Keele University

Mathematics at Keele

Mathematics at Keele Find a direction

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This prospectus was written and designed by Dr J Preater and Neil Turner, School of Computing and Mathematics, Keele University. The details contained in this prospectus were correct at the time of publication. All photographs, except aerial view of Keele, ÂŠNeil Turner ÂŠ 2012

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Contents

Mathematics at Keele 5 Applications and Admissions 6 Bursaries and Scholarships 6 Degree Programmes 7 Degree Structure 10 Degree Content 12 Teaching and Support 19 Assessment 20 Degree Classification 20 Books 21 Student Ambassadors 22 What some of our students say â€Ś

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Careers 23

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Mathematics combines the beautiful and the useful. It is intellectually satisfying, sometimes shocking, persistently relevant.

Mathematics has well-established, yet rapidly developing, applications in business, engineering, science, technology, economics and medicine. Moreover, it is invoked

increasingly in areas such as law, politics, sport, ecology, psychology and the arts. The well-qualified mathematics graduate has an invaluable education and is a potentially flexible and sought-after employee, with a high level of problem-solving skills.

Mathematics at Keele The Mathematics degree programmes available at Keele are: •

Single Honours Mathematics,

•

Dual Honours Mathematics,

•

Mathematics (Major),

•

Actuarial Science.

This booklet aims to familiarise you with these programmes. The Mathematics department is part of the School of Computing and Mathematics, and is housed on the second floor of the MacKay building. It has around twenty academic staff, specialising in widely varying branches of mathematics, together with a number of postgraduate students and researchers. Over the years we have prided ourselves on providing a friendly and supportive environment for students. In particular, by the end of your degree, you should feel that you know the staff well and that they know you.

transition as smooth as possible, and to this end we have: •

a very carefully designed first year curriculum,

•

a multi-layered student support structure,

•

rapid feedback on coursework,

•

considerable experience in teaching, examining and moderating A-level Mathematics and Further Mathematics, and in school/college liaison.

There is also an active Staff-Student Liaison Committee. Academic mathematicians are inquisitive and creative beings, who are involved in a great variety of pure and applied research projects. Students curious to know what we get up to can look at the department’s web page. This research work is of high repute, as witnessed by the latest external research assessment exercise, and invigorates our teaching. We aspire to passing on that enthusiasm for mathematics to our students.

Understandably, many students regard the transition from A-level (or equivalent) to University mathematics with some trepidation. On the one hand this is a healthy attitude! University mathematics is conceptually stretching and presents many challenges. On the other hand, however, we recognise the need for making this

The MacKay Building 5

Applications and Admissions

English language requirements

Applications to study Mathematics at Keele should be made through the Universities and Colleges Admissions Service (UCAS). In the majority of cases, a conditional offer will be made on the basis of information contained in the UCAS application.

All applicants must have an approved qualification in English language, the usual qualification being grade C at GCSE. Other qualifications may be deemed suitable, and details can be obtained from the Admissions Tutor.

Entry by A-level qualifications Entrance requirements will depend on the degree programme you wish to pursue. For admission in 2012/13 our requirements are expected to be: •

Single Honours: minimum grades AAB, with grade A in Mathematics;

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Dual Honours: the full offer will depend on the other subject and will be either AAB or ABB;

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Mathematics (Major): as for Dual Honours;

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Actuarial Science: minimum grades AAB, with grade A in Mathematics.

Bursaries and Scholarships Keele offers a £1,000 per year cash bursary for all students in receipt of the full maintenance grant. In addition the University offers Excellence Scholarships dependent on A-level performance; £1,000 cash per year for students achieving grades AAB and £2,000 cash per year for students achieving grades AAA or better. Full details of the above are available at: http://www.keele.ac.uk/studentfunding/

We do not require A-Level Further Mathematics (at either AS or A2), though of course we are pleased to receive students with this additional mathematics qualification.

Entry by other qualifications For entry by way of the Scottish Certificate of Education or International Baccalaureate, enquiries should be made of the Admissions Tutor. We welcome applications from mature and overseas students. Applicants from overseas can obtain details about financing and the application process from offices of the British Council. With regard to mature students, we accept that it is not always practicable to obtain the formal qualifications set out above; in such cases we will seek evidence of ability to succeed on our degree programmes, and such applicants are encouraged to speak to the Admissions Tutor prior to completing the UCAS application. 6

ADMISSIONS TUTOR The undergraduate Admissions Tutor for all degree programmes involving Mathematics is Neil Turner. You may contact Neil Turner: e-mail: r.n.turner@maths.keele.ac.uk telephone: 01782 733739. For more general admissions enquiries, please refer to the current University prospectus.

