# MMath Master of Mathematics

- UCAS Course Code
- G103
- A-Level typical
- AAB (2018/9 entry) See All Requirements

## About this course

Studying with us means that you’ll benefit from internationally recognised, research-led teaching and a high academic staff/student ratio. You’ll graduate with a deep understanding of mathematics and great career prospects (86% of our graduates were in work or study within six months), whether you specialise in pure maths, applied maths, or a mix of topics from the wide range of optional modules we offer.

This four-year programme offers advanced study of mathematics preparing you for either further study or a career beyond university. Your lectures are complemented by small-group teaching that provides you with quality contact time with our world class lecturers – we were ranked 7th in the UK for the quality of our research outputs (REF 2014).

This four-year programme offers advanced study of mathematics preparing you for either further study or a career beyond university. Your lectures are complemented by small-group teaching that provides you with quality contact time with our world class lecturers – we were ranked 7th in the UK for the quality of our research outputs (REF 2014).

## Course Profile

### Overview

This prestigious four-year Master of Mathematics programme allows you to develop your interests in pure and applied mathematics, and in greater depth of study than a three-year programme.

One of the advantages of studying with us is that we offer a great deal of flexibility in what you can study, enabling you to specialise in either pure or applied mathematics, or a combination of these topics. Apart from engaging in study of essential mathematical theory and technique, you will also have the opportunity to carry out a substantial research project in the final year, allowing you to experience the challenge of independent study and discovery. Furthermore this helps you to develop skills that are essential for many future career paths, including experience of report writing and oral presentation.

If you finish your studies with distinction you may want to join our active group of postgraduate students, as the programme is also excellent preparation for a career in research, either in industry or in a university. However research is just one pathway out of the wide range of challenging careers open to Master of Mathematics students.

### Course Structure

The first two years run in parallel with the three-year BSc programme, before more specialised content is covered in the third and fourth year. In these final two years you will learn advanced principles through a huge range of optional subjects, as well as undertaking an independent project.

**Year 1**

In the first year you will undertake a set of compulsory modules to consolidate a broad knowledge of mathematical disciplines, primarily algebra and calculus. This is supplemented by classes on the applications of mathematics, problem solving and analysis. The skills you gain from these courses will be revisited throughout the degree and should help inform your future module choices.

**Year 2**

**The modules on offer to you are the same as those available on the BSc programme.**

As you progress into your second year, you will continue to learn essential algebraic principles through compulsory modules whilst taking a selection of optional modules to suit your personal interests. The optional modules offered change each year but for instance you may choose to study the theory of special relativity or take a cryptography module to see how number theory studied elsewhere can be applied to the theory of coding. In addition, you will be introduced to mathematical software which will be invaluable in your individual fourth year MMath project.

**Year 3**

At this stage there are no compulsory modules and you will choose 6 modules from a range of approximately 15 offered. Each year the particular topics on offer vary to mirror the research interests of our lecturers. By this stage we anticipate that you will have found the areas of mathematics which most appeal to you, and you will use this year to focus on these topics, laying the foundations for a successful final-year research project.

**Year 4**

You will undertake a substantial individual project during your final year, working under the close supervision of a lecturer whose expertise matches your subject. Each of the lecturers proposes project titles covering a very wide range of current mathematical research, but many of our students come up with their own topics in conjunction with one of our lecturers. Recent topics have ranged from “The Mobius function of Finite Groups” to “The Aerodynamics of Golf Balls” (a topic suggested by the student). The project is assessed through a written report as well as a short oral presentation to lecturers and fellow Masters students on your findings. For the remainder of your final year, you will choose from a range of Master’s-level modules that explore topics such as lie algebra, fluid structure interaction and mathematical biology. The topics on offer typically change every year.

### Assessment

A variety of assessment methods are applied across the different mathematics modules, ranging from 100% coursework to 100% examination. Most mathematics modules are assessed 80% by examination and 20% by coursework. The coursework component is made up of problems set from an example sheet, to be handed in, marked and returned together with solutions. For some modules there are also programming assignments and/or class tests. In Year 4, the research project makes up one-third of the final assessment.

