MMath MASTER OF MATHEMATICS WITH A YEAR ABROAD

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)

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. 

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. 

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.

MTHF5035A  (20 Credits)

Cryptography is the science of coding and decoding messages so as to keep these messages secure. It has been used in different forms throughout history. In the past, encryption was mainly used by a small number of individuals in positions of authority. Nowadays the universal presence of the internet and e-commerce means that we all have transactions that we need to be keep secret. 

The speed of modern home computers means that an encrypted message that would have been perfectly secure (that is, would have taken an inordinately long time to break) a few decades ago can now be broken in seconds. But as decryption methods have advanced, the methods of encryption have also become more sophisticated. Modern cryptosystems depend on mathematics, in particular Number Theory and Algebra. The most famous example of a public key cryptosystem, RSA, relies on the fact that it is 'hard' to factor a large number into a product of primes. 

In this module, you will examine the mathematics underpinning both classical and modern methods of cryptography and consider how these methods can be applied. You will compare material on symmetric key cryptography and public key cryptography. Examples of both will be given, along with discussion of their strengths and weaknesses, with the emphasis being on the mathematics. You will look at how prime numbers can be used in cryptography, with material on primality testing and factorisation. You will also define and study elliptic curves in order to investigate the relatively new field of elliptic curve cryptography. 

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-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-5043B  (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.

ENV-5009B  (20 Credits)

This module will build upon material covered in Meteorology I,  by covering topics such as synoptic meteorology, weather hazards, micro-meteorology, further thermodynamics and weather forecasting. The module includes a major summative coursework assignment based on data collected on a UEA meteorology fieldcourse in a previous year.

EDUB5012A  (20 Credits)

This module will provide you with an introduction to key areas of psychology with a focus on learning and teaching in education.  By the end of the module you should be able to: - Discuss the role of perception, attention and memory in learning; - Compare and contrast key theories related to learning, intelligence, language, thinking and reasoning; - Critically reflect on key theories related to learning,intelligence, language, thinking and reasoning in the practical context; - Discuss the influence of key intrapersonal, interpersonal and situational factors on pupils learning and engagement in educational settings.

PHY-4003A  (20 Credits)

In this module, you will learn about the methods used to model the physics of the Earth and Universe. You will explore the energy, mechanics, and physical processes underpinning Earth's systems. This includes the study of its formation, subsequent evolution and current state through the understanding of its structure and behaviour - from our planet's interior to the dynamic surface and into the atmosphere. In the second part of this module, you will study aspects of astrophysics including the history of astrophysics, radiation, matter, gravitation, astrophysical measurements, spectroscopy, stars and some aspects of cosmology. You will learn to predict differences between idealised physics and real life situations. You’ll also improve your skills in problem solving, written communication, information retrieval, poster design, information technology, numeracy and calculations, time management and organisation.

NBS-4108B  (20 Credits)

This module provides a foundation in the theory and practice of accounting and an introduction to the role, context and language of financial reporting and management accounting. The module assumes no previous study of accounting. It is be taken to provide a foundation to underpin subsequent specialist studies in accounting.

MTHX6014Y  (120 Credits)

You will spend an academic year studying abroad at one of our partner institutions. You will be expected to successfully complete this year of study according to the host institution’s regulations and assessment schedule. In order to continue with the MMath degree back at the UEA, you will need to have achieved an overall pass mark by the host institution’s regulations, and the equivalent of a UEA mark of 60% or above. If you do not fulfil these progression requirements, then you may be offered the opportunity to transfer to a BSc degree course. Under these circumstances, the final year spent at UEA would count as the third year of the BSc degree. 

MTHA7029Y  (40 Credits)

In this module you will undertake an individually supervised research project on a mathematical topic of interest to you. Topics can be chosen from a booklet of suggestions, or you can propose your own. Throughout the year, you will have regular meetings with your supervisor, to discuss your progress, ask questions and get feedback. The focus of the module is on independent research and study, and the presentation of mathematical ideas to others. The module is assessed via a formal written report and an oral presentation. As well as learning about your particular topic, you will also gain a number of useful transferable skills from this module. 

MTHE7033A  (20 Credits) 

This module is about further topics in algebra. It builds on the knowledge obtained on groups, rings and vector spaces in the first two years. Groups can be studied directly, or via objects called algebras (which have the structures of both rings and vector spaces). On the other hand, algebras can also be studied in their own right. Some of these concepts will be explored in this module. 

