BSc (Hons) MATHEMATICS AND STATISTICS

MTHA4008A  (20 Credits)

Calculus is the mathematical study of rates of change. It underpins much of applied mathematics, since we are often interested in determining how quickly things will change, whether that is looking at changes in space, time, or some other variable. 

You will start this module by studying complex numbers and vectors. (These topics are not strictly calculus but will be needed in calculus later on). You will then study differentiation – how to determine the rate of change of one variable or function as another variable changes. You will meet the formal definition of the derivative, and discover how to differentiate from first principles, before learning various rules and techniques for differentiating more complex functions. You will also learn about applications to curve sketching and power series 

Next you will study integration (the inverse of differentiation). You will learn various techniques, including substitution and integration by parts, and consider applications to finding areas and arc lengths. You will then learn about methods for solving first and second-order ordinary differential equations. Techniques covered here include reduction of order. and integrating factors. You will learn how to use the computer program Maple to solve differential equations numerically. 

MTHA4001A  (20 Credits)

The module provides an introduction to various fundamental mathematical concepts and techniques that you will need to study more advanced mathematics later in your degree. 

You will gain a thorough introduction to some systems of numbers commonly found in Mathematics: natural numbers, integers, rational numbers, modular arithmetic. You will also learn about common set-theoretic notation and terminology, and a precise language with which to talk about functions. There is emphasis on precise definitions of concepts and careful proofs of results. You will study different techniques of mathematical proof, including: proof by induction, direct proofs, proof by contradiction, contrapositive statements, equivalent statements and the role of examples and counterexamples. 

In addition, you will also learn how to produce mathematical documents using a typesetting system called “LaTeX”, and how to solve mathematical problems computationally using both symbolic algebra software and Excel. 

MTHA4003A  (20 Credits)

This module provides an introduction to two key areas of Pure Mathematics: algebra and real analysis, both of which will be needed as a foundation for further study in your degree. 

In algebra, we start by considering linear equations. This module will provide you with a thorough introduction and develop this theory from first principles. You will learn about the theory of matrices, mainly (though not exclusively) over the real numbers. You will study matrix operations, linear equations, determinants, eigenvalues and eigenvectors, diagonalization and geometric aspects. 

In real analysis, we will explore the mathematical notion of a limit. You will see the precise definition of the limit of a sequence of real numbers, and learn how to prove that a sequence converges to a limit. After studying limits of infinite sequences, we move on to series, which capture the notion of an infinite sum. 

MTHA4008B  (20 Credits)

The module combines two important topics in preparation for your future studies.  

Probability is the study of the chance of events occurring. It has important applications to understand the likelihood of multiple events happening together in order to make rational decisions. You will be introduced to Kolmogorov’s modern axiomatic theory of probability and the concept of random variables. You will study both discrete and continuous random variables. Finally, will explore two applications of probability: reliability theory (which looks at the likelihood of the failure of a piece of equipment at some point in the future) and Markov chains (which model how the likelihood of being in different states evolves in time). 

Multivariable calculus extends concepts of calculus to functions of more than one variable. This leads to the idea of partial derivatives. You will learn how to differentiate functions of more than one variable, and how to do integrals of such functions along curves and over areas. You will also learn how to change coordinates in multiple integrals using Jacobians, and study Green's theorem in the plane.

MTHA4003B  (20 Credits)

This module continues the study of real analysis from the previous semester, and will introduce you to another important area of pure mathematics – group theory. 

In the real analysis thread, you will learn about limits of functions and continuity before moving on to study the formal definitions of the derivative and integral of a function. This then leads to the Fundamental Theorem of Calculus, which proves that integration and differentiation are essentially inverses of each other. 

In the group theory thread, you will focus on the study of symmetry and the axiomatic development of the theory. The concepts you will cover include: subgroups, Lagrange’s theorem, factor groups, group actions and the Isomorphism Theorem. 

MTHA4007B  (20 Credits)

Computation and modelling are essential skills for the modern mathematician. While many applied problems are amenable to analytic methods, many require some numerical computation to complete the solution. The synthesis of these two approaches can provide deep insight into highly complex mathematical ideas. 

This module will introduce you to the art of mathematical modelling, and train you in the computer programming skills needed to perform numerical computations. You will be introduced to the Python programming language and study algorithms for problems such as root finding. A particular focus of mathematical modelling is classical mechanics, which describes the motion of solid bodies. Central to this is Newton’s second law of motion, which states that a mass will accelerate at a rate proportional to the force imposed upon it. This leads to an ordinary differential equation to be solved for the velocity and position of the mass. In the simplest cases, the solution can be constructed using analytical methods, but in more complex situations, for example motion under resistance, you will use numerical methods to find the motion of a particle. 

