Group theory is the mathematical study of symmetry, a concept which appears throughout mathematics, science and nature. Physicists use group theory to describe the properties of fundamental particles of matter, Chemists use it to investigate crystal structure and it plays a central role in our every-day lives in cryptography. In mathematics, groups are found everywhere, in algebraic objects such as rings, field, vector spaces and matrices, in analysis, and in solutions to equations and differential equations.

This module will study groups and their actions on sets and geometric objects.  Highlights include the Jordan Holder Theorem, which allows us to break down groups into their fundamental building blocks, called the simple groups; and Sylow's theorems which are of fundamental importance to understanding the structure of finite groups. Topics covered may include:

• Groups
• Subgroups
• Cosets, normal subgroups, quotient groups
• Homomorphisms and the Isomorphism Theorems
• The Classification of finite abelian groups
• Group actions and the Orbit-Stabiliser Theorem
• The Jordan Holder Theorem
• Simple Groups
• Sylow Theory
• Solvable Groups

Vector calculus and partial differential equations are indispensable tools in science. These topics are essential to our understanding of electromagnetism and quantum mechanics in physics, and also to the modelling of physical phenomenon such that fluid flow and heat conduction. This course is divided into two halves. The first half will cover topics related to three-dimensional geometry and vector calculus including: vector-valued functions, parametrised curves, line integrals and conservative vector fields, multiple integrals, surface integrals, the theorems of Green and Stokes, and the Divergence Theorem. The second half focusses on the theory of and techniques for solving partial differential equations. Topics covered will include: first order PDEs, classification and techniques for solving linear second order PDEs, Fourier series, the method of separation of variables applied to the heat, wave and Laplace equations, and Fourier transforms.

Quality Control is an important application of statistical theory and methods. Since the industrial revolution, factories have needed to find ways to manufacture high quality products while minimising cost to maintain a competitive edge. From monitoring an automotive production line to stop out-of-control processes, to sampling techniques in the food production industries designed to minimise food waste, quality control procedures form a key part of the modern manufacturing process.

In this module you will learn the techniques companies employ to achieve the above. Topics for this module may include:

• Introduction to quality control: summary of relevant probability theory and probability distributions.
• Methods and philosophy of statistical process control.
• Control charts for variables: mean, median and range charts, special charts.
• Control charts for attributes: p-charts, np-charts, u-charts, c-charts.
• Process capability studies.
• Acceptance control charts.
• Acceptance sampling: single, double, and sequential sampling plans, rectifying inspections.
• Modified control limits.
• Cumulative sum techniques: CUSUM Charts, V-Masks, Tabular CUSUM.

Numerical analysis is the exploration of algorithms that rely on numerical approximation; this is invaluable for the study of many complicated problems, where an exact solution is very difficult, or even impossible to obtain. Its applications extend across mathematics, physical sciences and engineering, and, in the 21st century, have expanded into life sciences, social sciences, medicine, business, and even the arts. This module serves as an introduction to the fundamental techniques and methodologies for solving mathematical problems using numerical methods and equips students with the skills needed to analyse, design, and implement numerical algorithms effectively across a broad range of mathematical problems.

The topics include:

• Solving nonlinear equations, finding roots, Newton iteration and related methods.
• Introduction to optimisation, optimisation of functions of several variables, with and without constraints.
• Interpolation and approximations: Lagrange interpolation, Hermite interpolations,
• Numerical integration and differentiation: Trapezoidal method, Simpson method.
• Solutions of ordinary differential equations: Euler method, Runge-Kutta method, multistep methods, stability, convergence.
• Solution techniques for partial differential equations including the heat equation.
• Implementation and programming, e.g., Python, MATLAB.
• Error analysis: developing an understanding of the sources of error in numerical computations and methods for analysing and controlling numerical errors.

The project is an opportunity for you to explore deeply an area of mathematics of your choice.  This may be a topic not already covered in the degree, or you may explore in more depth a topic covered before.

You will engage with mathematical literature, seeking out material and learning for yourself, supported by regular meetings with your project supervisor.  You will develop skills of self-discipline and project management, and study the art of mathematical writing and communication.

You will negotiate with tutors the precise title and objectives for the project.