Degree Programmes This section outlines the four Mathematics degree programmes on offer. The detailed structure, and a glimpse at some of the content, of these programmes are given in the later sections. The Single Honours Mathematics, Dual Honours Mathematics and Mathematics (Major) programmes all supply an opportunity for breadth and depth of study in pure, applied and statistical mathematics, while allowing students to concentrate on areas of individual interest, especially in Year 3. By its very nature Actuarial Science has a more focused curriculum. Throughout, there is rigorous development of particular branches of mathematics, as well as stress on the applicability of the theory and techniques being studied. In particular, the power of mathematical models is demonstrated for solving real-world problems. We recognise that students will enter University with varying backgrounds from A-level (or equivalent) and we have designed the first year programme accordingly. In particular, students need not be anxious if they have not taken Further Mathematics at A-level. Furthermore, some students may feel, initially, that they wish to avoid either mechanics or statistics, and this is possible in our structures (though, naturally, Actuarial Science students are required to study statistical courses). However, we often find that, with more experience, students re-evaluate their preferences and choose to explore unexpected areas.

Single Honours Mathematics UCAS code: G100

Duration: 3 years

Degree awarded: BSc Mathematics Typical intake: 70 The Single Honours Mathematics programme is for students with a single-minded interest in mathematics. No other discipline need be studied, but in Year 1 there is an option to take a lecture course in a related or completely different subject;

moreover, it may be possible to link mathematics with another academic field in the optional Year 3 project. In Year 1 and the first semester of Year 2 you will study the major areas of mathematics: calculus, algebra, geometry and analysis. You will also see how mathematical models are constructed and analysed, especially those based on differential equations, and how mathematicians bring precision to the study of uncertainty. Furthermore, there is instruction in the state-of-the-art mathematical software MathematicaÂŽ and in techniques for solving problems numerically. Starting in Year 2 and emphatically in Year 3, having acquired a grounding in core areas of mathematics, you are free to design either a wideranging or a more focused programme, depending on your personal interests.

Dual Honours Mathematics UCAS code: various (see below) Duration: 3 years Degree awarded: BSc Mathematics and â€Ś Typical intake: 100 Keele University has been a leading proponent of Dual Honours degrees and retains its commitment to breadth in Higher Education. In a Dual Honours degree at Keele, each of two Principal subjects is studied separately and equally (with a few minor exceptions). This means, of course, that the student must be fully committed to both subjects. On successful completion of the degree, having studied two areas extensively, the graduate has particular flexibility with regard to career choice and employability. Dual Honours programmes in which one Principal subject is Mathematics are listed below. 7

Mathematics and â€Ś

UCAS code

Accounting

NG41

American Studies

GT17

Applied Environmental Science

FG71

Applied Psychology

CG8C

Astrophysics

FG51

Biology

CG11

Business Management

GN19

Chemistry

FG11

Computer Science

GG14

Creative Computing

GG41

Economics

GL11

Educational Studies

GX13

Film Studies

PG3C

Finance

GN13

Forensic Science

FG41

Geography

LG71

History

GV11

Human Biology

CG1C

Human Geography

GL17

Information Systems

GG15

International Business

NG11

International Relations

GL12

Law

GM11

Marketing

GN15

Media, Communications and Culture

PG31

Medicinal Chemistry

FGC1

Music

GW13

Neuroscience

BG11

Philosophy

GV15

Physical Geography

FG81

or applied mathematics, or explore probability and statistics. There is a large selection of lecture courses available in Year 3; these courses are generally common to Dual and Single Honours students. You can also engage in a substantial project, giving you experience in report writing and oral presentation as well as in more independent mathematical study. Dual Honours Mathematics students may elect to specialise entirely in Mathematics in Year 3, subject to permission by the School, no matter what their other subject. This, in effect, transfers them to the Mathematics (Major) programme (see below). Similarly, Dual Honours Mathematics students may be permitted to specialise in their other discipline in Year 3 (see the Keele University prospectus for details). Finally, Dual Honours Mathematics students who have demonstrated a particular aptitude for the subject may be allowed to transfer to Single Honours Mathematics at the end of their first semester.

