MMath MASTER OF MATHEMATICS

Code MTHA4001A Credits 20

The unit provides you with a thorough introduction to some systems of numbers commonly found in Mathematics: natural numbers, integers, rational numbers, modular arithmetic. It also introduces you to common set theoretic notation and terminology and a precise language in which to talk about functions. There is emphasis on precise definitions of concepts and careful proofs of results. Styles of mathematical proofs you will discuss include: proof by induction, direct proofs, proof by contradiction, contrapositive statements, equivalent statements and the role of examples and counterexamples. In addition, this unit will also provide you with an introduction to producing mathematical documents using Latex, and an introduction to solving mathematical problems computationally using both Symbolic Algebra packages and Excel.

Code MTHA4001B Credits 10

Probability is the study of the chance of events occurring. It has important applications to understand the likelihood of multiple events happening together and therefore to rational decision-making. This module will give you an introduction to the modern theory of probability developed from the seminal works of the Russian mathematician A.N. Kolmogorov in 1930s. Kolmogorov’s axiomatic theory describes the outcomes (events) of a random experiment as mathematical sets. Using set theory language you will be introduced to the concept of random variables, and consider different examples of discrete random variables (like binomial, geometric and Poisson random variables) and continuous random variables (like the normal random variable). In the last part of the module you will explore two applications of probability: reliability theory and Markov chains. Aside of the standard lectures and workshop sessions, there will be two computer-lab sessions of (2 hours each) where you will apply probability theory to specific everyday life case studies. The only pre-requisites for this module are a basic knowledge of set theory and of calculus that you would have acquired during the Autumn semester. If you have done probability or statistic at A-level you will rediscover its contents now taught using a proper and more elegant mathematical formalism.

Code MTHA4003A  Credits 20

Algebra plays a key role in pure mathematics and its applications. We will provide you with a thorough introduction and develop this theory from first principles. We develop the theory of matrices, mainly (though not exclusively) over the real numbers. The material covers matrix operations, linear equations, determinants, eigenvalues and eigenvectors, diagonalization and geometric aspects. Another topic underlying all mathematics is Real Analysis. We will explore the mathematical notion of a limit and 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.

Code MTHA4003B Credits 20

This module extends the material studied in the first semester module “Linear algebra, sequences and series” (MTHA4003A) in studying Analysis and Algebra – key topics underpinning all of Mathematics. In the Real Analysis thread, we learn about limits of functions and continuity before moving on to study precise definitions of differentiation and integration. This then leads to the Fundamental Theorem of Calculus. At the heart of group theory is the study of symmetry and the axiomatic development of the theory. The basic concepts are subgroups, Lagrange’s theorem, factor groups, group actions and the Isomorphism Theorem.

Code MTHA4007B  Credits 20

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. A particular focus 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, numerical methods may be required. Iterative methods for solving nonlinear algebraic equations are fundamental and will also be studied.

Code MTHA4008Y  Credits 30

In this module you will study: (a) Complex numbers. (b) Vectors. (c) Differentiation; power series. (d) Integration: applications, curve sketching, area, arc-length. (e) First and 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 coordinates by Jacobians. Green's Theorem in the plane.

Code MTHA5002A  Credits 20

n this module, building on knowledge from Calculus, you will develop skills in a variety of mathematical techniques for solving differential equations, and how they can be applied to model a range of applications. As a particular focus, you will consider how we can describe mathematically how a fluid behaves. Techniques for solving differential equations will consider both Ordinary Differential Equations, including series solutions and the method of Frobenius, and Partial Differential Equations, where the method of separation of variables will be introduced. Fourier series (representations of functions as infinite series in Sin and Cos) are also considered. You will also discover the fundamentals of Vector Calculus, how differentiation can be applied to vector fields such as fluid velocity. You will encounter a variety of important Partial Differential Equations from applied mathematics, including deriving the heat equation and the wave equation. The knowledge of vector calculus will also be applied to formulating the differential equations that govern fluid flows, and solving problems such as the flow out of a reservoir.

