Module Catalogue - Mathematics

Mathematics
School of Computing and Mathematics
Faculty of Natural Sciences

Last Updated 19 September 2013

Principal Course Timetable Blocks 3

Mathematics Dual Honours - Level 1 Modules

Semester 1			C/O	TYP	ECTS	CATS
	MAT-10038	Algebra I	C	M	7.5	15
		This module is intended to help students with the transition from the methods based approach of A level mathematics to the higher levels of understanding and rigour expected at degree level. It begins by discussing mathematical statements and the meaning and basic strategies of proof. This is followed by a short exposition of naive set theory and by a careful treatment of the notion of a function. The remainder of the module covers the algebraic development of number systems and their properties. The module ends with a brief discussion of the properties of polynomials, including the Fundamental Theorem of Algebra. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.
	MAT-10039	Calculus I	C	M	7.5	15
		This module is intended to help students with the transition from the methods based approach of A level Mathematics to the higher levels of understanding and rigour expected at degree level. The module will develop students' skills in mathematical techniques, mainly in differentiation and integration. It begins by revising the standard elementary functions and their properties and continues with the revision of basic results and techniques in differentiation. It then revises basic results and techniques in integration. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.

Semester 2			C/O	TYP	ECTS	CATS
	MAT-10040	Algebra II	C	M	7.5	15
		This module continues from Algebra I and provides an introduction to matrix and vector algebra. Topics will include matrix algebra, elementary row operations, solving linear equations, determinants, eigenvalues and eigenvectors, diagonalisation of matrices, three dimensional vector algebra and geometry, lines and planes, general vector algebra, linear independence and bases. Some of this material will be applied in a treatment of linear programming, a widely-used optimization technique. In particular the Simplex method and an introduction to duality theory are covered. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.
	MAT-10041	Calculus II	C	M	7.5	15
		Many physical problems are governed by ordinary or partial differential equations, the solution of which can help us understand their properties and characteristics. For instance, the oscillation frequency of a pendulum, the transfer time for sending a spaceship from the Earth to Mars, and the population evolution of a fish species in a lake can all be determined by solving ordinary differential equations. This module, which is a prerequisite for a number of other modules in the second and third years, will introduce some of the basic techniques for solving ordinary differential equations, familiarize students with partial differentiations, and explain how double integrals can be evaluated and used to compute areas and volumes. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.

