Compulsory modules

Sets, Functions and Proofs
The aim of this module is to support your progress from the methods-based approach to the higher levels of understanding and rigour expected at degree level Mathematics. You will be introduced to mathematical statements, the meaning of proof, and some proof strategies. The concepts of sets and functions will then be discussed. Finally, the module will cover the algebraic development of number systems and their properties.

Limits, Series and Calculus
This module aims at deepening your understanding and appreciation of the origins of the essential techniques of elementary calculus, explaining why these methods work. Starting with developing important preliminary ideas such as the real numbers, functions, sequences and series, on a more formal basis this leads to a fundamental concept of a limit, serving as a basis for further advances in differential and integral calculus.

Mathematical Methods
This module aims to provide you with a self-contained and intensive summary of all the essential methods and techniques required at the start of an undergraduate programme in mathematics. The topics covered include algebra, calculus, functions, trigonometry, and vectors and, as such, the module provides a useful refresher of these. The module also ensures that all students cover the topics to the same depth and overcomes the differences in syllabi of various pre-University qualifications.

Mathematical Communication, Investigations and Problem Solving
This module is designed specifically to help you develop mathematical communication and problem-solving skills. In this module, you will be presented with unfamiliar and less structured problems that may be open to several different approaches. This module will enhance your employability skills through teamwork on several group projects and focus on mathematical communication and presentation of your results.  It will also help you appreciate mathematics as a powerful method for describing and solving real-world problems. 

Linear Algebra
This module will begin with an introduction to number theory by looking at topics such as divisibility, greatest common divisors and modular arithmetic with a strong emphasis on rigour. Then you will study core concepts from linear algebra including matrix algebra, solving linear equations, determinants, eigenvalues and eigenvectors. You will also cover aspects of vector algebra with a particular emphasis of on three-dimensional vector algebra and geometry with a discussion on lines and planes, linear independence and bases.

Differential Equations and Multivariable Calculus
Many real-world problems are governed by differential equations, the solution of which can help us understand their properties. For instance, oscillation of a pendulum, and the population evolution of a fish species, can all be described by ordinary differential equations. This module, being a prerequisite for several modules in the second and third years, will introduce you to the basic techniques for solving differential equations and familiarize you with multivariable calculus, including partial derivatives, double integrals, and their applications. 

Optional modules

Statistics with Applications in R
This module aims to provide you with a sound knowledge of core statistical concepts, skills and techniques. It will enable you with applying statistical ideas to solve problems from a wide range of disciplines; developing competency in interpreting and explaining solutions of problems in non-technical language; giving fluency in procedural skills, common problem-solving skills and strategies; using statistical tools to analyse real data sets in the statistical software environment R. 

Introduction to Algorithms
In this module you will learn to apply a variety of computational problem-solving techniques to solve problems from the real world. You will be introduced to algorithmic design strategies and learn how to analyse an algorithm to determine its correctness and efficiency, to solve problems from a variety of real-world domains. You will learn transferable skills such as interdisciplinary expertise, creativity, and problem-solving. 

Compulsory modules

Linear Differential Equations
Differential equations are used as mathematical models describing various processes in the real world. This helps investigating problems related to population dynamics, heat dissipation in objects, vibrations of engineering structures and wave propagation, among many others. Some situations require more than one differential equation to be solved at once, e.g., modelling dynamics of two populations (e.g., foxes and rabbits). Within this module you will study a variety of methods for solving differential equations.

Probability
Probability is the mathematics of uncertainty and randomness. You will begin with classical notions of probability associated with the analysis of games of chance using cards, dice, etc., then moving to treatment of probability of random events and further to definitions of statistical independence and conditional probability. Then, you will study discrete and continuous, univariate and bivariate, random variables, covering key concepts of expectation, variance, covariance. Applications to a wide range of theoretical and real-world problems will also be discussed. 

