# Mathematics BSc (Hons) (with Foundation Year)

UCAS Code: **G110**|Duration: **4 years**|Full Time|Hope Park

UCAS Campus Code: **L46**

Accredited|Work placement opportunities|International students can apply|Study Abroad opportunities

## About the course

Mathematics is a fascinating and exciting subject. It is the language of modern Business and Commerce, Engineering, Science and Technology and is as old as mankind. As the universal language of science it is the best tool we have to describe reality. At Liverpool Hope, you will develop a passion and enthusiasm for mathematics and its applications. Mathematics encompasses many analytical and numerical methods that are used to solve scientific and industrial problems.

Mathematics at Liverpool Hope has been designed to help you develop strong analytical and numerical abilities, and skills so that you learn how to look at problems, break them down into simpler questions and then solve them. Mathematics at Liverpool Hope can be taken as a single honours degree or a combined honours degree with a related subject.

The degree will cover all areas of mathematics including pure mathematics, applied mathematics and statistics. By the end of the degree, you will be confident in tackling real world problems mathematically. By studying with us, you can expect to be given not only first class tuition and teaching, but first class support. We pride ourselves on providing an excellent student experience, and the tutors at Liverpool Hope work hard to ensure that you get the most from your degree.

## Course structure

Teaching on this degree is structured into lectures, where all students are taught together, seminars of smaller groups of around 15-20 students, and tutorials which typically have no more than 10 students.

If you are studying Mathematics as a Single Honours degree, in your first year of study there are approximately 12 teaching hours each week, which reduces to approximately 10 teaching hours in your second and third years. If you are studying Mathematics as a Combined Honours degree, in your first year of study there are approximately 6 teaching hours each week, which reduces to approximately 5 teaching hours in your second and third years.

On top of teaching hours, you are also expected to spend a number of hours studying independently each week, as well as studying in groups to prepare for any group assessments you may have.

## Accreditation

IThis Single Honours BSc degree has been accredited by the Institute of Mathematics and its Applications. This degree will meet the educational requirements of the Chartered Mathematician designation, awarded by the Institute of Mathematics and its Applications, when it is followed by subsequent training and experience in employment to obtain equivalent competences to those specified by the Quality Assurance Agency (QAA) for taught masters degrees.

## Assessment and feedback

There are a number of assessments across your three years of study, including written exams, portfolios, and coursework.

You will be given feedback on your assessments, and you will have the opportunity to discuss this with your tutor in more detail.

### Foundation Year

The Foundation Year is a great opportunity if you have the ability and enthusiasm to study for a degree, but do not yet have the qualifications required to enter directly onto our degree programmes. A significant part of the Foundation Year focuses upon core skills such as academic writing at HE level, becoming an independent learner, structuring academic work, critical thinking, time management and note taking.

Successful completion of the Foundation Year will enable you to progress into the first year (Level C) of your chosen honours degree. Further details can be found here.

### Year One

### 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 backgrounds, 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, probability and combinatorics**

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. We will also look at probabilistic techniques and how to deal with discrete data using combinatorics.

**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**

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 applications 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 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.

**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.

### Year Two

### 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 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 transformation**

Laplace transformation 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**

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

**Differential 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, making it easier to solve. The inverse z-transform is then applied to get the solution to the original difference equation.

### Year Three

### 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 methods**

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.

**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**

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.

**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.

## Entry requirements

There may be some flexibility for mature students offering non-tariff qualifications and students meeting particular widening participation criteria.

## Careers

As a Liverpool Hope Mathematics graduate, you will be highly competent in abstraction, analysis of structure and logical thinking. You will also have expertise in formulating and solving problems. You will have focused on the use of mathematics in solving real-world problems that arise in industrial, commercial, physical, biological and educational contexts and, as a highly numerate graduate, you will be in great demand from employers.

In the 2019 Complete University Guide, 93% of our mathematics graduates were either in employment or in further study within 6 months of graduating, placing us 3rd in the country for mathematics graduate employment. Some of our past graduates have gone on to further study, and there are opportunities to study at Liverpool Hope for taught Masters such as MSc Mathematics, as well as Doctoral research-based qualifications.

## Enhancement opportunities

### SALA

The Service and Leadership Award (SALA) is offered as an extra-curricular programme involving service-based experiences, development of leadership potential and equipping you for a career in a rapidly changing world. It enhances your degree, it is something which is complimentary but different and which has a distinct ‘value-added’ component. Find out more on our Service and Leadership Award page.

### Study Abroad

As part of your degree, you can choose to spend either a semester or a full year of study at one of our partner universities as part of our Study Abroad programme. Find out more on our Study Abroad page.

## Tuition fees

The tuition fees for the 2024/25 academic year are £9,250 for full-time undergraduate courses.

If you are a student from the Isle of Man or the Channel Islands, your tuition fees will also be £9,250.

The University reserves the right to increase Home and EU Undergraduate and PGCE tuition fees in line with any inflationary or other increase authorised by the Secretary of State for future years of study.

## Additional costs

As well as your tuition fees, you need to consider the cost of books, software, and general computer consumables such as discs and printing. We estimate this to cost around £300.

There is a small cost for Student IMA membership, and once you graduate, there is an annual fee for Associate Membership – full details of costs can be found on the IMA website.

You will also need to consider the cost of your accommodation each year whilst you study at university. Visit our accommodation pages for further details about our Halls of Residence.

## Scholarships

We have a range of scholarships to help with the cost of your studies. Visit our scholarships page to find out more.

## International tuition fees

The International Tuition fees for 2024/25 are £12,500.

Visit our International fees page for more information.

## Course combinations

This course is also available with Foundation Year as a Combined Honours degree with the following subjects:

Please note that the following courses are not accredited by the Institute of Mathematics and its Applications.