MTHA5003A  (20 Credits)

This module comprises two distinct parts, one in analysis and one in algebra. 

The first part, in analysis, will introduce you to the basic theory of the complex plane. The topics you will study include continuity, power series and how they represent functions for both real and complex variables, differentiation, holomorphic functions, Cauchy-Riemann equations. 

The second part, in algebra, follows on from the linear algebra you studied in Year 1. You will be introduced the concept of a vector space over a field. You will learn about vector subspaces. You will see the definition of a basis of a vector space, why this construction is useful, and how we can then talk about the dimension of the space. You will then look at linear transformations between pairs of vector spaces, which will lead to the definitions of the kernel and the image of a linear transformation and hence to the rank-nullity theorem. You will see how by fixing bases, a linear transformation can be encoded in matrix form and how changing the bases changes that matrix, which will lead on to the study of eigenvectors and the diagonalization of matrices. Throughout the module you will see examples of different vector spaces which will illustrate the results presented.

MTHA5005A  (20 Credits)

In applied mathematics, you will often need to solve equations involving derivatives of the unknown function that you are trying to find. Such equations are known as “differential equations”, and you will learn about various techniques for solving them in this module. 

When the unknown quantity is a function of one variable then the equation is known as an “ordinary differential equation”. You will already have seen some techniques of solving ordinary differential equations. In this module, you will study new techniques such as series solutions and the method of Frobenius.  When the unknown quantity is a function more than one variable then the equation is known as a “partial differential equation”. You will learn how to use the method of “separation of variables” to solve such equations, and study the use of Fourier series (representations of functions as infinite series involving trigonometric functions). You will encounter a variety of important partial differential equations from applied mathematics, including the heat equation, the wave equation, and Laplace’s equation.  

You will then move on to study two methods that can be used to solve various problems that occur in applied mathematics. Fourier transforms can be used to solve ordinary differential equations, partial differential equations, and integral equations. The method of characteristics is a particular technique for solving partial differential equations by converting them to ordinary differential equations.   

Finally, you will learn about techniques for analysing collections of coupled differential equations known as “dynamical systems”, which describe how certain variables evolve in time. The techniques here will help you analyse and understand the behaviour of nonlinear differential equations and acts as a starting point for the study of chaos. 

MTHA5003B  (20 Credits)

In this module you will continue your studies in pure mathematics, with two more topics; one in analysis and one in algebra. 

In analysis, you will learn about integration in the complex plane. This will include consideration of the topology of the complex plane along with proof of the Cauchy and Laurent theorems, along with applications including residue calculus. 

In algebra, you will study Ring Theory. You will first be introduced to the concept of a ring, using the integers as an example. You will then develop the theory further, with many examples related to familiar concepts such as substitution and factorisation. Important examples of commutative rings include fields, domains, polynomial rings, and their quotients. 

CMP-5034A  (20 Credits)

This module introduces the essential concepts of mathematical statistics deriving the necessary distribution theory as required. In consequence in addition to ideas of sampling and central limit theorem, it will cover estimation methods and hypothesis-testing.

CMP-5042B  (10 Credits)

This module considers both the theory and practice of statistical modelling of time series. Students will be expected to analyse real data using R.

CMP-5043  (10 Credits)

This is a module designed to give you the opportunity to apply linear regression techniques using R. While no advanced knowledge of probability and statistics is required, we expect you to have some background in probability and statistics before taking this module. The aim is to provide an introduction to R and then provide the specifics in linear regression.

MTHA5005B  (20 Credits)

In this module you will consider how to model the flow of liquids and gases using mathematics. Understanding fluid flows is important for weather predictions, the aerodynamics of air flow round a car, and understanding why planes are able to stay in the air. 

You will first study the fundamentals of “vector calculus”, which deals with how differentiation can be applied to vectors fields (vectors that vary in space), such as the velocity of a fluid. You will then apply your knowledge of vector calculus to see how we formulate the differential equations that govern fluid flows. You will go on to solve simple fluid-flow problems, such as the determining the rate of flow out of a reservoir.  

You will then examine how computers can solve differential equations and approximate continuous functions. This will involve studying the underlying algorithms relevant to understanding fluid flow, and also some practical programming using Python to study the motion of systems of vortices through a fluid. 

In the final part of the module, you will learn about how complex variables and functions can be used to solve problems in inviscid fluid flow, using what is known as “complex potentials”. This represents a nice application of some theory from pure mathematics in an applied mathematics context.  The methods you will study can be used, for example, to estimate the lift on an aerofoil. 