### Course Modules 2017/8

Students must study the following modules for 120 credits:

Name | Code | Credits |
---|---|---|

CALCULUS AND MULTIVARIABLE CALCULUS (a) Complex numbers. (b) Vectors. (c) Differentiation. Taylor and Maclaurin series. (d) Integration: Applications: curve sketching, areas, arc length. (e) First-order, second-order, constant coefficient ordinary differential equations. Reduction of order. Numerical solutions using MAPLE. Partial derivatives, chain rule. (f) Line integrals. Multiple integrals, including change of co-ordinates by Jacobians. Green's theorem in the plane. (g) Euler type and general linear ODEs. (h) Divergence, gradient and curl of a vector field. Scalar potential and path independence of line integral. Divergence and Stokes' theorems. (i) Introduction to Matlab. | MTHA4005Y | 40 |

LINEAR ALGEBRA In the first semester we develop the algebra of matrices: Matrix operations, linear equations, determinants, eigenvalues and eigenvectors, diagonalization and geometric aspects. This is followed in the second semester by vectors space theory: Subspqaces, basis and dimension, linear maps, rank-nullity theorem, change of basis and the characteristic polynomial. | MTHA4002Y | 20 |

MATHEMATICAL PROBLEM SOLVING, MECHANICS AND MODELLING STUDENTS FROM YEARS 2 OUTSIDE SCHOOL OF MATHEMATICS CAN TAKE THIS MODULE IF THEY HAVE ALREADY TAKEN MTHA4005Y, MTHB4006Y OR ENV-4015Y AND HAVE NOT TAKEN MTHB4007B. The first part of the module is about how to approach mathematical problems (both pure and applied) and write mathematics. It aims to promote accurate writing, reading and thinking about mathematics, and to improve students' confidence and abilities to tackle unfamiliar problems. The second part of the module is about Mechanics. It includes discussion of Newton's laws of motion, particle dynamics, orbits, and conservation laws. This module is reserved for students registered in the School of Mathematics or registered on the Natural Sciences programme. | MTHA4004Y | 20 |

REAL ANALYSIS This module is concerned with the mathematical notion of a limit. We 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. We then learn about limits of functions and continuity. Finally, we will learn precise definitions of differentiation and integration and see the Fundamental Theorem of Calculus. | MTHA4003Y | 20 |

SETS, NUMBERS AND PROBABILITY Basic set-theoretic notation, functions. Proof by induction, arithmetic, rationals and irrationals, the Euclidean algorithm. Styles of proof. Elementary set theory. Modular arithmetic, equivalence relations. Countability. Probability as a measurement of uncertainty, statistical experiments and Bayes' theorem. Discrete and continuous distributions. Expectation. Applications of probability: Markov chains, reliability theory. | MTHA4001Y | 20 |

Students must study the following modules for 80 credits:

Name | Code | Credits |
---|---|---|

ALGEBRA We introduce groups and rings, which together with vector spaces are the most important algebraic structures. At the heart of group theory in Semester I is the study of symmetry and the axiomatic development of the theory, groups appear in many parts of mathematics. The basic concepts are subgroups, Lagrange's theorem, factor groups, group actions and the Isomorphism Theorem. In Semester II we introduce rings, using the Integers as a model and develop the theory with many examples related to familiar concepts such as substitution and factorisation. Important examples of commutative rings are fields, domains, polynomial rings and their quotients. | MTHA5003Y | 20 |

ANALYSIS This module covers the standard basic theory of the complex plane. The areas covered in the first semester, (a), and the second semester, (b), are roughly the following: (a) Continuity, power series and how they represent functions for both real and complex variables, differentiation, holomorphic functions, Cauchy-Riemann equations, Moebius transformations. (b) Topology of the complex plane, complex integration, Cauchy and Laurent theorems, residue calculus. | MTHA5001Y | 20 |