Mathematics (Major) UCAS code: G105

Duration: 3 years

Degree awarded: BSc Mathematics with â€Ś Typical intake: 5

The Mathematics (Major) programme is identical to the Dual Honours Mathematics programme in Years 1 and 2 and the Single Honours Mathematics Physics FG31 programme in Year 3. It is currently available for Psychology CG81 students of Mathematics together with any of the Smart Systems GG71 disciplines listed above. As in all Dual Honours Sociology GL13. programmes at Keele, the two Principal subjects (of which one is Mathematics) are studied separately and equally in Years 1 and 2; so a student on this programme must be fully committed to both In Year 1 you will study core material in calculus subjects. and algebra, which all mathematicians should know. Then in Year 2, alongside the study of If, on reflection, you wish to continue to divide differential equations, advanced calculus and your time in Year 3 equally between Mathematics complex variable theory, you can begin to tailor and your other discipline, it is normally possible your programme according to your own strengths to transfer into the appropriate Dual Honours or interests; for instance, you can specialise in pure Mathematics programme. 8

Actuarial Science UCAS code: N323

Duration: 3 or 4 years

Degree awarded: BSc Actuarial Science Typical intake: 10 Actuaries are professionals who evaluate risk and develop strategies for its management. They work principally in the fields of finance, insurance, pension funds and investment, though their skills are applicable more broadly wherever risk is crucial. The Actuarial Science programme is delivered jointly by the Keele Management School (KMS) and the Mathematics department. In Years 1 and 2 you will divide your time equally between Mathematics and modules provided by KMS. In Mathematics you study core material in calculus, algebra, differential equations, probability, statistics and stochastic processes, all of which are central to actuarial practice. In Year 3, further options are available in Mathematics, but the majority of the programme is provided by KMS. Between academic Years 2 and 3 an Actuarial Science student may, subject to approval, choose to undertake a one-year placement, so that the degree is completed in four rather than three years. Career possibilities and details of those parts of the programme provided by KMS are given in the Keele University prospectus.

Spring on Campus

Spring on Campus

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Degree Structure Each degree programme is made up of a collection of distinct lecture courses called modules. Every module covers a particular branch of mathematics. As your studies progress the subject matter of the modules becomes more specialised and your options increase.

Single Honours Year 3

Year 1 Autumn semester Calculus I

Graph Theory

Group Theory

Algebra I

Complex Variable II

Computational Mathematics

Logic

Elective *

Functional Analysis Codes and Cryptography

Spring semester

Number Theory

Calculus II

Ring and Field Theory

Algebra II Analysis

Partial Differential Equations

Geometry or Applicable Mathematics

Nonlinear Differential Equations

choose eight**

Perturbation Methods

Year 2

Waves

Autumn semester

Relativity

Fluids

Differential Equations

Numerical Analysis

Abstract Algebra

Mathematical Biology

Probability

Probability Models

Linear Algebra

Medical Statistics

Operational Research

choose one**

Numerical Methods

Project

Spring semester Complex Variable I and Vector Calculus Mathematical Modelling Metric Spaces Dynamics Stochastic Processes

choose two**

Linear Statistical Models 10

* An Elective can be a module from another discipline: there is plenty of choice. Alternatively, there are the two Mathematics Electives, namely ‘Making Sense of Statistics’ and ‘Mathematical Methods’.

** Subject to timetabling and pre-requisite constraints.

Degree Structure Dual Honours Mathematics

Mathematics with...

Year 1

Years 1 & 2

Autumn semester

As Dual Honours Mathematics

Calculus I

Algebra I

Year 3

Spring semester

As Single Honours Mathematics

Calculus II Algebra II

Actuarial Science

Year 2

Year 1

Autumn semester

As Dual Honours Mathematics

Differential Equations Abstract Algebra Linear Algebra

Year 2 choose one **

Probability

Autumn semester

Operational Research

Differential Equations

Numerical Methods

Probability

Spring semester

Spring semester

Complex Variable I & Vector Calculus

Stochastic Processes Linear Statistical Models

Real Analysis

Mathematical Modelling Dynamics

choose one **

Stochastic Processes Linear Statistical Models

constraints.

Partial Differential Equations Numerical Analysis

Choose four ** modules from the Year 3 list given in the Single Honours Mathematics programme above.

**Subject

Year 3

Nonlinear Differential Equations

Year 3

to

timetabling

and

choose up to two**

Probability Models Medical Statistics

pre-requisite 11

Degree Content This section gives a brief indication of the content of each of the modules listed earlier.

Year 1 Modules Calculus I This module forms a bridge between A-level and University level mathematics. Many of the topics covered will be familiar, but the emphasis will be different, focusing on understanding and tackling some of the more technical issues necessarily left unresolved at A-level. Starting with a brief look at the real number system, the module then examines real-valued functions and, in particular, the trigonometric, exponential and logarithmic functions. Moving on to the notion of a â€˜limitâ€™, the module then discusses infinite series, differentiation and integration in a rather more careful and precise way than at A-level. The module closes with an introduction to differential equations.

