Mathematics BSc (Hons)

This module will build on the earlier module, Calculus. We will extend the ideas from the first year to dealing with both scalar fields and vector fields, the concepts with wide applications spanning from theory of fluids through meteorology to traffic control, as well as to functions of complex variables that allow for a very elegant and succinct solution of many classical problems of calculus. 

Compulsory

This module continues linear algebra and differential equations from the first year with the overlap between the areas emphasised. In both cases, the general goal will be to explore ways in which one can find a ‘suitable basis’ for a given problem. This could include orthonormal bases, bases of eigenvectors or bases of generalised eigenvectors. It will conclude with Singular Value Decomposition which will bring together bits of linear algebra seen over the first two years. The module will also cover second order ODEs (linear, but with non-constant coefficients), systems of first order ODEs, series solutions and self-adjoint operators. 

Compulsory

The module will introduce you to fundamental concepts of modern abstract algebra starting with groups. Key examples will be recalled before the theory is explored in detail. This will include fundamental results such as Lagrange’s theorem and its partial converse due to Sylow, alongside key ideas such as quotient groups, isomorphism theorems, direct products and group actions. Next there will be an analogous discussion of rings, where you will encounter ideals, quotient rings and homomorphisms. Finally, by considering the ring of polynomials over a field, you will see how to construct finite fields.  

Compulsory

The module acts as an introduction to the vast fields of Partial Differential Equations (PDEs) and Analytical Mechanics. You will be introduced to standard techniques to solve a range of PDEs. The module also provides keys elements relating to variational calculus and focuses on Lagrangian and Hamiltonian formalisms for Analytical Mechanics, which are naturally formulated via PDEs. 

Compulsory

By starting with only the basic properties of real numbers, a rigorous approach will be pursued whereby the main results in elementary differential calculus are proven. This will include the fundamental epsilon-delta definition. You will see sequences, series, continuity and differentiability of functions. This will further develop your powers of logical thinking and expand upon the notion of formal definition and rigorous proof seen in first-year algebra. This is a key aspect of modern mathematics and one which takes time and practice to succeed at. In doing so, you should improve your understanding of calculus and become more comfortable in formal proof. 

Compulsory

This module will introduce two of the most commonly used statistical techniques, multiple regression and analysis of variance (ANOVA). The statistical inference concepts of hypothesis testing and estimation will be extended within the statistical modelling framework. Statistical computing environments and packages will be used throughout. The methods taught are used extensively in industry, commerce, government, research and development. This module is particularly relevant for those intending to go on placement year, for graduate jobs, and for final year statistics projects². 

Compulsory

This module² provides you with an opportunity to reflect upon and gain experience for an approved placement undertaken during your programme. A placement should usually be at least 26 weeks or equivalent; however, each placement will be considered on its own merits, having regard to the ability to achieve the learning outcomes. 

Optional

This module² provides you with an opportunity to reflect upon and gain experience for an approved international study/work placement undertaken during your programme. A work/study placement should usually be at least 26 weeks or equivalent; however, each placement will be considered on its own merits, having regard to the ability to achieve the learning outcomes. 

Optional

This module will present to you a foundational treatment of number theory, covering many important practical results such as the Chinese Remainder Theorem and Hensel’s Lemma. You will gain experience performing calculations in modular arithmetic and build on your ability to write rigorous proofs from the second-year modules Groups and Rings, and Real Analysis. The pure concepts from number theory will then be directly applied to modern problems of cryptography. You will see how these ideas are put into practice, for example in the RSA cryptosystem. 

Compulsory

In this module, you will be introduced to a selection of advanced mathematical methods such as asymptotic analysis and Green’s functions to solve a range of complex real-world problems well beyond the scope of previous modules.

Compulsory

This is an introductory module in fluid dynamics. The module aims to build and develop the fundamentals for incompressible flows of fluids. In this module, we will cover a range of different fluid flow problems including irrotational flow, viscous flow, and boundary layer flows. In this module you will have an opportunity to apply mathematical techniques to solve real-world problems, use analytical and numerical methods to determine physically relevant solutions, and gain an understanding of current research topics in fluid dynamics. 

Compulsory

This module forms a major individual study at the honour’s level in areas related to mathematics or applied mathematics. You will take the responsibility of managing such a study through all its stages. The project will build upon your knowledge and skills developed in previous years. It will further look to develop your skills of enquiry, research and innovation and will enhance your critical and communication skills. 

Compulsory

Choose two out of the following four modules:

Financial Mathematics – 20 credits
The module serves two goals. First, the module introduces you to the main instruments that are traded in the financial markets including their practical uses for investment, hedging and speculation. Second, it equips you with an understanding of mathematical models and solution techniques that are currently used in financial engineering. Practical calculations with financial data illustrate the theory.

Topology and Applications – 20 Credits
The purpose of the module is to provide you with an elementary introduction to the methods of algebraic topology via homology of simplicial complexes. There will be an emphasis on computation thereby enabling further study in fields which are benefitting greatly from tools and ideas in topology, such as topological data analysis. This will build off the methods of linear algebra which you have seen in years one and two.

Quantum Information and Quantum Computation – 20 credits
In this module you will be introduced to fundamental concepts of quantum information and quantum computation; discuss applications with and without quantum advantage; engage with contemporary software. This module offers a comprehensive introduction to the main ideas and techniques of the field of quantum computation and quantum information. The required physical, mathematical, and computational frameworks are briefly introduced and effects such as fast quantum algorithms, quantum teleportation, quantum cryptography, quantum error-correction, quantum Fourier transform, and quantum searches are discussed. You will acquire a working understanding of the fundamental tools and results in the field and will be able to engage with contemporary software and applications.

Advanced Topics in Statistics – 20 credits
This module introduces you to several important topics in the areas of advanced statistics. This module is useful if you are interested in a Statistics postgraduate study or in careers involving Statistics or Data Science.

Optional