Mathematics Dual Honours - Level 2 Modules

Semester 1			C/O	TYP	ECTS	CATS
	MAT-20008	Differential Equations	C	M	7.5	15
		This module focuses on methods for solving ordinary differential equations. The topics include: solutions to first-order equations, higher order linear equations, power series methods, graphical aspects of differential equations, Fourier series and Laplace transforms.
	MAT-20011	Mathematics - Study Abroad I	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20012	Mathematics - Study Abroad II	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20015	Numerical Methods	O	M	7.5	15
		This module studies numerical methods as applied in linear algebra and to differential equations. Topics include: the solution of algebraic equations, numerical quadrature, the numerical solution of differential equations including the Euler and Runge-Kutta methods, the solution of linear equations by direct and iterative methods, and numerical methods for determining eigenvalues and eigenvectors.
	MAT-20015	Numerical Methods	EP	M	7.5	15
		This module studies numerical methods as applied in linear algebra and to differential equations. Topics include: the solution of algebraic equations, numerical quadrature, the numerical solution of differential equations including the Euler and Runge-Kutta methods, the solution of linear equations by direct and iterative methods, and numerical methods for determining eigenvalues and eigenvectors.
	MAT-20023	Probability	O	M	7.5	15
		Probability is the mathematics of uncertainty and randomness. The module begins with classical notions of probability associated with the analysis of games of chance using cards, dice, etc. It then moves to the axiomatic treatment of the probability of random events. This leads to the definitions of statistical independence and conditional probability. The remainder of the module is concerned with a systematic study of discrete and continuous, univariate and bivariate, random variables, covering expectation, variance, covariance and the moment generating function. The theory is applied to a wide range of theoretical and applied problems. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20023	Probability	EP	M	7.5	15
		Probability is the mathematics of uncertainty and randomness. The module begins with classical notions of probability associated with the analysis of games of chance using cards, dice, etc. It then moves to the axiomatic treatment of the probability of random events. This leads to the definitions of statistical independence and conditional probability. The remainder of the module is concerned with a systematic study of discrete and continuous, univariate and bivariate, random variables, covering expectation, variance, covariance and the moment generating function. The theory is applied to a wide range of theoretical and applied problems. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20024	Operational Research	O	M	7.5	15
		This module presents a variety of topics, techniques and models in operational research. 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. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20024	Operational Research	EP	M	7.5	15
		This module presents a variety of topics, techniques and models in operational research. 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. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20025	Abstract Algebra	O	M	7.5	15
		This module introduces and studies the abstract algebraic structure known as a group. Starting 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, Lagrange'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. The module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 2. An appreciation of the development and value of Mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.
	MAT-20025	Abstract Algebra	EP	M	7.5	15
		This module introduces and studies the abstract algebraic structure known as a group. Starting 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, Lagrange'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. The module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 2. An appreciation of the development and value of Mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.
	MAT-20026	Linear Algebra	O	M	7.5	15
		This module introduces the concept of an abstract vector space The module builds on the knowledge of vectors and matrices gained from MAT-10040. 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 just different specialisations of the same general concept. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity, and independence of thought. 2. An appreciation of the development and value of mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.
	MAT-20026	Linear Algebra	EP	M	7.5	15
		This module introduces the concept of an abstract vector space The module builds on the knowledge of vectors and matrices gained from MAT-10040. 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 just different specialisations of the same general concept. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity, and independence of thought. 2. An appreciation of the development and value of mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.