Algebra I The aim of the first part of this module is to introduce students to the fundamental mathematical concepts of logic, numbers, sets and functions in a setting that will be unfamiliar in its level of abstraction, but that will provide an essential grounding for later modules. Progressing to complex numbers, polynomials and divisibility in the set of integers, this module also introduces students to the Fundamental Theorems of Arithmetic and Algebra. The concept of rigorous proof is central to all of the material in this module.

Computational Mathematics This module will expose students to numerical methods for solving algebraic and differential equations, iteration techniques and numerical integration. One aim of the module is to familiarise students with appropriate technology for solving problems numerically and, in particular, to introduce and provide a working knowledge of MathematicaÂŽ, a powerful software package useful in many diverse areas of mathematics.

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Calculus II Building on the foundations of Calculus I, this module is largely techniques based, with much of the material being essential for second and third year applied and methods modules. Divided into three parts, the first topic of study is that of ordinary differential equations, including linear and nonlinear first-order equations and certain classes of second-order linear equations. The second part of the module studies the theory and application of Taylor series, whilst the third part introduces students to functions of two variables, including partial differentiation and double integration techniques. There is also a brief introduction to partial differential equations.

Algebra II The first two-thirds of this module concentrates on an introduction to linear algebra. Topics studied are systems of linear equations, matrices and their algebra, determinants, vectors in 2-, 3-, and n- dimensional Euclidean space and a brief introduction to some of the basic concepts of general vector spaces. The emphasis is on precise derivation of results and drawing together apparently disparate areas of mathematics. The final third combines elements of geometry and linear algebra to examine the important optimisation technique of linear programming. This is used extensively in organisational and manufacturing contexts.

Applicable Mathematics This module is designed to assist studentsâ€™ appreciation of mathematics as a tool for describing and solving real-world problems. In addition it aims to help students to make the transition from A-level to undergraduate mathematics; as such it concentrates on mathematical problem solving that moves away from the examples-based methods encountered at A-level towards the more sophisticated approaches expected at degree level and in employment. Physical or computer experiments will be used to motivate the study of a number of phenomena. The mathematical and problem solving ideas will be developed throughout the module by means of group projects.

plane, as first explored by the mathematicians of ancient Greece. It then progresses to examine straightedge and compasses constructions, constructible numbers, and isometries of the plane, together with some non-Euclidean geometries, such as spherical geometry and hyperbolic geometry. A study of the symmetries of polygons and polyhedra will introduce students, in an intuitive way, to an algebraic structure called a group. The module concludes with a brief treatment of knots.

Analysis This module is compulsory for Single Honours students and is available as a second semester option for Dual Honours students in their second year. The module introduces students to formal analysis, and its core concept of a limit, in the context of sequences, infinite series and functions. Through extension of these ideas to the definition of continuity of functions, students will see how a rigorous foundation for differential calculus is achieved.

The Mathematics Lab

Geometry This module in pure mathematics, which is available to Single Honours students only, explores a subject that has become virtually extinct in the school curriculum. The module begins with a detailed examination of the geometry of the Euclidean 13

Year 2 Modules Differential Equations This module builds on the first year module Calculus II. It aims to develop skills in mathematical techniques, focusing on methods for solving ordinary differential equations. The topics covered include: solutions to first-order differential equations, unforced and forced linear differential equations with constant coefficients, the harmonic oscillator, power series methods of solution and graphical aspects of differential equations. Students will also be introduced to Fourier series and Laplace transforms in a problem-solving context.

Abstract Algebra This module introduces and studies the abstract algebraic structure known as a group. Beginning with the axiomatic and theoretical foundations, the module progresses, through a study of subgroups, to the proof of one of the most important theorems in group theory, namely Lagrangeâ€™s Theorem. The module also examines applications of group theory, and the often beautiful way in which it interacts with geometry and number theory. It concludes with a preliminary exploration of other, closely related algebraic structures, namely rings and fields.

Concepts such as linear independence, span and scalar products of vectors are generalised from Euclidean space to other vector spaces, such as function spaces, in such a way that seemingly disparate results from different branches of mathematics are sometimes seen to be just different specialisations of the same general concept.

Operational Research This module presents a variety of topics, techniques and models in operational research.Â The material falls into four main areas: mathematical programming, combinatorial optimization, inventory modelling and multicriteria optimization. Throughout there is an emphasis on the portability of techniques from one area to another; on applying principles and concepts imaginatively and with sound judgement; and on modelling issues. The module builds on and uses first year work on linear programming and the Simplex method.

