Mathematics with a Year Industry BSc (Hons)

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 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

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, and Newton’s 2^nd law.

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.

Probability I

This topic will cover the basics of probability theory, such as binomial theorem, Bayes theorem, distributions, and random variables.

Numerical analysis I

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 and integration.

Mathematical 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

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 study 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 test including chi squared, ANOVA, and t-tests. We also analyse data using a programme called R.

Number theory

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.

Explorations in Mathematics

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 II

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

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

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

Although we have covered statistical methods in years 1 and 2, in year 3 we look at how to use statistics for modelling purposes and applying them to physical problems. We look at a variety of techniques, including Markov processes and conditional probability techniques such as Bayes theory.

Symmetries, 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.

Galilean and special relativity

Although very much a physics problem, we look at the mathematics behind this famous theory. We look at Galilean transformations, hyperbolic geometry and the Minkowski spacetime.

Group theory

Group theory is about collections of mathematical objects that share similar properties. This part of the course will look at the pure mathematical properties of groups and will use examples such as symmetry and alternating groups to illustrate the methods.

Topology

This is a pure mathematical topic that looks at geometrical objects and how such objects behave when they are deformed somehow. We cover methods such as metric spaces, Cauchy sequences, and Hausdorff spaces to study such objects.

Complex analysis

This expands on complex numbers covered in earlier years. We look at how complex functions behave, their differentiability, and apply theorems such as Cauchy and Cauchy-Riemann to such functions.

Advanced Studies in Mathematics

As with the 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 the analysis of various systems describing electrical circuits, traffic flow, signals in fibre optics, human tissue, etc, and their study is vital to understand such systems. We introduce methods to study waves as solutions to certain equations, and discuss the properties and applications of stationary waves, travelling waves, and dispersive waves.

Integrable systems

These are systems of nonlinear equations which have some very remarkable properties. As there is no universal definition for integrability, we present various criteria for integrability (Lax pairs, Darboux and Bäcklund transformations, symmetries, soliton solutions) and explore their relations, as well as we employ certain methods to construct special solutions to integrable systems.

Hamiltonian systems

The evolution of some physical systems, such as planetary systems, can be described using Hamiltonian dynamics. We consider continuous and discrete Hamiltonian systems, discuss the notion of Louiville integrability, conserved densities, as well as the Lax representation of such systems and its relation to integrability.

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. Student 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.