Mathematics (with Year Industry) BSc (Hons)

This year is designed to ensure that you have all the fundamentals you need for a Maths degree, from mathematical logic and proof to calculus and linear algebra. The year consists of four modules with combined students taking Core Mathematics and Calculus, and the single honours additionally taking Mathematical Modelling and Applications of Mathematics.

Core Mathematics Introduction to Mathematics

The purpose of this course is to cover the fundamentals of mathematics that a new undergraduate should know. Therefore, the topics covered are broad in nature but sufficiently different from A-Level or equivalent mathematics. It is appreciated that students come from different mathematical backgrounds, both in the UK and abroad, and therefore there are topics that we teach from scratch, such as linear algebra and complex numbers. All students will cover mathematical logic, proof, numbers and set theory within this course.

Calculus

Although this subject is studied at school, we take a different approach to calculus and look at this as the study of functions and the way they behave. We will look at the basics of functions, differentiation, integration and basic differential equations. In the second half of the course, we look at multivariable calculus and vector calculus.

Mathematical modelling

Taking a real-world problem and using mathematics to solve it is one of the sought-after skills for mathematics graduates. This part of the course will introduce students to the theory of mathematical modelling, and how to approach real world problems that need solving mathematically. We look at the various types of modelling that we can do from discrete modelling with difference equations, to continuous modelling with differential equations.

Application of mathematics

This module looks at three fundamental areas that we believe are important for mathematical graduates to be skilled in.

- Mathematical Communication: We teach you the fundamentals of how to communicate your mathematics with others. We show you how to write mathematics using a typesetting programming language called LaTeX, before moving on to presentation of your mathematics in the form of posters and slides presentations. Throughout this part, you will be working on a mini project as part of a group, and will have to present your findings at the end of the year.

- Graph theory: We look at how mathematical objects are linked. It takes a set of objects and gives precise instructions on how to get from one element of that set to another, if it is possible. We look at various methods of creating graphs and look at some famous graph theory problems such as the Chinese postman problem.

- Financial Mathematics: We look at how mathematics is used to address problems in finance, from the simple banking interest problems to utilising probabilistic methods to predict how a financial situation will evolve over time. This course will comprise not only a theoretical approach but will also a practical side by analysing data.

Explorations in Mathematics (Core 1)

With the fundamentals covered at Year 1, we keep the Year 2 topics quite broad but start to focus in on some areas of mathematics. Statistics is a hugely sought-after skill currently, and so we cover the main statistical modelling methods within this course, and analyse the data using R, which is also a very sought-after statistical programming language.

We also cover a variety of other topics, introducing some new topics, and expanding on topics that were covered at Year 1.

Multivariable and Vector Calculus

We extend the calculus covered at year 1 to include functions of several variables. We look at how these functions behave, how to differentiate them, and look at the several methods of integration. Various other topics include surface area and the basics of vector calculus.

Differential and Algebraic Geometry

For differential geometry, we look at some geometrical techniques that utilise calculus. We introduce vector functions and their properties, before looking at curvature and arc length of curves, and the Frenet-Serret equations. Algebraic geometry is a branch of mathematics that studies the solutions of systems of polynomial equations using geometric methods. It combines techniques from both algebra and geometry to understand the structure and properties of these solutions, which can form complex shapes known as algebraic varieties.

Linear algebra

We extend the linear algebra topics covered in first year and look at vector spaces, matrix factorisation, orthogonal spaces and applications of linear algebra.

Statistics & R programming

We look at distributions, regression analysis, and a variety of statistical tests including chi squared, ANOVA, and t-tests. We also analyse data using a programme called R. This topic is taught such that at the end of it, you will be proficient and confident in analysing data with the correct tests, and interpreting the results of the analyses. 

Number theory and abstract algebra

Number theory is a vast area of mathematics, and we look at a small part of it and its applications to cryptography. We begin by looking at relations on sets, equivalence classes, modulo mathematics, and RSA cryptography. We will then look at the group theory and its applications to real world situations, such as puzzle solutions.

Explorations in Mathematics (Core 2)

For those doing single honours Mathematics, we have designed this part of the course to give students exposure to areas of mathematics that can be applied to other areas of science and technology. Topics such as ordinary differential equations and partial differential equations are fundamental to the modelling of continuous systems in science and technology, and applications of Fourier analysis and Laplace transformation are found in engineering and physics. Other topics such as probability and numerical analysis ensure that the students have a solid grounding in many areas of mathematics that can be taken and applied to the challenging topics of year 3.

Ordinary differential equations

Following on from topics covered in the first year, we now look at differential equations that are classed as systems (more than one equation to be solved simultaneously). Methods from solving single equations are now extended to systems and we look at areas such as eigenvalues, steady states, phase portraits and applications such as Lotka-Volterra.

Partial differential equations

Partial differential equations are differential equations that involve functions of several variables. They are generally difficult to solve, but we demonstrate techniques which allow us to solve certain classes of such equations.