Probability Probability is the mathematics of uncertainty. Motivated by a discussion of classical probability based on cards, coins, dice, etc., an axiomatic approach to probability theory is developed. This is a sound framework within which to handle independence of several events, conditional probability and Bayesâ€™ Theorem. Thereafter the module focuses on discrete and continuous random variables and their probability distributions, concluding with a treatment of moment generating functions and the central limit theorem.

Linear Algebra This module introduces the concept of an abstract vector space. The module builds on the knowledge of vectors and matrices gained from Algebra ll. 14

Library Numerical Methods Aiming to furnish students with a working knowledge of a fundamental set of numerical methods and the use of available technology, this module studies such methods, concentrating primarily on solving linear algebraic equations. Topics covered include: algebraic equations of the form f (x) = 0, numerical quadrature, the Euler and Runge-Kutta methods for solving differential equations, iteration techniques and numerical methods for determining eigenvalues and eigenvectors.

Complex Variable I and Vector Calculus

Real Analysis

The first half of the module is an introduction to vector calculus, which provides a framework for solving physical and geometric problems. First developing familiarity with the basic ideas, language and operators of vector calculus, the module then progresses to the classical integral theorems of Green, Gauss and Stokes. The second half of the module is concerned with functions of a complex variable. After a quick revision of complex numbers from Algebra I, students will study elementary functions of complex variables before examining analytic functions and the CauchyRiemann equations. The module then moves on to the fundamental results embodied in Cauchy’s Theorem, Cauchy’s Integral Formula and Cauchy’s Residue Theorem, providing some elementary applications along the way.

This module is intended as an introduction to mathematical analysis, which can be described as the study of the infinitesimally small and infinitely large. The module studies the real number system, sequences, series, limits of functions, continuity and Taylor series. This module is available only as a Year 2 option for Dual Honours students.

Metric Spaces Motivated by examples from real analysis, numerical analysis and geometry, this module explores the idea of a metric space, i.e. a set and associated ‘distance measure’ between elements of the set. Expanding on ideas from Year 1 Analysis, the module examines the core concepts of convergence, completeness, continuity, connectedness and compactness in the context of a metric space. The Contraction Mapping Theorem will be studied and students will have their first introduction to the ideas of topological spaces.

Mathematical Modelling The aim of this module is to demonstrate how realworld problems can be modelled mathematically. The mathematical modelling process is introduced through a six-step, problem-solving approach. Mathematical tools used in the model construction and solution process will include ordinary differential equations and their solution methods (including phase-plane analysis), dimensional analysis and difference equations. The modelling ideas will be developed through novel and innovative case studies of real-world scenarios and through individual/group projects.

Dynamics This module aims to demonstrate the successful application of mathematics in the modelling of physical systems, in particular those systems which can be modelled by particle dynamics. The topics investigated will include Newton’s laws, momentum, kinetic and potential energy, projectiles, simple harmonic motion, springs, pendulums, rocket motion, planetary and satellite orbits together with the linear theory of oscillations, normal modes, and forced oscillations with or without damping. A key aim of this module is the development of students’ skills in translating a verbal problem into mathematics, solving the resulting equations and interpreting the answer.

Stochastic Processes This module introduces students to the mathematical description of random processes that evolve in time. Initially, the theory from firstsemester Probability is used to derive properties of simple random walks. The module then moves on to Markov processes, examining discrete time Markov chains, and continuous time Markov chains and, finally, to reliability theory. Throughout the module, the theory is illustrated with practical examples.

Linear Statistical Models This is a module in the theory of statistical data analysis. The first step in such an analysis is to adopt a formal model for the data, which will very often be expressed in terms of linear relationships between several variables, some of which are random. In this setting the analyst can make wellfounded inferences about the population from which the data are drawn. The module includes the study of ANOVA (analysis of variance) and regression. 15

Year 3 Modules In the third year, all students have a wide choice of modules, though Single Honours students must take the module in Partial Differential Equations. The availability of modules, and their content, may change from time to time subject to student uptake, staff availability and timetabling restrictions.

Graph Theory This module introduces the concept of a graph as a pictorial representation of a symmetric relation. A variety of topics are investigated and, for each one, at least one of the major theorems is proved. The emphasis is on pure graph theory, although a significant number of applications are explored.

Group Theory Building on the concept of a group introduced in Abstract Algebra, this module develops some of the mathematics underlying the classification of finite groups, culminating in a proof of Sylow’s First Theorem. The module also develops some applications of group theory.

