Mathematics BSc

This module involves the application of mathematics to a variety of practical, open-ended problems - typical of those that mathematicians encounter in industry and commerce.

Specific projects are tackled through workshops and student-led group activities. The real-life nature of the problems requires you to develop skills in model development and refinement, report writing and teamwork. There are various streams within the module, for example:

• Pure Mathematics
• Applied Mathematics
• Data Analysis
• Mathematical Physics

This ensures that you can work in the area that you find most interesting.

In this module you will apply the quantum mechanics that you learned in Year 2 to more general problems. New topics will be introduced such as the quantum theory of the hydrogen atom and aspects of angular momentum such as spin.

During this module you will build on your theoretical knowledge of statistical inference by a practical implementation of the generalised linear model. You will progress to enhance your understanding of statistical methodology including the analysis of discrete and survival data. You will also be trained in the use of a high-level statistical computer program.

This module provides an introduction to coding theory in particular to error-correcting codes and their uses and applications. You’ll learn cryptography, including classical mono- and polyalphabetic ciphers.  There will also be a focus on modern public key cryptography and digital signatures, their uses and applications.

This module provides an opportunity for mathematics students to gain first-hand experience of being involved with providing mathematical education. Students will work at local schools alongside practising mathematics teachers in a classroom environment and will improve their skills at communicating mathematics. Typically, each student will work with a class (or classes) for half a day a week for about fifteen weeks. Students will be given a range of responsibilities from classroom assistant to leading mathematical activities. As well as half a day a week spent in local schools, students will typically attend 4 timetabled seminars during the year, as outlined in the “Method and frequency of class” section below.

The assessment is carried out by a variety of means: writing a collection of reflective logs (on-going throughout the year); contributing to a reflective group seminar; giving an individual oral presentation; and writing a research project report.

Please note that applications for the module are made during the Spring semester for the following academic year, and information about the application process will be sent to eligible students. Students who select this module during module enrolment without confirmation from the module convenor of their successful application will automatically have their choice rejected.

The aim of Discrete Mathematics is the study of discrete and finite rather than continuous quantities. This includes counting problems, graphs and other quantities parametrised by integers.

As such Discrete Mathematics is of great importance for various branches of Pure Mathematics, Mathematical Physics, Statistics and Computer Sciences.

The course will cover a range of Discrete Mathematics topics, including:

• an introduction to advanced aspects of combinations and permutations
• ordinary and exponential generating functions
• recurrence relations and applications to counting problems
• graphs
• digraphs
• Eulerian and Hamiltonian trails
• planar graphs
• graph colouring
• trees
• minimum spanning tree algorithms

Game theory contains many branches of mathematics (and computing); the emphasis here is primarily algorithmic. The module starts with an investigation into normal-form games, including strategic dominance, Nash equilibria, and the Prisoner’s Dilemma. We look at tree-searching, including alpha-beta pruning, the ‘killer’ heuristic and its relatives. It then turns to mathematical theory of games; exploring the connection between numbers and games, including Sprague-Grundy theory and the reduction of impartial games to Nim.

You will explore the concepts of discrete time Markov chains to understand how they used. We will also provide an introduction to probabilistic and stochastic modelling of investment strategies, and for the pricing of financial derivatives in risky markets.

You will gain well-rounded knowledge of contemporary issues which are of importance in research and workplace applications.

Mathematics can be usefully applied to a wide range of applications in medicine and biology.

Without assuming any prior biological knowledge, this module describes how mathematics helps us understand topics such as:

• population dynamics
• biological oscillations
• pattern formation
• nonlinear growth phenomena

There is considerable emphasis on model building and development.

This module is concerned with the analysis of multivariate data, in which the response is a vector of random variables rather than a single random variable. Key topics to be covered include:

• Vector and matrix algebra ideas relevant to multivariate statistics, including the eigen and singular value decompositions
• Methods for dimension reduction including principal components analysis, canonical correlation analysis, and factor analysis
• Multivariate regression methods.
• Classification methods, such as linear discriminant analysis.
• Multivariate normal distributions and their use in multivariate analysis of variance tests
• Exploratory data analysis methods such as clustering tools, and multi-dimensional scaling, as well as other data visualisation techniques.
• Statistical software for conducting analysis of multivariate data

In this module a variety of techniques and areas of mathematical optimisation will be covered including Lagrangian methods for optimisation, simplex algorithm linear programming and dynamic programming. You’ll develop techniques for application which can be used outside the mathematical arena. 

You will be introduced to Einstein’s theory of general and special relativity. The relativistic laws of mechanics will be described within a unified framework of space and time. You’ll learn how to compare other theories against this work and you’ll be able to explain exciting new phenomena that occur in relativity.

Scientific Computation and Numerical Analysis covers advanced numerical methods, and its analysis, for the approximate solution of important mathematical problems, such as solving partial differential equations (PDEs), finding solutions to overdetermined systems, rapidly solving large linear systems of equations, and finding matrix approximations. Methods discussed have important applications in modern-day society, including PDEs in engineering and applied sciences, Google's PageRank algorithm, model reduction and machine learning in data science. Specific techniques include:

• The finite element method (FEM) for PDEs, including weak formulations and its Galerkin discretization, and a priori error analysis
• Advanced methods for linear systems, including QR factorization, perturbation and conditioning theory, and the innovative Conjugate Gradient (CG) method
• Methods for low-rank approximation, including computation of eigenvalues, the singular value decomposition (SVD), and the fundamental Eckart-Young theory for best rank-k matrix approximation.'

This module is concerned with the two main theories of statistical inference, namely classical (frequentist) inference and Bayesian inference.

You will explore the following topics in detail:

• sufficiency
• estimating equations
• likelihood ratio tests
• best-unbiased estimators

There is special emphasis on the exponential family of distributions, which includes many standard distributions such as the normal, Poisson, binomial and gamma.

This module will develop your knowledge of discrete-time Markov chains by applying them to a range of stochastic models. You will be introduced to Poisson and birth-and-death processes. You will then move onto more extensive studies of epidemic models and queuing models, with introductions to component and system reliability.