DIFFERENTIAL EQUATIONS AND APPLIED METHODS (a) Ordinary Differential Equations: solution by reduction of order; variation of parameters for inhomogeneous problems; series solution and the method of Frobenius. Legendre's and Bessel's equations: Legendre polynomials, Bessel functions and their recurrence relations; Fourier series; Partial differential equations (PDEs): heat equation, wave equation, Laplace's equation; solution by separation of variables. (b) Method of characteristics for hyperbolic equations; the characteristic equations; Fourier transform and its use in solving linear PDEs; (c) Dynamical Systems: equilibrium points and their stability; the phase plane; theory and applications. | MTHA5004Y | 20 |

FLUID DYNAMICS - THEORY AND COMPUTATION (a) Hydrostatics, compressibility. Kinematics: velocity, particle path, streamlines. Continuity, incompressibility, streamtubes. Dynamics: Material derivative, Euler's equations, vorticity and irrotational flows. Velocity potential and streamfunction. Bernoulli's equation for unsteady flow. Circulation: Kelvin's Theorem, Helmholtz's theorems. Basic water waves. (b) Computational methods for fluid dynamics; Euler's method and Runge-Kutta methods and their use for computing particle paths and streamlines in a variety of two-dimensional and three-dimensional flows; numerical computation and flow visualisation using Matlab; convergence, consistency and stability of numerical integration methods for ODEs. (c) Theory of Irrotational and Incompressible Flows: velocity potential, Laplace's Equation, sources and vortices, complex potential. Force on a body and the Blasius theorem. Method of images and conformal mappings. | MTHA5002Y | 20 |

Students will select 20 - 40 credits from the following modules:

Name | Code | Credits |
---|---|---|

MATHEMATICAL STATISTICS It 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. Some Bayesian ideas will be also introduced. | CMP-5034A | 20 |

TOPICS IN APPLIED MATHEMATICS This module is an optional Year long module. It covers two topics, Lagrangian Systems and Special Relativity, one in each semester. Lagrangian Systems involves reformulation of problems in mechanics allowing solution of problems such as the osci llation of a double pendulum. Some discussion of Hamiltonian systems will also be included. Special Relativity is concerned with changes in time and space when an observer is moving at a speed close to the speed of light. | MTHF5200Y | 20 |

TOPICS IN PURE MATHEMATICS This module provides an introduction to two selected topics within pure mathematics. These are self-contained topics which have not been seen before. The topics on offer for 2017-18 are the following. Topology: This is an introduction to point-set topology, which studies spaces up to continuous deformations and thereby generalises analysis, using only basic set theory. We will begin by defining a topological space, and will then investigate notions like open and closed sets, limit points and closure, bases of a topology, continuous maps, homeomorphisms, and subspace and product topologies. Computability: This is an introduction to the theoretical foundation of computability theory. The main question we will focus on is "which functions can in principle (i.e., given unlimited resources of space and time) be computed?". The main object of study will be certain devices known as unlimited register machines (URM's). We will adopt the point of view that a function is computable if and only if i is computable by a URM. We will identify large families of computable functions and will prove that certain naturally occurring functions are not computable. | MTHF5100Y | 20 |

Students will select 0 - 20 credits from the following modules:

Name | Code | Credits |
---|---|---|

APPLIED GEOPHYSICS What lies beneath our feet? This module addresses this question by exploring how wavefields and potential fields are used in geophysics to image the subsurface on scales of metres to kilometres. The basic theory, data acquisition and interpretation methods of seismic, electrical, gravity and magnetic surveys are studied. A wide range of applications is covered including archaeological geophysics, energy resources and geohazards. This module is highly valued by employers in industry; guest industrial lecturers will cover the current 'state-of-the-art' applications in real world situations. Students doing this module are normally expected to have a good mathematical ability, notably in calculus and algebra before taking this module (ENV-4015Y Mathematics for Scientists A or equivalent). | ENV-5004B | 20 |

APPLIED STATISTICS A This is a module designed to give students the opportunity to apply statistical methods in realistic situations. While no advanced knowledge of probability and statistics is required, we expect students to have some background in probability and statistics before taking this module. The aim is to teach the R statistical language and to cover 3 topics: Linear regression, and Survival Analysis. | CMP-5017B | 20 |