Code MTHA5003A Credits 20

One thread of this module covers the standard basic theory of the complex plane. The areas covered include continuity, power series and how they represent functions for both real and complex variables, differentiation, holomorphic functions, Cauchy-Riemann equations. The second thread follows on from the Linear Algebra studied in Year One. We introduce the concept of a vector space over a field. Throughout the module we will see examples of different vector spaces which will illustrate the results presented. We will learn about vector subspaces. We 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. We 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. We 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.

Code MTHA5002B  Credits 20

Students will study a range of methods for solving problems in applied mathematics, including the method of characteristics for first-order partial differential equations, and Fourier transforms. This is followed by an introduction to dynamical systems – understanding the behaviour of nonlinear differential equations. In the other part of this module, we shall consider both numerical and analytical methods for solving the the equations of fluid flow.

Code MTHA5003B  Credits 20

Study of complex integration 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. The other area of mathematics studied in this module is Ring Theory. After an introduction to rings using integers as a model, we develop the theory 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.

Code CMP-5034A  Credits 20

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.

Code MTHF5030Y  Credits 20

This module introduces you to quantum mechanics and special relativity. In quantum mechanics focus will be on: 1. Studying systems involving very short length scales – eg structure of atoms. 2. Understanding why the ideas of classical mechanics fail to describe physical effects when sub-atomic particles are involved. 3. Deriving and solving the Schrodinger equation. 4. Understanding the probabilistic interpretation of the Schrodinger equation. 5. Understanding how this equation implies that certain physical quantities such as energy do not vary continuously, but can only take on discrete values. The energy levels are said to be quantized. For special relativity, the general concept of space and time drastically changes 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. The module will also explain the basis 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. The module will end with the implications of special relativity and quantum mechanics on a relativistic theory of quantum mechanics.

Code MTHF5034Y Credits 20

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. You 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. Logic: This is an introduction to various aspects of mathematical logic. We will cover selected topics from truth and propositional logic, proofs and deductions, computability, countability of sets, ordered sets, Boolean algebras, and connections with Topology.

Code CMP-5020B  Credits 20

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.

Code CMP-5042B  Credits 10

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

Code CMP-5043B  Credits 10

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.

Code ECO-4006Y  Credits 20

The aim of this module is to introduce students to the economic way of reasoning, and to apply these to a variety of real world macroeconomic issues. Students will begin their journey by learning how to measure macroeconomic aggregates, such as GDP, GDP growth, unemployment and inflation. The module will establish the foundations to conduct rigorous Macroeconomics analysis, as students will learn how to identify and characterize equilibrium on the goods market and on the money market. The module will also introduce students to policy-making, exploring and evaluating features and applications of fiscal and monetary policy. Students will grow an appreciation of the methods of economic analysis, such as mathematical modelling, diagrammatic representation, and narrative. The discussion of theoretical frameworks will be enriched by real world applications, and it will be supported by an interactive teaching approach.

Code EDUB5012A  Credits 20

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.

Code ENV-5043A  Credits 20

The weather affects everyone and influences decisions that are made continuously around the world. From designing and siting a wind farm to assessing flood risk and public safety, weather plays a vital role. Have you ever wondered what actually causes the weather we experience, for example why large storms are so frequent across north western Europe, especially in Winter? In this module you will learn the fundamentals of the science of meteorology. We will concentrate on the physical processes that underpin the radiation balance, thermodynamics, wind-flow, atmospheric stability, weather systems and the water cycle. We will link these to renewable energy and the weather we experience throughout the Semester. Assessment will be based entirely on a set of practical reports that you will submit, helping you to spread your work evenly through the semester. You will learn how Weather is a rich fusion of descriptive and numerical elements and you will be able to draw effectively on your own skill strengths while practising and developing others, guided by Weatherquest’s Meteorologists.

Code NBS-4108B  Credits 20

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. 

Code PHY-4001Y  Credits 20

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.

Code CMP-6004A  Credits 20

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.

Code MTHE6003B  Credits 20

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. Therefore, second year Analysis is a desired prerequisite.

Code MTHE6007B  Credits 20

The ocean is an important component of the Earth's climate system. This module covers mathematically modelling of the large-scale ocean circulation and oceanic wave motion. This module builds upon the techniques in fluid dynamics and differential equations that you developed in year two. It then uses these techniques to explain some interesting phenomena in the ocean that are relevant to the real world. We begin by examining 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.