Number Theory Number Theory studies the properties of the natural numbers and the integers. It is one of the oldest and most beautiful areas of pure mathematics, first studied by the Ancient Greeks, and yet has a surprising number of modern applications, such as cryptography. It is a topic which is famous for a large number of results which are extremely simple to state, but turn out to be difficult to prove. Indeed, many of these problems remain unsolved, and so Number Theory is one of the most active areas of modern research.

Ring and Field Theory This module gives an introduction to fundamental topics and concepts in modern abstract algebra via the systems of rings and fields. The first half of the module covers elementary topics in commutative ring theory while the second half covers Galois theory – a highlight in the history of 16

pure mathematics where field theory and group theory come together to answer some of the oldest questions about polynomials and their roots. In particular, Galois theory allows us to prove that there is no general formula for finding the roots of a quintic polynomial. Several applications of the theory will be given, including the application to classical proofs concerning ruler and compasses constructions.

Logic By introducing mathematical concepts for examining philosophical questions about the nature of mathematics as a whole, this module aims to present a sophisticated perspective on mathematics that is accessible to undergraduates. Some of the topics covered are the ZermeloFraenkel axioms of set theory, construction of number systems, the construction of formal languages and predicate calculus.

Codes and Cryptography This module will illustrate how abstract ideas from group theory, number theory and vector spaces can be brought together to solve problems concerning reliable, efficient and secure communication of information. Coding theory studies methods and algorithms for reliably and efficiently transmitting information in a situation where there is a risk of “noise” disrupting the communication. Cryptography adds the requirement that this communication should be secure. Both of these areas depend on the study of finite fields and linear spaces over them. In addition to studying theory, there will be opportunities to implement the techniques and algorithms via group activities.

Functional Analysis

Numerical Analysis

This could be considered to be a fusion of vector and metric spaces. The main thrust of the module is the theory of Hilbert spaces, which provides a natural setting for generalised Fourier series. These series are fundamental in, for example, the mathematical foundations of quantum mechanics.

This module is concerned with the analysis of numerical methods, concentrating on how to select a suitable method and analysing the results produced, rather than on programming the methods themselves.

Complex Variable II Extending the material from Complex Variable I, students will first study further applications of contour integration and the Residue Theorem. The module then examines conformal mappings, in particular their application in determining harmonic functions in two-dimensional regions. Certain analytical aspects of complex functions are also covered. Complex variable analysis has major applications in fluid dynamics.

Nonlinear Differential Equations

Perturbation Methods This module introduces students to the asymptotic techniques which are indispensable tools in dealing with mathematical problems arising in the real world. These ‘raw’ problems very rarely resemble simple models which can be solved exactly. Fortunately, very often the ‘raw’ problems have either large or small parameters, or there is a way to determine an approximate solution which is sufficient for most practical purposes. The module will present an overview of the main analytical perturbation techniques based upon use of a small/large parameter or small/large values of a coordinate.

The great variety of behaviour in physical systems is reflected in the solutions of the differential equations used to model them. The majority of these equations are nonlinear, very few of which have exact solutions. The module focuses on methods for obtaining approximate solutions and phase plane representations of dynamics.

Partial Differential Equations Consider a function, f, that depends on two variables, x and y. A derivative of this function, with respect to one of these variables whilst the other variable is held fixed, is called a partial derivative. Partial differential equations (PDEs) are equations that contain partial derivatives, and many physical processes, such as wave propagation and heat conduction, are governed by PDEs. This module discusses basic solution techniques for PDEs. A typical question that can be answered by the theory of PDEs is “Where should we place the radiators in a room such that the temperature is as uniform as possible?”

Hornbeam and Dorothy Hodgkin Waves This module aims to give an account of the underlying mathematical theory that describes the behaviour of waves in diverse physical settings. These include waves on beams, membranes and stretched strings, sound waves and waves in liquids with a free surface.

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Fluid Mechanics Fluids are everywhere â€“ your blood is a fluid, as is the air in your lungs, the water in the oceans and the burning gas in the sun. Despite this diversity, most fluid motion can be explained by a tiny number of core principles. The module derives these core principles, and illustrates them in some simple examples of fluid flow.

Relativity Students will study both Special Relativity (Minkowski space-time, relativistic mass and energy, time dilation) and General Relativity (the Einstein field equations, curvature of space, Schwarzschild space-time) in a module designed to introduce students to the mathematical foundations of two of the most important developments in 20th century physics.