DYNAMICS AND VIBRATION | ENG-5004B | 20 |

ELECTROMAGNETISM, OPTICS, RELATIVITY AND QUANTUM MECHANICS This module gives an introduction to important topics in physics, with particular, but not exclusive, relevance to chemical and molecular physics. Areas covered include optics, electrostatics and magnetism and special relativity. The module may be taken by any science students who wish to study physics beyond A Level. | PHY-4001Y | 20 |

INTRODUCTION TO BUSINESS (2) Introduction to Business is organised in thematic units across semesters 1 and 2, aiming to provide a platform for understanding the world of management and the managerial role. The module explores the business environment, key environmental drivers and functions of organisations, providing an up-to-date view of current issues faced from every contemporary enterprise such as business sustainability, corporate responsibility and internationalisation. There is consideration of how organisations are managed in response to environmental drivers. To address this aspect, this module introduces key theoretical principles in lectures and seminars are designed to facilitate fundamental study skills development, teamwork and practical application of theory. By the end of this module, students will be able to understand and apply key concepts and analytical tools in exploring the business environment and industry structure respectively. This module is for NON-NBS students only. | NBS-4008Y | 20 |

INTRODUCTION TO FINANCIAL AND MANAGEMENT ACCOUNTING (2) 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 may be taken as a standalone course for those students following a more general management pathway or to provide a foundation to underpin subsequent specialist studies in accounting. This module is for NON-NBS students only. | NBS-4010Y | 20 |

INTRODUCTORY MACROECONOMICS This is a compulsory module for all ECO students and it is a prerequisite for later economic modules. The aim of the module is to introduce you to the fundamental principles, concepts and tools of macroeconomics and to apply these to a variety of real world macroeconomic issues. There is some mathematical content - you will be required to interpret linear equations and solve simple linear simultaneous equations. The module will introduce students to core macroeconomic indicators such as income, inflation, unemployment and the stance of the balance of payments. Thus, focussing predominantly on the short-run, the module will consider: (1) models for equilibrium in the goods market and the money market, (2) applications of such models to discuss the role of fiscal and monetary policy, (3) the trade-off between inflation and unemployment, and (4) the role of expectations in macroeconomic analysis. | ECO-4006Y | 20 |

INTRODUCTORY MICROECONOMICS This is a compulsory module for all ECO students and it is a prerequisite for later economic modules. The aim of the module is to introduce you to the fundamental principles, concepts and tools of microeconomics. The aim of the module is apply these to a variety of real world economic issues. There is some mathematical content - you will be required to interpret linear equations, solve simple linear simultaneous equations and use differentiation. The module is primarily concerned with: (1) the ways individuals and households behave in the economy; (2) the analysis of firms producing goods and services; (3) how goods and services are traded or otherwise distributed - often but not exclusively through markets; and (4) the role of government as provider and/or regulator. | ECO-4005Y | 20 |

METEOROLOGY I This module is designed to give a general introduction to meteorology, concentrating on the physical processes in the atmosphere and how these influence our weather. The module contains both descriptive and mathematical treatments of radiation balance, fundamental thermodynamics, dynamics, boundary layers, weather systems and meteorological hazards. The assessment is designed to allow those with either mathematical or descriptive abilities to do well; however a reasonable mathematical competence is essential, including a basic understanding of differentiation and integration. | ENV-5008A | 20 |

PROGRAMMING FOR NON-SPECIALISTS The purpose of this module is to give the student a solid grounding in the essential features programming using the Java programming language. The module is designed to meet the needs of the student who has not previously studied programming. | CMP-5020B | 20 |

UNDERSTANDING THE DYNAMIC PLANET Understanding of natural systems is underpinned by physical laws and processes. This module explores energy, mechanics, physical properties of Earth materials and their relevance to environmental science using examples from across the Earth's differing systems. The formation, subsequent evolution and current state of our planet are considered through its structure and behaviour - from the planetary interior to the dynamic surface and into the atmosphere. Plate Tectonics is studied to explain Earth's physiographic features - such as mountain belts and volcanoes - and how the processes of erosion and deposition modify them. The distribution of land masses is tied to global patterns of rock, ice and soil distribution and to atmospheric and ocean circulation. We also explore geological time - the 4.6 billion year record of changing conditions on the planet - and how geological maps can used to understand Earth history. This course provides an introduction to geological materials - rocks, minerals and sediments - and to geological resources and natural hazards. | ENV-4005A | 20 |