Computational Mathematics with Python
The aim of this module is to introduce you to the three elements of scientific computing; numerical analysis, programming and modelling.  The programming element provides you with valuable transferrable skills in Python. No prior knowledge of the language is assumed or required for the module. We also aim to give you a broad appreciation of the Jupyter environment for writing Python code, displaying computational results and annotating the algorithmic developments with the underpinning mathematics using markdown.

Exploring Algebra and Analysis
Algebra and analysis are two of the main branches of pure mathematics. Linear algebra generalises vector and matrix algebra encountered at Level 4. Mathematical analysis, dealing with the infinitely small and large, aims to provide rigorous theoretical foundations for calculus. You will study abstract vector spaces, focussing on the role of bases in describing elements. You will also cover rigorous proofs for calculus results. Each topic will be introduced via guided investigations, followed by lectures to develop the formal theory.

Complex Variable and Vector Calculus
This module will introduce you to both vector calculus and complex variable. Vector calculus studies vector functions of one or more real variables and its techniques and results are essential tools in many areas of applied mathematics, such as fluid mechanics. Complex variable studies functions mapping complex numbers to complex numbers. In this module, you will achieve elegant results in integration which can often be used to evaluate definite integrals of real-valued functions, without any knowledge of the indefinite integral.

Abstract Algebra
Abstraction is a powerful idea in Mathematics: by focusing on the underlying properties of Mathematical systems we can identify connections and similarities that are not visible on the surface. In this module you will explore these techniques and styles of thinking. The central concept is that of a group. You will study some of the fundamental abstract results concerning groups. You will also consider other algebraic structures, e.g., rings and fields, and discuss applications in combinatorics, geometry, and number theory.

Professional Mathematics and Data Analysis
This module gives the opportunity to further develop skills of mathematical problem solving and application of mathematical results to real-world scenarios, especially those involving the analysis and interpretation of large data sets. This will contribute to further development of your professional skills that are needed when undertaking employment or research, such as working in depth on a problem over an extended period, writing reports, communicating mathematical results to different audiences, and working in collaboration with others.

Optional modules

Dynamics
In this module you will focus on examples of successful application of mathematics in the modelling of various physical systems, within the context of Newtonian dynamics. You will investigate projectiles and simple harmonic motion, and also more advanced laws of planetary motion, problems involving variable mass, such as, for example, rocket motion.

Real Analysis
This module is aimed at equipping you with further core knowledge in analysis, building upon the material covered in MAT-20035 (Exploring Algebra and Analysis), providing a mathematically rigorous justification for methods in calculus. The module is focused on infinite number series, limits of functions and their continuity, followed by differentiability and related problems for extremum of functions of one variable, as well as analysis of power and Taylor series. 

Professional Experience: Flexible Work Placement (Level 5)
In a competitive job market, practical skills and understanding workplace dynamics are crucial.  This module allows you to gain hands-on experience, enhancing your employability, and the opportunity to gain valuable professional insights via a 75-hour placement. You will be supported by various activities to secure a role and can source your own placement or apply to university-sourced roles.  This module offers flexibility on when and how you complete your placement, enabling you to balance your placement around existing demands. 

Compulsory modules

Partial Differential Equations 
A large variety of physical phenomena, e.g. wave propagation, heat conduction, traffic flows, and even the price evolution of the stock market, are governed by partial differential equations (PDEs). This module introduces the basics of PDEs and equips you with some fundamental analytical skills (e.g. the methods of separation of variables and Fourier transform/series).  This module would be useful if you intended to do a Master’s or PhD degree in financial mathematics or continuum mechanics.

Optional modules

Nonlinear Differential Equations 
Systems evolving in time are often modelled by differential equations, with numerous applications including climate change, stock markets, population dynamics, mechanical systems, etc. You will study nonlinear differential equations, for which very few exact solutions are known, hence, introducing qualitative methods for describing various types of behaviour. In particular, you will focus on geometrical representations of dynamics and systematic approximation techniques. The consideration will then progress to dynamics of maps, setting the path for studying multiple periods and chaos. 