Mathematical Biology One of the fastest growing fields in which mathematics can explain and predict behaviour is within the biological sciences. Game theory has supplied new mathematical tools to study the evolution of animal behaviour. Mathematical models of population growth, and particularly of disease transmission, have provided new insights into the spread of MRSA. Moreover, in the last few years, there have been advances in the application of mathematics to the study of animal locomotion. The module discusses these and other applications.

particular emphasis placed on cancer studies. No prior knowledge of medicine or biology is required. The module commences with a revision of hypothesis testing procedures. This is followed by three main topics: clinical trials, survival analysis and epidemiology. Clinical trials are immensely important for evaluating the relative effectiveness of different treatments, and their design and analysis are considered in depth. Survival analysis looks at the features and analysis of data from studies of patients with a potentially fatal disease. Epidemiology explores the distribution of disease in a population and discusses studies for assessing whether there is a possible association between a factor (such as, smoking, eating beef, using a mobile phone) and the subsequent development of a disease.

Project It is possible for students to replace one taught module with a project module. In this case the student will be assigned an appropriate supervisor who will provide guidance throughout. The project itself may comprise an in-depth account of a body of theory new to the student, the solution of an applied problem, or an analysis of a substantial data set using advanced statistical techniques. The project is assessed by way of an oral presentation given to staff and students, a brief interim report, and a final report submitted towards the end of the spring semester.

Probability Models This module deepens the study of probability theory, surveying a range of random processes and problems with relevance to meteorology, economics, engineering, etc. A particular focus is the beautiful Brownian motion process, which was originally used as a mathematical description of the chaotic motion of particles under molecular bombardment, but which is now central to physical and financial modelling.

Medical Statistics This module illustrates the application of statistical techniques to health related research. Methods are applied using data from real-life studies, with 18

Medical Centre

Teaching and Support Each academic year has two teaching semesters. In each semester the teaching period is followed by formal examinations. All students take the equivalent of four modules in each semester (two Mathematics modules for Dual Honours students). A typical Mathematics module will have three lectures and one examples class each week. The purpose of the examples classes is to consolidate the material presented in the lectures, by way of solving specific problems related to that material. The studentsâ€™ understanding is then further enhanced by the completion of coursework or class tests, the marks for which comprise a component of the final module mark. Whilst lecture classes can be large we endeavour to limit the size of examples classes to no more than twenty students. As already stated, the transition from A-level to undergraduate mathematics can be demanding, though we seek to make that transition as smooth as possible. Coping with that transition will depend, partly, on the attitude of the individual student. In order to foster an appropriate work ethic, attendance at lectures and examples classes is deemed compulsory. We also recommend that for each hour of contact time, you spend around two hours studying your lecture notes and completing any assignments. We recognise, of course, that students may well need additional support in their studies and we pride ourselves on the support network that we provide. Students are encouraged to seek advice from the relevant module lecturer or examples class tutor. To this end, staff normally operate an open-door policy, whereby students may call in for advice at any time that the member of staff is available. All students have access to each of the three Year Tutor Teams, each of which comprises two or three members of staff dedicated to looking after our students during their time at Keele. Moreover, each Single Honours student is allocated a personal tutor in Mathematics who will be willing to talk over any difficulties, academic or otherwise

Keele Hall

Feedback From a studentâ€™s perspective, good feedback on assignments is an important and valuable part of the learning experience. We take pride in the quality of our feedback and the speed with which we provide it. Generally, we aim to return marked assignments within one week of submission. In addition, solutions to assignments, and explanatory notes, are often provided electronically. Students also have recourse to their personal tutor (or relevant Tutor Team in the case of Dual Honours students) for provision of feedback on their general academic progress. Of course, feedback is not a one-way process. We place great weight on the views of students, both with respect to individual modules and our teaching in general; results of student questionnaires are fed back from a working group to the Staff-Student Liaison Committee to consider the implementation of suggestions. In this way, we hope to maintain our high standards of teaching.

To support student learning further there is a range of material available on the Universityâ€™s web-based learning system. 19

Assessment Student performance on each module is assessed separately. The standard assessment pattern is: 20% from continuous assessment during the module, and 80% from a written examination taken after the end of the module. To inculcate good working habits there is a variation from the standard pattern in Year 1. Furthermore, in a few modules in Years 2 and 3, the nature of the subject-matter is reflected by a higher coursework weighting. For example, the optional Year 3 Numerical Analysis module involves substantial problem-solving using sophisticated computer algebra systems, so the assessment scheme is adjusted accordingly. The pass mark on each module is 40%. Students who fail a module in Years 1 or 2 may be offered reassessment for that module, which is normally in the form of a written examination. Re-assessment marks are capped at 40%.