MTHF5032B  (20 Credits)

Mathematical modelling is concerned with how to convert real problems, such as those arising in industry or other sciences, into mathematical equations, and then solving them and using the results to better understand, or make predictions about, the original problem. 

In this module, you will study various techniques used in mathematical modelling, and see examples of how mathematics can be applied to a variety of real-life problems. The techniques will include approximation and non-dimensionalising, and discussion of how a mathematical model is created. We will then apply this theory to a variety of situations, such as traffic flow, population modelling, chaos, aerodynamics, and other problems arising from industry. 

CMP-5020B  (20 Credits)

You will be introduced to a number of programming concepts at the start of your programming career, using a modern programming language common to many digital industries, with specific focus on applications within STEM fields. We structure learning through lectures, delivering core materials, and tutor supported exercises to reinforce learning, and to prepare you for programming in your following studies.

CMP-6047A  (10 Credits)

This module covers stochastic processes - including the random walk, Markov chains, Poisson processes, and birth and death processes.

MTHD6004A  (20 Credits)

Galois theory is one of the most spectacular mathematical theories. Named after the French mathematician Evariste Galois, who died in a duel aged 20, it gives a beautiful connection between the theory of solving polynomial equations and group theory. In fact, many fundamental notions of group theory originated in the work of Galois. For example, why are some groups called "solvable"? Because they correspond to the equations which can be solved (by some formula based on the coefficients and involving algebraic operations and extracting roots of various degrees). Galois theory explains why we can solve quadratic, cubic and quartic equations, but no similar formulae exist for equations of degree greater than four. In modern exposition, Galois theory deals with "field extensions", and the central topic is the "Galois correspondence" between extensions and groups. 

MTHD6015A  (20 Credits)

Mathematical logic analyses symbolically the way in which we reason formally, particularly about mathematical structures. The ideas have applications to other parts of Mathematics, as well as being important in theoretical computer science and philosophy. In this module, you will be presented with a thorough treatment of predicate and propositional logic, and an introduction to model theory. 

MTHD6021A  (20 Credits)

Mathematical Biology concerns the application of mathematics to problems in biology. It is a rapidly developing and hugely exciting field with many areas the focus of dedicated research. 

In this module, you will discover how to use the mathematics you have learned to date to understand a wide range of interesting biological problems. In many cases, important biological insights can be gained from quite simple mathematical models. Examples include the diffusion-limited growth of solid tumours, and the reasons why animal coats patterns are so widely varied - for example, why does the tiger have stripes and the leopard have spots? Mathematics has made fundamental contributions in these and many other areas which we will explore during the module. Further examples may include the propagation of wave-fronts in migrating animal populations, blood flow in arteries and veins and the onset of arterial disease, cochlear mechanics in the ear, and tear film dynamics on the human eye. 

No prior knowledge of biology is required to be able to take this module. 

MTHD6020A  (20 Credits)

Fluid dynamics concerns the mathematical modelling of the flow of liquids and gases. It has wide ranging applications across nature, engineering, and biology. Applications include understanding the behaviour of ocean waves and weather, designing efficient aircraft and ships, and describing the flow of blood around the body.  

In this module, you will consider mathematical models of fluids, particularly including viscosity (or stickiness) of a fluid. Illustrated by practical examples throughout, you will develop the governing differential Navier-Stokes equations, and then consider their solution either finding exact solutions, or using analytical techniques to obtain solutions in certain limits (for example low viscosity or high viscosity). 

MTHD60--B  (20 Credits)

A “graph” here is a mathematical object comprising a set of point (vertices) joined in pairs by a number of lines (edges). Graph theory is the branch of mathematics that studies the properties of such objects. In this modules, you will be introduced to graph theory, and some of the numerous theorems and results in this area of mathematics. 

Combinatorics is one of the most applicable and accessible part of mathematics, yet it is also full of challenging problems. You will learn about a number of fundamental combinatorial concepts including counting arguments (enumerative combinatorics) and Ramsey theory.

MTHD6034B  (20 Credits)

Algebraic Topology uses techniques from abstract algebra to study topological spaces: spaces where we have a notion of ``shape’’ but no numerical notion of distance. Conversely, methods from topology can then be used to prove results in Algebra. 

You will study some basic notions and results in algebraic topology. In particular you will see how to associate certain groups with a topological space that capture important basic information about the shape of the space. Topics covered will include: CW complexes, elementary concepts of homotopy theory, fundamental groups, covering spaces, free products of groups and the van Kampen Theorem, and presentations of groups. 