Laplace Transforms

Laplace transform is used to convert differential equations into algebraic equations (polynomials). Once this is done, it enables the equations to be solved quicker and easier. We show how to derive such transforms and how to apply them to the differential equations.

Fourier analysis

In engineering and science, we often wish to approximate functions to make them easier to solve or implement for other equations. Fourier analysis allows us to do this.

Numerical analysis 

We consider methods for solving numerically differential equations, such as Euler, Runge-Kutta, and Taylor methods.

Z-Transforms

We will look closer at the relationship between sequences and difference equations for both first and second order. We also introduce the z-transform which converts difference equations into an algebraic equation, making it easier to solve. The inverse z-transform is then applied to get the solution to the original difference equation.

The third year is spent at a placement. Students on placement will be supported by staff from the university, both from the mathematics team and the careers service.

Advanced Studies in Mathematics (Core 1)

In year 4, we study topics that are at the forefront of the research interests of the staff currently teaching on the programme. This highlights how the mathematics that has been taught in years 1 and 2 is used in the more advanced mathematical topics.

Statistics and data modelling

Building up from the statistical methods learnt previous years, we look at some practical applications in the real world. Using techniques such as Markov chains and Chi-square distributions, we learn how to model some dataset.

Many physical problems look at the conservation of particular properties such as energy and momentum. This topic will look at some of those laws and mathematical methods and will cover Euler-Lagrange equations, Noether’s theorem, time invariance and conservation laws.

Mathematical Physics

We start by defining quantities known as Lagrangian and Hamiltonian, and we show how the Euler-Lagrange equations emerge naturally from the least-action principle. We then study the invariance of such equations under certain transformations (eg. translations and rotations). We show how these symmetries are related to the conservation of fundamental quantities such as energy or momentum. We study Noether’s theorem which epitomises the connection between Mathematics and Theoretical Physics.

We then introduce the principle of relativity (1632), which states that there is no physical way to differentiate between a body moving at a constant speed and an immobile body. We then show why and how Einstein had to modify this principle in accordance with the observation that the speed of light is independent of the observer. In particular we will derive the Lorentz transformations from first principle. We will then show why time is not universal with the famous twin paradox. Finally we will introduce some elements of hyperbolic geometry, pseudo-metric and Minkowskian product.

Group theory

Group theory is an important subject in mathematics that deals with algebraic structures known as groups.  In this topic we start with some basic definitions and examples of groups, such as groups of permutations, groups of symmetries of regular polygons, and  groups of congruence classes modulo an integer.  We go on to cover interesting results such as Lagrange's theorem and the classification of finitely generated abelian groups.

Complex analysis

A complex function is a mapping from the complex numbers to the complex numbers. Just as previously studied in the calculus of real functions, we can define limits, continuity, and differentiation for complex functions. Whilst the definitions are similar to the real variable case, the consequences of differentiability for complex functions are much stronger and lead to powerful and often surprising results such as Cauchy's Residue Theorem. 

Advanced Studies in Mathematics (Core 2)

As with its sister course Advanced Studies in Mathematics (Core 1), this course will cover topics that are at the forefront of the research interests of the staff currently teaching on the programme.

Symmetries of differential equations

A symmetry of a differential equation is a transformation which does not change the equation. They were introduced by Sophus Lie and provide us the means to study and classify differential equations and find particular solutions. In this topic, we present Lie's method and apply it in finding the symmetries of ordinary differential equations. We also cover how we can use symmetries of a given equation to construct particular solutions, to reduce the order of the equation, and to integrate it.

Hamiltonian systems

The evolution of some physical systems, such as planetary systems, can be described using Hamiltonian dynamics. In this topic, we cover methods and theorems, such as Louiville integrability, Louiville-Arnold theorem and Lax representation, to study the properties of these systems, as well as their solutions.

Chaos theory and fractal geometry

Sometimes, seemingly innocent looking equations can produce the most remarkable and unexpected behaviour. Chaos theory looks at how we can use mathematical techniques to study how equations that give us solutions that are deterministic exhibit what seems like random behaviour. We look at the logistic equation as the base example, and cover topics such as discrete dynamical systems, stability of fixed points including finding Lyapunov exponents, before finishing with some fractal geometry.

We round off this part of the course with a look at fractal geometry and the mathematics behind these fascinating objects. We look at fractals from their basic set theory properties and extend these ideas to fractal dimension and applications to dynamical systems, linking it to the chaos theory we have covered earlier.

Perturbation methods

Sometimes equations (algebraic or differential) can contain small valued parameters. We show techniques that handle these types of equations and enable us to extract solutions to such equations. These techniques can sometimes turn extremely difficult problems into much simpler ones.

Research Projects and Dissertations

All students will undertake project work either as a research project (for combined students) or as a dissertation (for single honours students). Students will be able to choose from a broad range of ready-made projects (from pure mathematical topics to applied mathematics), but can create their own in certain circumstances. Students will be allocated a supervisor, with whom they will meet on a regular basis. These projects are a chance for the students to be creative with mathematics, and to work on an area of mathematics that takes their interest.