Semester 2			C/O	TYP	ECTS	CATS
	MAT-20004	Complex Variable I and Vector Calculus	C	M	7.5	15
		This module contains a first course on vector calculus and a first course in functions of a complex variable. Complex variable leads to elegant results in pure mathematics and both complex variable and vector calculus provide a framework for solving physical and geometrical problems. The topics covered include: complex functions, analytic functions, Cauchy&�s theorems, complex power series, singularities, the residue theorem, contour integration, differentiation of vectors, differential operators, integration of vectors, the divergence theorem and Stokes&� theorem.
	MAT-20005	Dynamics	O	M	7.5	15
		The module is an introduction to dynamics with applications mainly to systems which can be modelled by particle dynamics. The topics investigated include: Newton&�s laws, momentum, kinetic and potential energy, projectiles, simple harmonic motion, springs, the pendulum, rocket motion, planetary and satellite orbits, linear theory of oscillations, and normal modes. 2 hours lectures & 1 hour examples class
	MAT-20005	Dynamics	EP	M	7.5	15
		The module is an introduction to dynamics with applications mainly to systems which can be modelled by particle dynamics. The topics investigated include: Newton&�s laws, momentum, kinetic and potential energy, projectiles, simple harmonic motion, springs, the pendulum, rocket motion, planetary and satellite orbits, linear theory of oscillations, and normal modes. 2 hours lectures & 1 hour examples class
	MAT-20006	Stochastic Processes	O	M	7.5	15
		This module examines the mathematics of random processes with particular reference to biological and social sciences. The course will include: revision of probability and distribution theory, difference equations, Markov chains, branching and Poisson processes, birth and death processes, queues, reliability and lifetime distributions. 2 hours lectures & 1 hour examples class
	MAT-20006	Stochastic Processes	EP	M	7.5	15
		This module examines the mathematics of random processes with particular reference to biological and social sciences. The course will include: revision of probability and distribution theory, difference equations, Markov chains, branching and Poisson processes, birth and death processes, queues, reliability and lifetime distributions. 2 hours lectures & 1 hour examples class
	MAT-20016	Mathematical Modelling	O	C	7.5	15
		The aim of the module is to demonstrate how real-world problems can be modelled mathematically. The mathematical modelling process will be introduced through a six-step problem-solving approach: identifying a suitable problem from the particular real-world scenario, making assumptions to simplify the problem, classifying variables influencing the problem, constructing a mathematical model to determine interrelationships among the variables, solving and interpreting the model and finally validating the model with real-world data. Mathematical tools that will be used in the model construction and solution process include: Ordinary Differential Equations and their solution methods, such as 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.
	MAT-20016	Mathematical Modelling	EP	C	7.5	15
		The aim of the module is to demonstrate how real-world problems can be modelled mathematically. The mathematical modelling process will be introduced through a six-step problem-solving approach: identifying a suitable problem from the particular real-world scenario, making assumptions to simplify the problem, classifying variables influencing the problem, constructing a mathematical model to determine interrelationships among the variables, solving and interpreting the model and finally validating the model with real-world data. Mathematical tools that will be used in the model construction and solution process include: Ordinary Differential Equations and their solution methods, such as 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.
	MAT-20017	Mathematics - Study Abroad Vi	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20018	Mathematics - Study Abroad VII	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20021	Real Analysis	O	M	7.5	15
		This module is intended as an introduction to Mathematical Analysis, which can be described as the study of the infinitely small and infinitely large. The module studies the real number system, sequences, series, limits of functions, continuity and Taylor series. Such a study is essential for understanding subsequent modules. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal forms.
	MAT-20021	Real Analysis	EP	M	7.5	15
		This module is intended as an introduction to Mathematical Analysis, which can be described as the study of the infinitely small and infinitely large. The module studies the real number system, sequences, series, limits of functions, continuity and Taylor series. Such a study is essential for understanding subsequent modules. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal forms.
	MAT-20027	Linear Statistical Models	O	M	7.5	15
		This module presents an introduction to statistical inference and illustrates the theory with practical applications to real life data sets. The first half of the module will encorporate hypothesis testing, confidence interval and parameter estimation. The second half of the module considers regression and anova methods, and demonstrates how statistical models can be used to draw conclusions from observations and experimental data collected in the physical and social sciences. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20027	Linear Statistical Models	EP	M	7.5	15
		This module presents an introduction to statistical inference and illustrates the theory with practical applications to real life data sets. The first half of the module will encorporate hypothesis testing, confidence interval and parameter estimation. The second half of the module considers regression and anova methods, and demonstrates how statistical models can be used to draw conclusions from observations and experimental data collected in the physical and social sciences. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.