Students will select 80 - 120 credits from the following modules:

Name | Code | Credits |
---|---|---|

ADVANCED MATHEMATICAL TECHNIQUES We provide techniques for a wide range of applications, while stressing the importance of rigor 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. We develop techniques for algebraic and differential equations. This includes asymptotic analysis. This provides approximate solutions when exact solutions can not be found an6d when numerical solutions are difficult. Integral transforms are useful for solving problems including integro-differential equations. This unit will include illustration of concepts using numerical investigation with MAPLE. | MTHD6032B | 20 |

ADVANCED STATISTICS This module covers two topics in statistical theory: Linear and Generalised Linear models and also includes Stochastic processes. The first two topics consider both the theory and practice of statistical model fitting and students will be expected to analyse real data using R. Stochastic processes including the random walk, Markov chains, Poisson processes, and birth and death processes. | CMP-6004A | 20 |

CRYPTOGRAPHY Cryptography is the science of coding and decoding messages to keep them secure, and has been used throughout history. While previously only a few people in authority used cryptography, the internet and e-commerce mean that we now all have transactions that we want to keep secret. The speed of modern computers means messages encrypted using techniques from just a few decades ago can now be broken in seconds; thus the methods of encryption have also become more sophisticated. In this module, we will explore the mathematics behind some of these methods, notably RSA and Elliptic Curve Cryptogrphy. | MTHD6025A | 20 |

DYNAMICAL METEOROLOGY 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. The fluid dynamical equations and some basic thermodynamics for the atmosphere are introduced. These are then 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. | MTHD6018B | 20 |

FERMAT'S LAST THEOREM This module looks at the Mathematics developed in attempts to prove Fermat's Last Theorem: that there are no natural number solutions to xn+yn=zn when n>2, This begins with Fermat's method of infinite descent, together with the property that any integer can be factorized uniquely into primes. However, to go beyond very small values of n, we must look at extensions of the integers, where unique factorization fails. Everntually, using tools from Abstract Algebra (rings and ideals) we will see Kummer's proof for so-calle regular primes n. | MTHD6024B | 20 |

FLUID DYNAMICS Fluid dynamics has wide ranging applications across nature, engineering, and biology. From understanding the behaviour of ocean waves and weather, designing efficient aircraft and ships, to capturing blood flow, the ability the understand and predict how fluids (liquids and gasses) behave is of fundamental importance. This Module considers mathematical models of fluids, particularly including viscosity (or stickiness) of a fluid. Illustrated by practical examples throughout, we 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). | MTHD6020A | 20 |

MATHEMATICAL BIOLOGY Mathematics finds wide-ranging applications in biological systems: including population dynamics, epidemics and the spread of diseases, enzyme kinetics, some diffusion models in biology including Turing instabilities and pattern formation, and various aspects of physiological fluid dynamics. | MTHD6021A | 20 |

MATHEMATICAL LOGIC The subject 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. We give a thorough treatment of predicate and propositional logic and an introduction to model theory. | MTHD6015A | 20 |

THEORY OF FINITE GROUPS Group theory is the mathematical study of symmetry. The modern treatment of this is group actions and these are a central theme of this course. We will begin with permutation groups, group actions and the orbit-stabilizer theorem with many applications. This is followed by a discussion of the Sylow theorems, the class equations and an elementary theory of p-groups. Further topics include the theorem of Jordan and Hoelder, solvable groups and simple. Simplicity of finite and infinite alternating groups. | MTHD6014A | 20 |

Students will select 0 - 40 credits from the following modules:

Name | Code | Credits |
---|---|---|

HISTORY OF MATHEMATICS 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 worsk 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. | MTHA6002B | 20 |

