Mathematics MMath (Hons)

Introduction to Mathematics (Core 1)

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 background, both in the UK and abroad, and therefore there are topics that we teach from scratch, such as linear algebra and complex numbers.

Set theory, Logic, Numbers, and proofs

This part of the course is aimed to get students thinking mathematically. The building blocks of mathematics lie in these topics, particularly set theory, and other topics studied will be intrinsically linked to these ones.

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.

Complex numbers

This topic is based around the square root of one, which we call the imaginary number. We look at these fascinating mathematical objects both from a numerical viewpoint and a geometric viewpoint, looking at their properties as that will be utilised throughout this degree.

Linear algebra

We begin this topic by looking at vectors and matrices before looking at how we can use such objects to solve several equations at once. We also look at eigenvalue problems related to matrices.

Statistics

Being able to effectively deal with data is a prerequisite for most scientific based employment, and this topic will introduce some of the basics of data handling.

MATLAB & programming

Modern mathematicians are expected to know some programming and we introduce students to the basics of structured problem solving from a programming perspective, before moving on to using MATLAB – an industrial piece of software that helps solve many mathematical and data-based problems.

Introduction to Mathematics (Core 2)

For those doing single honours, other mathematical topics are introduced that not only complement some of the topics in Core 1 but take certain topics further.

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.

Application of mathematics (mechanics, biology, physics, financial mathematics)

Following on from mathematical modelling, we look at several applications of mathematics. There are many topics we could cover here including applications to physics, biology, chemistry, engineering and financial mathematics. We look at simple mathematical models and how these are analysed.

Difference equations

Difference equations occur when we are looking at models in what is known as a discrete time frame, i.e. when we consider just certain points in time in an interval, and not all of time for that interval (which is known as continuous time). We look at the properties of these equations, their solutions and how to use this information in real-world problems.

Ordinary differential equations

These are equations that contain derivatives. We look at what these equations are, their properties and some of the basic techniques on how to solve them. We look at some of the application of these such as Newton’s model of heating and cooling, bimolecular reactions, to name just a couple.

Graph theory

Graph theory looks 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 Matematics

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.

Introduction to Numerical analysis

Sometimes an equation or a system of equations cannot be solved using usual methods, and so we have to look numerical methods to generate numerical values of the solutions which will then help us understand what the solutions look like. We introduce the basics of numerical methods here and look at certain numerical analysis topics, such as numerical differentiation.

Matematical communication

As mathematicians, we might at times be asked to perform some analysis for others who may not be mathematicians. Once we have done that, getting that information to the other people in a way that is understandable to them is a vital skill. In this topic, we look at how to write and present mathematical topics in a way that is understandable to non-mathematicians.

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 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 geometry

In this topic, we look at some geometrical techniques that utilise calculus. We look at curvature and arc length of curves, before studying the Frenet-Serret equations.

Linear algebra

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

Statistics & R programming

Expanding on topics covered in year 1, 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.

Number theory & 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 it’s applications to real world situations, such as puzzle solutions

Explorations in Mathematics (Core 2)

For those doing single honours, 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.

Systems of ordinary differential equations

Following on from topics covered in 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 Lokta-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 that allows us to solve certain classes of such equations.

Laplace transformation

Laplace transformation is used to convert differential equations into algebraic equations (polynomials). On doing this, 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 

Following from year 1, we consider methods for solving numerically differential equations, such as Euler, Runge-Kutta, and Taylor methods.

Different Equations

We follow on from the theory developed in year 1. 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, this making it easier to solve. The inverse z-transform is then applied to get the solution to the original difference equation.

Advanced Studies in Mathematics (Core 1)

In year 3, 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 in the first two 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, in particular how to estimate the uncertainties and to assess the quality of a given model.etries, groups and conservation laws

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.

Symmetries, groups and conservation laws

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 the conservation of fundamental quantities such as energy or momentum. We study Noether’s theorem which epitomise the connection between Mathematics and Theoretical Physics.

Galilean and special relativity

We introduce the principle of relativity (1632), which stated 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.  

Metric Spaces and Topology

In this topic we introduce the idea of a metric space, which is an abstract space in which we can define a meaningful notion of 'distance' between two points.  This enables us to generalise concepts from real analysis such as convergence, continuity and open sets.  In this second part we go on to look at topology which is the mathematical study of spaces which have no notion of distance, angles or other similar quantities and for this reason topology is sometimes referred to as 'rubber sheet' geometry.

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.