MTHD6018B  (20 Credits)

Dynamical meteorology is a core subject on which weather forecasting and the study of climate and climate change are based. This module applies fluid dynamics to the study of the circulation of the Earth's atmosphere. 

You will be introduced to the fluid dynamical equations and some basic thermodynamics for the atmosphere. You will then learn how these can be applied to topics such as geostrophic flow, thermal wind and the jet streams, boundary layers, gravity waves, the Hadley circulation, vorticity and potential vorticity, Rossby waves, and equatorial waves. Emphasis will be placed on fluid dynamical concepts as well as on finding analytical solutions to the equations of motion. 

MTHA6026A  (20 Credits)  

Further information will be provided shortly.

MTHD6032B  (20 Credits)

In applied mathematics, we will often model a real-life problem and obtain an integral or differential equation to solve. Sometimes we can find exact solutions to these equations, but we often need to resort to approximations. This module will teach you a range of useful techniques that can be used across a variety of applied mathematics problems. 

We provide techniques for a wide range of applications, while stressing the importance of rigour in developing such techniques. The Calculus of Variations includes techniques for maximising integrals subject to constraints. A typical problem is the curve described by a heavy chain hanging under the effect of gravity. Asymptotic analysis provides a method for solving equations, or evaluating integrals, which involve small parameters, when exact results can not be found and when numerical solutions are difficult. A range of integral transforms are discussed which are useful for solving problems including integro-differential equations. This unit will include illustration of concepts using numerical investigation with MAPLE but no previous experience of using this software is assumed.

MTHA6002B  (20 Credits)

We trace the development of mathematics from prehistory to the high cultures of old Egypt, Mesopotamia and the Valley of Ind, through Islamic mathematics onto the mathematical modernity through a selection of results from the present time. We present the rise of calculus from the first work of the Indian and Greek mathematicians’ differentiation and integration through at the time of Newton and Leibniz. We discuss mathematical logic, the ideas of propositions, the axiomatisation of mathematics, and the idea of quantifiers. Our style is to explore mathematical practice and conceptual developments in different historical and geographic contexts. 

MTHF5036A  (20 Credits)

This module comprises two distinct parts, covering additional topics in pure mathematics. 

Point-set topology is the study of spaces where we have a notion of “shape” and “closeness” but no numerical measure of distance: so, famously, a ring doughnut and a teacup are the same as topological spaces, since both are solid shapes with one hole (through the handle, for the teacup). This generalises Analysis, using only basic set theory. You will begin by defining a topological space and will then investigate how analytic notions like limit points and continuity can be interpreted in terms of open and closed sets, as well as seeing how new topological spaces can be built from old ones.  

Linear transformations underpin much of modern mathematics and are the key to many applications. In many of these applications there is more structure, for example the dot product of vectors, and this is reflected in the structure of the linear transformations which arise. You will learn about bilinear forms and symmetric operators on vector spaces, which leads on to the diagonalization of symmetric linear transformations and the spectral theorem. This theorem is key to many applications in Statistics and Physics. Other topics you will study include polynomials of linear maps, the Cayley-Hamilton theorem, and the Jordan normal form of a matrix. 

MTHF5030B  (20 Credits)

This module will introduce you to the theories quantum mechanics and special relativity. Quantum mechanics describes how atoms and sub-atomic particles behave on very small length-scales. Special relativity describes how objects behave at very high speeds. 

In quantum mechanics you will study systems with small lengthscales and understand why the ideas of classical mechanics fail to describe physical effects when sub-atomic particles are involved. You will then see how the famous “Schrödinger equation” is derived, and gain an understanding of its probabilistic interpretation. Finally, you will study how the solutions of the Schrödinger equation imply that certain physical quantities (e.g. energy) do not vary continuously, but can only take on discrete values. Such variables are said to be “quantized”. 

In special relativity, you will see how the general concepts of space and time drastically change for an observer moving at speeds close to the speed of light. For example, time undergoes a dilation and space a contraction. These counterintuitive phenomena are however direct consequences of physical laws. You will understand the basics of special relativity using simple mathematics and physical intuition. Important well-known topics like inertial and non-inertial frames, the Lorentz transformations, the concept of simultaneity, time dilation and Lorentz contraction, mass and energy relation will be explained. At the end of the module, you will see how special relativity and quantum mechanics can be combined in a relativistic theory of quantum mechanics. 

CMP-5020B  (20 Credits)

You will be introduced to a number of programming concepts at the start of your programming career, using a modern programming language common to many digital industries, with specific focus on applications within STEM fields. We structure learning through lectures, delivering core materials, and tutor supported exercises to reinforce learning, and to prepare you for programming in your following studies.