MATHEMATICS PROJECT | MTHA6005Y | 20 |

MODELLING ENVIRONMENTAL PROCESSES The aim of the module is to show how environmental problems may be solved from the initial problem, to mathematical formulation and numerical solution. Problems will be described conceptually, then defined mathematically, then solved numerically via computer programming. The module consists of lectures on numerical methods and computing practicals (using Matlab); the practicals being designed to illustrate the solution of problems using the methods covered in lectures. The module will guide students through the solution of a model of an environmental process of their own choosing. The skills developed in this module are highly valued by prospective employers. | ENV-6004A | 20 |

THE LEARNING AND TEACHING OF MATHEMATICS The aim of the module is to introduce students to the study of the teaching and learning of mathematics with particular focus to secondary and post compulsory level; to explore theories of learning and teaching of mathematical concepts typically included in the secondary and post compulsory curriculum and to explore mathematics knowledge for teaching. This module is recommended for anyone interested in Mathematics teaching as a career or, indeed, for anyone interested in mathematics education as a research discipline. | EDUB6014A | 20 |

Students will select 0 - 20 credits from the following modules:

Name | Code | Credits |
---|---|---|

APPLIED GEOPHYSICS What lies beneath our feet? This module addresses this question by exploring how wavefields and potential fields are used in geophysics to image the subsurface on scales of metres to kilometres. The basic theory, data acquisition and interpretation methods of seismic, electrical, gravity and magnetic surveys are studied. A wide range of applications is covered including archaeological geophysics, energy resources and geohazards. This module is highly valued by employers in industry; guest industrial lecturers will cover the current 'state-of-the-art' applications in real world situations. Students doing this module are normally expected to have a good mathematical ability, notably in calculus and algebra before taking this module (ENV-4015Y Mathematics for Scientists A or equivalent). | ENV-5004B | 20 |

APPLIED STATISTICS A This is a module designed to give students the opportunity to apply statistical methods in realistic situations. While no advanced knowledge of probability and statistics is required, we expect students to have some background in probability and statistics before taking this module. The aim is to teach the R statistical language and to cover 3 topics: Linear regression, and Survival Analysis. | CMP-5017B | 20 |

MATHEMATICAL STATISTICS It 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. Some Bayesian ideas will be also introduced. | CMP-5034A | 20 |

METEOROLOGY I This module is designed to give a general introduction to meteorology, concentrating on the physical processes in the atmosphere and how these influence our weather. The module contains both descriptive and mathematical treatments of radiation balance, fundamental thermodynamics, dynamics, boundary layers, weather systems and meteorological hazards. The assessment is designed to allow those with either mathematical or descriptive abilities to do well; however a reasonable mathematical competence is essential, including a basic understanding of differentiation and integration. | ENV-5008A | 20 |

OCEAN CIRCULATION This module gives you an understanding of the physical processes occurring in the basin-scale ocean environment. We will introduce and discuss large scale global ocean circulation, including gyres, boundary currents and the overturning circulation. Major themes include the interaction between ocean and atmosphere, and the forces which drive ocean circulation. You should be familiar with partial differentiation, integration, handling equations and using calculators. Shelf Sea Dynamics is a natural follow-on module and builds on some of the concepts introduced here. We strongly recommend that you also gain oceanographic fieldwork experience by taking the 20-credit biennial Marine Sciences fieldcourse. | ENV-5016A | 20 |

PROGRAMMING FOR NON-SPECIALISTS The purpose of this module is to give the student a solid grounding in the essential features programming using the Java programming language. The module is designed to meet the needs of the student who has not previously studied programming. | CMP-5020B | 20 |

SHELF SEA DYNAMICS AND COASTAL PROCESSES The shallow shelf seas that surround the continents are the oceans that we most interact with. They contribute a disproportionate amount to global marine primary production and CO2 drawdown into the ocean, and are important economically through commercial fisheries, offshore oil and gas exploration, and renewable energy developments (e.g. offshore wind farms). This module explores the physical processes that occur in shelf seas and coastal waters, their effect on biological, chemical and sedimentary processes, and how they can be harnessed to generate renewable energy. | ENV-5017B | 20 |