Linear and nonlinear waves

Waves occur in many natural systems (oceans, electrical circuits, human tissue) and the study of these is vital to understand such systems. We introduce methods to study waves as solutions to certain equations. We look at stationary waves, travelling waves, and waves that occur in nature such as transport waves.

Integrable systems

We consider particular types of systems that are classed as nonlinear, and look at their solutions using analytical techniques. We study the symmetries of these systems and apply techniques such as Bäcklund transformations to extract their solutions.

Hamiltonian systems

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

Chaos theory

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.

Calculus of variations

Many physical problems can be studied using variational techniques. We introduce the main methods using a very simple problem – the shortest distance between two points problem. We study the properties of functionals, extending the problem to the Euler-Lagrange equations before looking at further problems such as the brachistochrone and minimal surface of revolution.

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.  

The fourth year of MMath is split into four equal courses. There are three teaching courses, that are common with the courses on the MSc Mathematics programme, plus a dissertation.

Integrable Systems

This course is an introduction to integrable systems and their applications. It is split into three main parts with each part covering different area of integrable systems and all of them are interrelated. The area subjects which will be covered are the following ones.

Fourier transformations and generalized functions

We introduce Fourier transformation and use integration on the complex plane for its computation. We discuss distributions or generalised functions and their importance as fundamental solutions.

Integrable systems and solitons

We look at certain properties of nonlinear differential and difference equations which are related to the notion of integrability. Specifically, we consider ordinary differential and difference equations and their integrability in the Arnold-Liouville sense. We discuss partial differential and difference equations, their soliton solutions and how they are related to integrability.

Properties of integrable systems

We focus on certain properties of integrable systems. We look at Lax pairs and conservation laws, Bäcklund and Darboux transformations and their nonlinear superposition principles. Hierarchies of generalised symmetries and their relation to integrability are next discussed with integrability conditions and canonical conservation laws before looking at reductions.

Elements of Pure Mathematics

The overriding aim of this course is to expose students to areas of pure mathematics that students don’t normally see at Undergraduate level. This course will cover topics that will stretch student’s imagination and will cover areas of pure mathematics that are the forefront of mathematical research

Riemann Surfaces

The theory of Riemann surfaces studies holomorphic functions on surfaces. We start this course with a definition of what a holomorphic function is as well as holomorphic maps and associated theories from complex analysis. We then look at meromorphic differential before moving onto residues and integrals. Certain properties of holomorphic functions such as uniform convergence and analytic continuation are then discussed, before looking at the Riemann sphere and compact Riemann surfaces.

Lie Groups

A lie group is a group and also a manifold. Therefore, we start this course by looking at the definition a group and its related properties, before looking what a manifold is and its various properties. We look at a variety of problems with Lie groups as well as examples from topology.

Differential Geometry

We start the course by looking at the curvature and torsion properties for space curvces, before looking at the Serret-Frenet Basis for space curves. We then look at surfaces in three space together with the first fundamental form and curves on surfaces. We then move onto the second fundamental form before looking at Gauss curvature, mean curvature, asymptotic directions and curves, and the Gauss-Weingarten equation.

Elements of Applied Mathematics

This course takes a journey through some of the 

Fractal Geometry

Fractal geometry is an alternative look at what constitutes a geometrical object and their properties. Objects like coastline and trees can been approximated to be like classical mathematics shapes, but in fact they are not. Fractal geometry takes a look at the properties of these shapes, in particular what constitutes a shape to be a fractal, what is the fractal dimensions, and how fractals interact with each other. 

Applied Dynamical Systems

In this topic, we look at how systems of differential equations can be used to study how things change over time. We look at what a continuous dynamical system is, what properties it has, the various methods to analyse such systems, and how small changes to their solutions can lead to very different behaviours.

Bifurcation Theory and Finite Element Methods

We extend the work on continuous dynamical systems to take a look at how solutions to such system change when a parameter in the system changes. This is known as a bifurcation. We look at a variety of bifurcations (pitchfork, Hopf, saddle node) and see how they behave. In addition, we will also look at the basic of Finite Element Methods – a way of solving partial differential equations. We will use MATLAB to help to do this, and see how methods from other areas of mathematics are brought together to help solve these problems.

Advanced Dissertation

The dissertation will enable students to commence work on a topic of interest to them but will be under the research topics of the school staff at the time. The project will be a research based piece of work commensurate to Level M(7) expectations. Students will receive supervision from an experienced research active member of staff on a basis that is in line with the student's progress on the project.