Code MTHE6026B  Credits 20

The Mathematical Modelling of Finance is a relatively new area of application of mathematics yet it is expanding rapidly and has great importance for world financial markets. The module is concerned with the valuation of financial instruments known as derivatives. Introduction to options, futures and the no-arbitrage principle. Mathematical models for various types of options are discussed. We consider also Brownian motion, stochastic processes, stochastic calculus and Ito's lemma. The Black-Scholes partial differential equation is derived and its connection with diffusion brought out. It is applied and solved in various circumstances.

Code MTHE6030A Credits 20

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 tangent spaces and the first and second fundamental forms, Gaussian curvature, and further topics.

Code MTHE6031B  Credits 20

You will gain an introduction to the theory of waves. You will study aspects of linear and nonlinear waves using analytical techniques and Hyperbolic Waves and Water Waves will also be covered. It requires some knowledge of hydrodynamics and multi-variable calculus. The module is suitable for those with an interest in Applied Mathematics.

Code MTHE6032A Credits 20

This module covers the laws of physics described by quantum mechanics that govern the behaviour of microscopic particles. The module will focus on non-relativitic quantum mechanics that is described by the Schrodinger equation. Time-dependent and time-independent solutions will be presented in different contexts including an application to the hydrogen atom. Approximation schemes will also be discussed, with particular emphasis on variational principles, WKB approximation.

Code MTHE6033A  Credits 20

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.

Code MTHE6034A  Credits 20 

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.

Code MTHE6035B  Credits 20

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, Diophantine Geometry and has found surprising applications in modern life (notably in Cryptography). In this module, building on first year material on prime factorization and basic congruences, and second year material on groups, rings and fields, you will study various aspects of Number Theory, including certain diophantine equations, polynomial congruences and the famous theorem of Quadratic Reciprocity.

Code PHY-6002Y  Credits 20

On this module you will study a selection of advanced topics in classical physics that provide powerful tools in many applications as well as provide a deep theoretical background for further advanced studies in both classical and quantum physics. The topics include analytical mechanics, electromagnetic field theory and special relativity. Within this module you will also complete a computational assignment, developing necessary skills applicable for computations in many areas of physics.

Code EDUB6014A  Credits 20

The aim of the module is to introduce you to the study of the teaching and learning of mathematics with particular focus to secondary and post compulsory level. In this module, you will explore theories of learning and teaching of mathematical concepts typically included in the secondary and post compulsory curriculum. Also, you will learn about knowledge related to mathematical teaching. If you are interested in mathematics teaching as a career or interested in mathematics education as a research discipline, then this module will equip you with the necessary knowledge and skills.

Code ENV-6004A  Credits 20

Our aim 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; the practicals being designed to illustrate the solution of problems using the methods covered in lectures. We will guide you through the solution of a model of an environmental process of your own choosing. The skills developed in this module are highly valued by prospective employers.

Code MTHA6002B  Credits 20

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.

Code MTHA6005Y  Credits 20

This module is reserved for students who have completed an appropriate number of mathematics modules at levels 4 and 5. It is a project on a mathematical topic supervised by a member of staff within the school, or in a closely related school. The focus of the project is on independent study; you will have the opportunity to undertake research in an area which is interesting to you. You will write an in-depth report on your topic, using the mathematical typesetting system LaTeX. You will also give a short oral presentation on your topic.

Code BIO-6018Y  Credits 20

You will gain an understanding of how science is disseminated to the public and explore the theories surrounding learning and communication. You will investigate science as a culture and how this culture interfaces with the public. Examining case studies in a variety of different scientific areas, alongside looking at how information is released in scientific literature and subsequently picked up by the public press, will give you an understanding of science communication. You will gain an appreciation of how science information can be used to change public perception and how it can sometimes be misinterpreted. You will also learn practical skills by designing, running and evaluating a public outreach event at a school or in a public area. If you wish to take this module you will be required to write a statement of selection. These statements will be assessed and students will be allocated to the module accordingly.

Code CMP-5020B  Credits 20

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.

Code CMP-5034A  Credits 20

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.

Code EDUB5012A  Credits 20

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.