Group Theory
Group theory provides a language for the abstract study of symmetry found in various areas of Mathematics. Groups are mathematical objects defined by several rules and exhibiting a rich and fascinating amount of structure. In this module you will study the ways in which groups can act upon other objects such as sets, vectors, shapes in the plane, platonic solids, and even other groups. You will also explore the tools for comparing groups and for classifying groups of certain sizes. 

Number Theory and Cryptography
Number theory studies properties of the integers, with particular emphasis on the prime numbers. Being one of the oldest areas of Mathematics, it is also an active area of modern research. Recently, classical ideas from number theory have been used in the design of secure ciphers. In this module you will trace the subject development from its ancient beginnings to modern applications, including primality tests, factoring algorithms, modular arithmetic, the RSA and ElGamal ciphers, and techniques for attacking these ciphers. 

Financial Mathematics
Financial Mathematics' growing importance is shown by the 2008 financial crisis. It controls various aspects of our daily life. Over the years, the demands of the knowledge in financial mathematics have helped many graduates to gain positions in financial institutions successfully. This module aims to provide you with a mathematical perspective on the valuation of financial instruments (futures, options, etc.) and their risk-management. By the end of this module, you will have confidence solving various problems in stock modelling and derivatives pricing. 

Structured Investigations in Mathematics (SEM1)
The module offers you an opportunity to study a specialist Mathematical topic under the supervision of a member of staff. A wide range of topics may be available, such as advanced linear algebra, solid mechanics, information theory, numerical methods, and topics in the history of Mathematics. In addition to the furthering their subject knowledge you will have the opportunity to develop skills in independent research, time management, communication, and producing a structured report using specialist Mathematical typesetting software (Latex). 

Project
This module is an opportunity for you to explore a particular topic/ problem in mathematics, being supervised by a member of staff. The project will require engagement with appropriate mathematics literature. For students on specialized pathways, your project will need to be within the area of specialisation. The module will enable you to perform an in-depth study of a mathematical topic, develop your research skills and ability to communicate mathematical results to audiences using a variety of different communication methods. 

Fluid Mechanics
Fluid mechanics describes the mathematical theory of the flow of liquids and gasses, and thus underpins the dynamics of the atmosphere and oceans, hence weather and climate prediction, but also how an aeroplane flies, the forces acting on a wind-turbine, flow of blood around the body, and many other applications. This course introduces you to the equations of fluid motion and techniques for solving these equations. 

Waves
The module will provide you with knowledge of mathematical techniques for modelling wave propagation, typically accompanied by physical interpretations. The exposed mathematical theory is illustrated by the discussion of real-world applications including waves arising in musical instruments, such as violins, drums, and flutes. The module also involves small demonstrations and computer-generated animations. The main topics covered in the module include the mathematical background of wave phenomena, elastic waves on stretched strings, membranes and beams, sound waves, and waves in liquids. 

Medical Statistics
This module provides you with knowledge and fundamental skills to gain deeper understanding of statistical and epidemiological methods, and applications in public health, health policy, clinical medicine, and health economics. After completing this module, you will be able to choose and perform appropriate statistical techniques, interpret the results of statistical analysis, and communicate them in a clear, concise and appropriate manner. This module opens employment opportunities to develop career as medical statistician in the academia, research organisations, and pharmaceutical industry. 

Mathematical Modelling
In this module you will learn how real-world problems can be modelled mathematically. You will use appropriate mathematical tools in the model construction and solution process, including linear/nonlinear ordinary differential equations, partial differential equations and relevant solution techniques, such as phase-plane analysis, separation of variables and method of characteristics. Computational tools will involve programming in Python. The modelling ideas will be developed through innovative case studies of real-world scenarios, with individual/group projects contributing to further development of professional skills. 