Progression Progression from Year 1 to Year 2 and from Year 2 to Year 3 is subject to satisfactory performance in module assessments in the current year. Under certain circumstances students unable to progress may be permitted to repeat a year of study.

Tawney Building

Degree Classification All students who complete their degree programmes are awarded a single degree class. In addition, all students receive a detailed transcript showing their individual module marks. The University classifies its degree performance according to the usual UK scheme: First (I), Upper Second (IIi), Lower Second (IIii),

Honours

Third (III) Pass Fail The degree class awarded is derived from the profile of module marks in Years 2 and 3. Naturally, Year 3 marks carry more weight. Students are notified, when they enter the University, of exactly how their degree class will be calculated.

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Books At the outset of each module the lecturer will provide information on relevant textbooks. In some cases students will be required to purchase their own copy of a book: these should be available from the campus bookshop. In many cases no purchase is required, but books helpful as background reading, which are normally stocked in the University library, will be indicated. It is not necessary to purchase any books before arriving at Keele. However, it is very valuable to do some mathematical reading before beginning your degree programme. Almost anything mathematical that captures your interest is worthwhile. Here are some ideas:

For a wonderful survey of mathematics as a whole, at various levels of sophistication, try the bottom book in the photograph: The Princeton companion to mathematics. Ed. Timothy Gowers, Princeton. Judicious use of the internet is also useful; a great site is: www.cut-the-knot.org. Perhaps most important of all is to keep trying problems. The more you read and do, the more compelling it all becomes.

Mathematics: a very short introduction. Timothy Gowers, Oxford. 1089 and all that. David Acheson, Oxford. Game, set and math. Ian Stewart, Penguin. How to think like a mathematician. Kevin Houston, Cambridge. An introduction to mathematical reasoning. Peter J. Eccles, Cambridge. Mathematical puzzles and diversions. Martin Gardner, Penguin. The Penguin book of curious and interesting puzzles. David Wells, Penguin. The Mathematical Olympiad handbook. Tony Gardner, Oxford. Browse the ‘popular science’ shelves of any large bookstore – there should be plenty to interest you. 21

Student Ambassadors A number of our second and third year students assist the department at Open and Visit Days. This generally requires talking to prospective students and their families about the student experience of studying Mathematics at Keele, as well as escorting

them on tours of the campus. In addition, some of these student ambassadors become involved with our outreach activities, such as the Maths Days, provided for local schools and colleges.

Some of our students have said …

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“The Maths department at Keele has a great atmosphere about it and if you ever have any problems there is always someone willing to help. The course is challenging but incredibly interesting and you get to have a go at lots of different aspects of maths.”

The lecturers have an open-door policy and are always willing to help and advise students, even if they’re having their lunch!”

“Choosing the right university can be quite overwhelming, but I am not regretting choosing Keele. I chose Keele partly because of the homely feel on campus, when I visited on the Open Day. The campus is really pretty and has a good community vibe, where everyone knows each other. I also chose Keele, because of the impression left on me by both the Mathematics and the Philosophy departments; the staff came across as very friendly and approachable.”

“Everyone at Keele is really friendly and welcoming and in my first year it wasn’t hard to make friends on my course.”

“The lecturers are very in tune with their subject and show a lot of passion for their area of mathematics. They try and make lectures interesting and whenever I have had a problem, they have been eager to help and point me in the right direction.

“The help you get from the lecturers is fantastic. If you have any problems then you can just go to them and ask, and they are more than willing to sit with you until you can really figure it out.”

“I’ve absolutely loved my time at Keele. The lecturers are fantastic and really helpful. ”

“The lecturers in Maths at Keele are very enthusiastic about their subject and I think that really makes students enthusiastic as well, and want to learn.” “The teaching is really good. We have an open-door policy so if we need help from the lecturers, and they are available, then they will always help. ”

Careers Mathematics graduates pursue a wide variety of careers, and options open to Dual Honours students can be especially broad. Some graduates go on to further study, either at Masters or PhD level, either at Keele or another institution. Others move directly into employment or professional training. Some specific examples are: accountancy

NHS accounts analyst

actuarial training

operational research

aircraft engine design

software developer

civil servant

scientific research

data analyst

statistician

defence modelling

teaching

Formula 1 car design

University lecturer.

NHS project manager

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Notes

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Notes

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Mathematics at Keele University School of Computing and Mathematics, Colin Reeves Building, Keele University, Keele, Staffordshire ST5 5BG Email: admin@maths.keele.ac.uk Tel: +44 (0)1782 733075 Fax:+44 (0)1782 734268 www.facebook.com/KeeleSCM

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