Code ENV-5009B  Credits 20

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.

Code ENV-5043A  Credits 20

The weather affects everyone and influences decisions that are made continuously around the world. From designing and siting a wind farm to assessing flood risk and public safety, weather plays a vital role. Have you ever wondered what actually causes the weather we experience, for example why large storms are so frequent across north western Europe, especially in Winter? In this module you will learn the fundamentals of the science of meteorology. We will concentrate on the physical processes that underpin the radiation balance, thermodynamics, wind-flow, atmospheric stability, weather systems and the water cycle. We will link these to renewable energy and the weather we experience throughout the Semester. Assessment will be based entirely on a set of practical reports that you will submit, helping you to spread your work evenly through the semester. You will learn how Weather is a rich fusion of descriptive and numerical elements and you will be able to draw effectively on your own skill strengths while practising and developing others, guided by Weatherquest’s Meteorologists.

Code NBS-5101A  Credits 20

What are the rules that dictate how company accounts should be prepared and why do those rules exist? This is the essence of this module. Whilst company directors may wish to present the financial condition of a business in the best possible light, rules have been developed to protect investors and users of the accounts from being misled. You’ll develop knowledge and skills in understanding and applying accounting standards when preparing financial statements. You will also prepare and analyse statements of both individual businesses and groups of companies. Large UK companies report using International Financial Reporting Standards, and these are the standards that you’ll use. You’ll begin by preparing basic financial statements and progress, preparing accounts of increasing complexity by looking at topics including goodwill, leases, cashflow statements, foreign currency transactions, financial instruments and group accounts. You will also deepen your analytical skills through ratio analysis. You’ll learn through a mixture of lectures, seminars and self-study, and be assessed by one three-hour examination. On successful completion of this module, you’ll have acquired significant technical skills in both the preparation and analysis of financial statements. This will give you a strong basis from which to build if you are planning on a career in business or accounting.

Code NBS-5104B  Credits 20

The module aims to develop students’ understanding of the theory and practice of management accounting. It develops underpinning competencies in management accounting and builds on topics introduced in the first year. It extends comprehension of the role and system of management accounting for performance measurement, planning, decision making and control across a range of organisations. Additionally, it introduces recent developments in management accounting practice, particularly those which underpin its growing strategic role.

Code PHY-6002Y  Credits 20

On this module you will study a selection of advanced topics in classical physics that provide powerful tools in many applications as well as provide a deep theoretical background for further advanced studies in both classical and quantum physics. The topics include analytical mechanics, electromagnetic field theory and special relativity. Within this module you will also complete a computational assignment, developing necessary skills applicable for computations in many areas of physics.

Code ENV-5004B  Credits 20

In this module you will learn about the processes that shape the Earth's shallow subsurface, and how to detect and map subsurface structures and resources. Physical properties of solid materials and subsurface fluids will be explored, including how fluid movement affects these properties. Methods to image the subsurface will be introduced using real datasets, collected by the class where possible. We will apply the theory to real-life problems including risk mitigation, engineering and resource exploration. This module will include fieldwork on campus where possible, specialist computer software, and some light mathematical analysis (trigonometry, rearranging linear equations, logarithms).

Code MTHF5031Y  Credits 20

Combinatorics is one of the most applicable and accessible part of mathematics, yet it is also full of challenging problems. We shall cover many basic combinatorial concepts including counting arguments (enumerative combinatorics) and Ramsey theory. Linear Algebra underpins much of modern mathematics and is the key to many applications. We will introduce bilinear forms and symmetric operators on vector spaces leading to the diagonalization of linear maps and the spectral theorem. This theorem is key to many applications in statistics and physics. Other topics covered will include polynomials of linear maps, the Cayley-Hamilton theorem and the Jordan normal form of a matrix.

Code MTHF5032Y  Credits 20

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, using the results to better understand, or make predictions about, the original problem. You will look at techniques of mathematical modelling, examining how mathematics can be applied to a variety of real problems and give insight in various areas, including approximation and non-dimensionalising, and discussion of how a mathematical model is created. You will then apply this theory to a variety of models, such as traffic flow, as well as examples of problems arising in industry.