TOPICS IN APPLIED MATHEMATICS This module is an optional Year long module. It covers two topics, Lagrangian Systems and Special Relativity, one in each semester. Lagrangian Systems involves reformulation of problems in mechanics allowing solution of problems such as the osci llation of a double pendulum. Some discussion of Hamiltonian systems will also be included. Special Relativity is concerned with changes in time and space when an observer is moving at a speed close to the speed of light. | MTHF5200Y | 20 |

TOPICS IN PURE MATHEMATICS This module provides an introduction to two selected topics within pure mathematics. These are self-contained topics which have not been seen before. The topics on offer for 2017-18 are the following. Topology: This is an introduction to point-set topology, which studies spaces up to continuous deformations and thereby generalises analysis, using only basic set theory. We will begin by defining a topological space, and will then investigate notions like open and closed sets, limit points and closure, bases of a topology, continuous maps, homeomorphisms, and subspace and product topologies. Computability: This is an introduction to the theoretical foundation of computability theory. The main question we will focus on is "which functions can in principle (i.e., given unlimited resources of space and time) be computed?". The main object of study will be certain devices known as unlimited register machines (URM's). We will adopt the point of view that a function is computable if and only if i is computable by a URM. We will identify large families of computable functions and will prove that certain naturally occurring functions are not computable. | MTHF5100Y | 20 |

Students will select 40 credits from the following modules:

Please note that CMP-6004A Advanced Statistics or equivalent is a prerequisite for CMP-7017Y.

Name | Code | Credits |
---|---|---|

MATHEMATICS DISSERTATION Reserved for courses G102, G103 and G106. A fourth year dissertation on a mathematical topic that is a compulsory part of some Master of Mathematics degrees. | MTHA7029Y | 40 |

MMATH PROJECT ONLY AVAILABLE TO STUDENTS REGISTERED ON MMATH IN SCHOOL OF MATHEMATICS. This module is modelled on the Mathematics MMath project module MTH-MA9Y. However, in this case it consists of a supervised dissertation on a topic in the general area of probability or statistics. It may involve some computation, this will depend on the topic chosen. | CMP-7017Y | 40 |

Students will select 20 credits from the following modules:

Name | Code | Credits |
---|---|---|

DIFFERENTIAL GEOMETRY This module will give an introduction to 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 more advanced topics including surfaces. | MTHD7030B | 20 |

QUANTUM FLUIDS Macroscopic Wavefunction for a Superfluid, Nonlinear Schrodinger Equation for Wavefunction, Madelung transformation and fluid equations of motion, Integral identities for Energy and Momentum, Travelling plane Wave and Vortex Solutions, Vortex Dynamics in 3D, Biot-Savart Law, Extensions to Bose-Einstein condensates and Superfluid Liquid Helium. | MTHD7027B | 20 |

Students will select 60 credits from the following modules:

Name | Code | Credits |
---|---|---|

ADVANCED MATHEMATICAL TECHNIQUES WITH ADVANCED TOPICS This unit provides a selection of techniques applicable to mathematical problems in a wide range of applications, while at the time stressing the importance of rigor in developing such techniques. Topics to be studied include calculus of variation, asymptotic analysis, Green's functions and Integral transforms. There will be in depth study of other aspects of asymptotic theory, including Matched Asymptotic Expansions and the WKB approximation. This unit will include illustration of concepts using numerical investigation with MAPLE. | MTHD7032B | 20 |

CRYPTOGRAPHY WITH ADVANCED TOPICS Cryptography is the science of coding and decoding messages to keep them secure, and has been used throughout history. While previously only a few people in authority used cryptography, the internet and e-commerce mean that we now all have transactions that we want to keep secret. The speed of modern computers means messages encrypted using techniques from just a few decades ago can now be broken in seconds; thus the methods of encryption have also become more sophisticated. In this module, we will explore the mathematics behind some of these methods, notably RSA and Elliptic Curve Cryptogrphy. | MTHD7025A | 20 |

