences, snake lemma and five lemma, long exact sequence of cohomology.

The project will involve guided reading, review of literature and writing up a report. There will be also be a minor computational element to the project.

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

Prerequisite 2nd year modules:
MA2008: Linear Algebra II
MA0213 Algebra I – Groups

Recommended 3rd year modules for concurrent study:
MA3013: Algebra II – Rings
MA0322: Algebra III – Fields

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

Title of project:
Spectrum of the Laplace (Schrodinger) Operator on an Interval and its Asymptotics for Small Perturbations

Code:
MC2021B

Supervisor: 
Dr M. Cherdantsev

Project description:

Prerequisite modules:
MA3005 Introduction to Functional and Fourier Analysis

Recommended Modules:
MA2010 Multivariable and Vector Calculus
MA3012 Ordinary Differential Equations
MA3303 Partial Differential Equations

Recommended 4th year modules for concurrent study:
MA4015 Advanced Topics in Analysis: Sobolev Spaces and Viscosity Solutions

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