Code EDUB6006A  Credits 20

This module will introduce you to key issues in mathematics education, particularly those that relate to the years of compulsory schooling. Specifically, in this module, we: • introduce the mathematics curriculum and pupils' experiences of learning key mathematical topics; • discuss public and popular culture discourses on mathematics, mathematical ability and mathematicians as well as address ways in which these discourses can be challenged; • outline and discuss specific pedagogical actions (focused on challenge and motivation) that can be taken as early as possible during children’s schooling and can provide a solid basis for pupils' competence, confidence and and appreciation of mathematics. In this module, you will: • gain insight into key curricular, pedagogical, social and cultural issues that relate to the teaching and learning of mathematics, a crucial subject area in the curriculum; • reflect on pedagogical action that aims to address those issues, particularly in the years of compulsory schooling; • be informed and able to consider the potential of pursuing a career in mathematics education, either as a teacher, educational professional or researcher with particular specialisation in the teaching and learning of mathematics.

Code ENV-6025B  Credits 20

What sets the mean global temperature of the world? Why are some parts of the world arid whilst others at the same latitudes are humid? This module aims to provide you with an understanding of the processes that determine why the Earth’s climate (defined, for example, by temperature and moisture distribution) looks like it does, what the major circulation patterns and climate zones are and how they arise. You will study why the climate changes in time over different timescales, and how we use this knowledge to understand the climate systems of other planets. This module is aimed at you if you wish to further your knowledge of climate, or want a base for any future study of climate change, such as the Meteorology/Oceanography.

Code PPLP5175B  Credits 20

Epistemology examines what knowledge is. Science is concerned with the acquisition of secure knowledge, and philosophy of science considers what counts as science, what objects the scientist knows about, and what methods can be used to attain such knowledge; logic uses formal tools to investigate different forms of reasoning deployed to acquire knowledge. You will be given an opportunity to explore a selection of these areas of philosophy, through teaching informed by recent and ongoing research: which ones will be explored on this occasion will be selected in the light of the lecturers’ current research interests and their general appeal.

Code MTHA7029Y  Credits 40

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. We give a thorough treatment of predicate and propositional logic and an introduction to model theory.

Code MTHD7015A  Credits 20

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. We give a thorough treatment of predicate and propositional logic and an introduction to model theory.

Code MTHD7020A  Credits 20

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 to understand and predict how fluids (liquids and gasses) behave is of fundamental importance. 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).

Code MTHD7021A  Credits 20

Mathematical Biology 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. The main part of the module description is the same as for the Level 6 variant of the module - the Advanced Topic work will be announced during lectures.

Code MTHD7025A  Credits 20

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.

Code MTHD7032B  Credits 20

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. The Advanced level topics are both related to asymptotic methods applied to different classes of differential equations. In particular we will look at the method of Matched Asymptotic Expansions and the WKB approximation.

Code MTHD7033B  Credits 20

This course will cover normed spaces; completeness; functionals; Hahn-Banach theorem; duality; and operators. Time permitting, we shall discuss Lebesgue measure; measurable functions; integrability; completeness of L-p spaces; Hilbert space; compact, Hilbert-Schmidt and trace class operators; as well as the spectral theorem. The advanced topic will be Lebesgue measure, studied in depth

Code MTHD7034A  Credits 20

This module will be an introduction to some basic notions and results in algebraic topology. In this area tools from abstract algebra are used to study topological spaces, and conversely methods from topology can be used to prove results in algebra. In particular we will see how we can 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. In the advanced topic we will investigate some additional concepts and results including: fundamental groups of graphs and the proof that every subgroup of a free group is free, K(G,1)-spaces, free products with amalgamation, and HNN extensions.

Code MTHE7004B  Credits 20

A prerequisite of this module is that you have studied the Algebra module. Galois theory is one of the most spectacular mathematical theories. It gives a beautiful connection between the theory of 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 that can be solved (by some formula based on the coefficients, 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 4. In modern exposition, Galois theory deals with field extensions , and the central topic is the Galois correspondence between extensions and groups. The advanced topic concerns the so-called Inverse Galois problem : does every group correspond to some polynomial, and is the answer dependent on the base field?

Code MTHD7018B  Credits 20

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: Barotropic and baroclinic instability.