DYNAMICAL METEOROLOGY WITH ADVANCED TOPICS 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. The fluid dynamical equations and some basic thermodynamics for the atmosphere are introduced. These are then 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. Advanced Topic: Advanced Rossby wave propagation. | MTHD7018B | 20 |

FERMAT'S LAST THEOREM WITH ADVANCED TOPICS This module looks at the Mathematics developed in attempts to prove Fermat's Last Theorem: that there are no natural number solutions to xn+yn=zn when n>2, This begins with Fermat's method of infinite descent, together with the property that any integer can be factorized uniquely into primes. However, to go beyond very small values of n, we must look at extensions of the integers, where unique factorization fails. Everntually, using tools from Abstract Algebra (rings and ideals) we will see Kummer's proof for so-calle regular primes n. Finally, we will look at a sketch of Wiles's proof of the general case. | MTHD7024B | 20 |

FLUID DYNAMICS WITH ADVANCED TOPICS Fluid dynamics has wide ranging applications across nature, engineering, and biology. From understanding the behaviour of ocean waves and weather, designing efficient aircraft and ships, to capturing blood flow, the ability the understand and predict how fluids (liquids and gasses) behave is of fundamental importance. This Module considers mathematical models of fluids, particularly including viscosity (or stickiness) of a fluid. Illustrated by practical examples throughout, we 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). | MTHD7020A | 20 |

MATHEMATICAL BIOLOGY WITH ADVANCED TOPICS Mathematics finds wide-ranging applications in biological systems: including population dynamics, epidemics and the spread of diseases, enzyme kinetics, some diffusion models in biology including Turing instabilities and pattern formation, and various aspects of physiological fluid dynamics. | MTHD7021A | 20 |

MATHEMATICAL LOGIC WITH ADVANCED TOPICS The subject 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. We give a thorough treatment of predicate and propositional logic and an introduction to model theory. The Advanced Topic will be Further model theory. | MTHD7015A | 20 |

THEORY OF FINITE GROUPS WITH ADVANCED TOPICS Group theory is the mathematical study of symmetry. The modern treatment of this is group actions and these are a central theme of this course. We will begin with permutation groups, group actions and the orbit stabilizer theorem with many applications. This is followed by a discussion of the Sylow theorems, the class equations and an elementary theory of p groups. Further topics include the theorem of Jordan and Hoelder, solvable groups and simple. Simplicity of finite and infinite alternating groups. Advanced Topic: Finite Reflection Groups. | MTHD7014A | 20 |

#### Disclaimer

Whilst the University will make every effort to offer the modules listed, changes may sometimes be made arising from the annual monitoring, review and update of modules and regular (five-yearly) review of course programmes. Where this activity leads to significant (but not minor) changes to programmes and their constituent modules, there will normally be prior consultation of students and others. It is also possible that the University may not be able to offer a module for reasons outside of its control, such as the illness of a member of staff or sabbatical leave. Where this is the case, the University will endeavour to inform students.## Essential Information

### Entry Requirements

### Fees and Funding

__Undergraduate University Fees and Financial Support: Home and EU Students__

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Please see our webpage for further information on the current amount of tuition fees payable for Home and EU students and for details of the support available.

**Scholarships and Bursaries**

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__Undergraduate University Fees and Financial Support: International Students__

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**Scholarships**

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### How to Apply

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**UCAS Apply** is a secure online application system that allows you to apply for full-time Undergraduate courses at universities and colleges in the United Kingdom. It is made up of different sections that you need to complete. Your application does not have to be completed all at once. The system allows you to leave a section partially completed so you can return to it later and add to or edit any information you have entered. Once your application is complete, it must be sent to UCAS so that they can process it and send it to your chosen universities and colleges.

The UCAS code name and number for the University of East Anglia is **EANGL E14**.

### Further Information

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**Undergraduate Admissions Office (Mathematics)**

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Email: admissions@uea.ac.uk

Please click here to register your details online via our Online Enquiry Form.

International candidates are also actively encouraged to access the University's International section of our website.

## Next Steps

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