Structured Investigations in Mathematics (SEM2)
The module offers you an opportunity to study a specialist Mathematical topic under the supervision of a member of staff. A wide range of topics may be available, such as advanced linear algebra, solid mechanics, information theory, numerical methods, and topics in the history of Mathematics. In addition to the furthering their subject knowledge you will have the opportunity to develop skills in independent research, time management, communication, and producing a structured report using specialist Mathematical typesetting software (Latex). 

Professional Experience: Flexible Work Placement (Level 6)
In a competitive job market, practical skills and understanding workplace dynamics are crucial.  This module allows you to gain hands-on experience, enhancing your employability, and the opportunity to gain valuable professional insights via a 75-hour placement. You will be supported by various activities to secure a role and can source your own placement or apply to university-sourced roles.  This module offers flexibility on when and how you complete your placement, enabling you to balance your placement around existing demands.

Professional Experience in Education
This module allows undergraduates to gain academic credit for their work in schools and colleges. The undergraduate will develop a range of skills and be presented with an opportunity to experience an early taste of teaching for those interested in pursuing teaching or training as a career. The undergraduate will work in a school or college for a total of 64 hours, to be agreed between the placement host and student.  The timing of the placement is flexible to support students’ home, study and work life balance, and to meet the needs/requirements of the placement host, but ideally within a concentrated block rather than dispersed over the whole year. Suggested options include 8 × 8-hour (whole day) sessions, or 16 × 4-hour (half day) sessions. Students will have the flexibility to source their own placement, a useful skill for seeking out future employment opportunities, but also can apply for a range of roles sourced by the university. 

Compulsory modules

Masters Project
This module gives you the opportunity to substantially develop research skills required for researching mathematics., such as working in depth on a problem over an extended period, writing reports aiming closer at journal style, presenting and communicating mathematical results with a certain degree of originality to academic audience (poster and seminar talk). You will produce a dissertation at the end of the module. This module will provide employability skills targeted at future academic work but also relevant for graduate employment. 

Optional modules

Hydrodynamic Stability Theory 
Flows of liquids and gasses can become turbulent. Understanding this complicated, chaotic motion is widely regarded as the outstanding problem of classical mechanics. Turbulence can be initiated by instabilities of simple steady flows. You will be introduced to analytical techniques for predicting a flow's stability properties. These methods can predict the approximate size of dominant flow structures in the early stages of the transition to turbulence, and underpin methods for predicting aerodynamic drag, and mixing in the atmosphere and oceans. 

Continuum Mechanics 
If you have heard about the wave equation or heat equation and have ever wondered how they were derived, then Continuum Mechanics will be of interest to you. This module introduces the basic principles that should be followed when formulating the equations governing the static or dynamic behaviour of any continuum whether it is a fluid and solid. You will derive the governing equations for linear elastic materials and Newtonian fluids as special cases and solve  typical boundary value problems. 

Perturbation Methods 
This module introduces you to the asymptotic techniques, which are of vital importance in mathematical models arising from the real-world problems. These 'raw' problems very rarely resemble simple models which can be solved exactly. Fortunately, very often 'raw' problems involve a small parameter, or there is a method for determining an approximate solution which is sufficiently accurate. The module will present an overview of the main analytical perturbation techniques leading to finding approximate solutions of various mathematical problems. 

Field and Galois Theory 
Galois theory is named for the 19th century French mathematician Evariste Galois, who studied symmetries in the roots of polynomial equations. This approach can be used to obtain analogues of the quadratic formula for polynomials of degree 3 or 4, and to prove that it is impossible to find such a formula for degree 5 or higher. In this module you will study these ideas from a modern perspective, employing tools from abstract algebra such as groups, rings, and fields. 

Topics in Group Theory 
Building upon the group theoretical concepts explored in earlier modules, in this module you will focus on techniques for classifying finite groups. Powerful theorems are proved that allow a group to be decomposed into smaller, more manageable pieces. Conversely, methods will be presented for constructing new groups from existing ones. The full range of ways in which this can be done is not fully understood; the module includes some open questions and current research topics.