MTHE7030A  (20 Credits)

This module will introduce you to the fundamental ideas of differential geometry. Key examples will be curves and surfaces embedded in 3-dimensional Euclidean space. We will start with curves and will study the curvature and torsion, building up to the fundamental theorem of curve theory. From here we move on to tangent spaces and the first and second fundamental forms, Gaussian curvature, and further topics. 

MTHE7032A  (20 Credits)

Quantum mechanics is a physical theory that describes the behaviour of microscopic particles. The module will focus on non-relativistic quantum mechanics that is described by the Schrodinger equation. 

You will learn about the laws of physics that lead to the Schrodinger equation. You will then consider time-dependent and time-independent solutions in different contexts, including an application to the hydrogen atom. Approximation schemes will also be discussed, with particular emphasis on variational principles, and the WKB approximation. 

For the advanced topic, you will study quantum fluids such as ultra-cold Bose gases, and the behaviour of superfluids in terms of the Gross-Pitaevskii equation. 

MTHE7034A  (20 Credits)

Partial Differential Equations (PDEs) are ubiquitous in applied mathematics. They arise in many models of physical systems where there is coupling between the variation in space and time, or more than one spatial dimension. Examples include fluid flows, electromagnetism, population dynamics, and the spread of infectious diseases. It is therefore important to understand the theory of PDEs, as well as different analytic and numerical methods for solving them. 

This module will provide you with an understanding of the different types of PDE, including linear, non-linear, elliptic, parabolic and hyperbolic; and how these features affect the required boundary conditions and solution techniques. We will study different methods of analytical solution (such as greens functions, boundary-integral methods, similarity solutions, and characteristics); as well as appropriate numerical methods (with topics such as implicit versus explicit schemes, convergence, and stability). Examples and applications will be taken from a variety of fields. 

For the advanced topics, you will study similarity solutions and implicit numerical methods for non-linear PDES. 

MTHE7003B  (20 Credits)

This module is concerned with foundational issues in mathematics and provides the appropriate mathematical framework for discussing sizes of infinity. On the one hand we shall cover concepts such as ordinals, cardinals, and the Zermelo-Fraenkel axioms with the Axiom of Choice. On the other, we shall see how these ideas come up in other areas of mathematics, such as graph theory and topology. Familiarity with and a taste for mathematical proofs will be assumed. 

MTHE7035B  (20 Credits)

Number Theory is the study of arithmetical properties of the integers: properties of, and patterns in, prime numbers, integer solutions of equations with integer coefficients, etc. Gauss called Number Theory "The Queen of Mathematics" and, following on from work of Fermat and Euler, is responsible for the emergence of Number Theory as a central subject in modern mathematics. Since then, Number Theory has developed in many directions, including Algebraic, Analytic and Probabilistic Number Theory, and Diophantine Geometry, and has found surprising applications in modern life (notably in Cryptography).  

In this module, building on previous material on prime factorization and congruences, and second year material on groups, rings and fields, you will study various aspects of Number Theory, including certain Diophantine equations and patterns in primes. 

For the advanced topics, you will study applications of Number Theory to Cryptography. Important cryptographic methods like RSA rely on the computational difficulty of factorizing large integers into primes. You will learn some techniques, based on the Number Theory you have learnt, to attack this factorizing problem. 

MTHE7031B  (20 Credits)

Waves are a general physical phenomenon that allow the transfer of energy from one place to another without the net transfer of matter. Typically, there will be a medium that supports oscillations, in which oscillations in one location induce further oscillations nearby. This module concerns the mathematical modelling of such oscillations and waves that result. 

You will gain an introduction to the theory of waves. You will study aspects of linear and nonlinear waves using analytical techniques. Hyperbolic waves and water waves will also be covered. 

MTHE7007B  (20 Credits)

The ocean is an important component of the Earth's climate system. This module covers mathematical modelling of the large-scale ocean circulation and oceanic wave motion, building upon techniques in fluid dynamics and differential equations covered earlier in your degree. 

You will begin the module by considering the effects of rotation on fluid flows. This naturally leads to the important concept of geostrophy, which enables ocean currents to be inferred from measurements of the sea surface height or from vertical profiles of seawater density. Geostrophy also plays a key role in the development of a model for the global scale circulation of abyssal ocean. The role of the wind in driving the ocean will be examined. This enables us to model the large-scale circulation of the ocean including the development of oceanic gyres and strong western boundary currents, such as the Gulf Stream. The module concludes by examining the role of waves, both at the sea surface and internal to the ocean. The differences between wave motion at mid-latitudes and the Equator are examined, as is the roll of the Equator as a wave-guide. The equatorial waves that you will study are intimately linked with the El Niño phenomenon that affects the climate throughout the globe.