Mathematics Dual Honours - Level 3 Modules

Semester 1			C/O	TYP	ECTS	CATS
	MAT-30002	Non-linear Differential Equations	O	M	7.5	15
		Systems that evolve in time can often be modelled by differential equations. There are countless examples of such systems from the physical world including the weather, climate change, stock markets, the economy, population dynamics, mechanical systems, etc. The great variety of behaviours exhibited by these systems is reflected in the solutions to the corresponding differential equations. This module introduces a number of methods for identifying and classifying various types of behaviour in various types of differential equation. While linear differential equations model some processes, the majority are described by nonlinear equations, and it is these that display the greatest diversity of behaviour. However, very few nonlinear differential equations have exact solutions. Nevertheless, a great deal of insight can be obtained from qualitative methods. This module focuses on geometric methods for constructing phase plane representations of dynamics and perturbation methods for obtaining approximate solutions. With these tools it is then possible to examine the changes in behaviour that can occur when a parameter is varied, and bifurcation theory is introduced to describe this. The relation between the evolution of differential equations and the evolution of maps is explained, and more exotic behaviour, like period doubling and chaos, are then studied in terms of the dynamics of maps.
	MAT-30002	Non-linear Differential Equations	EP	M	7.5	15
		Systems that evolve in time can often be modelled by differential equations. There are countless examples of such systems from the physical world including the weather, climate change, stock markets, the economy, population dynamics, mechanical systems, etc. The great variety of behaviours exhibited by these systems is reflected in the solutions to the corresponding differential equations. This module introduces a number of methods for identifying and classifying various types of behaviour in various types of differential equation. While linear differential equations model some processes, the majority are described by nonlinear equations, and it is these that display the greatest diversity of behaviour. However, very few nonlinear differential equations have exact solutions. Nevertheless, a great deal of insight can be obtained from qualitative methods. This module focuses on geometric methods for constructing phase plane representations of dynamics and perturbation methods for obtaining approximate solutions. With these tools it is then possible to examine the changes in behaviour that can occur when a parameter is varied, and bifurcation theory is introduced to describe this. The relation between the evolution of differential equations and the evolution of maps is explained, and more exotic behaviour, like period doubling and chaos, are then studied in terms of the dynamics of maps.
	MAT-30003	Partial Differential Equations	O	M	7.5	15
		This module provides an introduction to analytical methods for solving partial differential equations (PDEs). Throughout the module focuses on PDEs in two independent variables, although generalisation to three, or more, independent variables is briefly discussed. The module begins by introducing the method of characteristics for solving first order linear and quasi-linear PDEs. This is followed by the classification of linear second order PDEs into hyperbolic, parabolic or elliptic type, with a detailed treatment of the reduction to canonical form in each case. Thereafter, the module introduces the separation of variables technique and transform methods to solve the wave, diffusion, Laplace and Poisson equations. These PDEs are frequently encountered in many branches of applied mathematics, including fluid dynamics, mathematical biology, financial mathematics and electromagnetism. Finally the module introduces the technique of eigenfunction expansions for solving inhomogeneous PDEs, and the powerful technique of Green&�s functions.
	MAT-30003	Partial Differential Equations	EP	M	7.5	15
		This module provides an introduction to analytical methods for solving partial differential equations (PDEs). Throughout the module focuses on PDEs in two independent variables, although generalisation to three, or more, independent variables is briefly discussed. The module begins by introducing the method of characteristics for solving first order linear and quasi-linear PDEs. This is followed by the classification of linear second order PDEs into hyperbolic, parabolic or elliptic type, with a detailed treatment of the reduction to canonical form in each case. Thereafter, the module introduces the separation of variables technique and transform methods to solve the wave, diffusion, Laplace and Poisson equations. These PDEs are frequently encountered in many branches of applied mathematics, including fluid dynamics, mathematical biology, financial mathematics and electromagnetism. Finally the module introduces the technique of eigenfunction expansions for solving inhomogeneous PDEs, and the powerful technique of Green&�s functions.
	MAT-30005	Relativity	O	M	7.5	15
		Relativity is a theory of space and time. It consists of two fundamentally different areas, that is, Special Relativity and General Relativity. Special Relativity was first proposed by Einstein in 1905 and is concerned with the connection between observations made in frames of reference that move with constant velocity relative to each other. It has many practical applications in situations which involve extremely high speeds. General Relativity was first proposed by Einstein in 1916 and incorporates non-uniform motion. Its main applications lie in gravitation theory, including astrophysics and cosmology. This module gives an introduction to both Special and General Relativity from a mainly mathematical point of view. The standard applications to physical problems are given, but experimental details are kept to a minimum.
	MAT-30005	Relativity	EP	M	7.5	15
		Relativity is a theory of space and time. It consists of two fundamentally different areas, that is, Special Relativity and General Relativity. Special Relativity was first proposed by Einstein in 1905 and is concerned with the connection between observations made in frames of reference that move with constant velocity relative to each other. It has many practical applications in situations which involve extremely high speeds. General Relativity was first proposed by Einstein in 1916 and incorporates non-uniform motion. Its main applications lie in gravitation theory, including astrophysics and cosmology. This module gives an introduction to both Special and General Relativity from a mainly mathematical point of view. The standard applications to physical problems are given, but experimental details are kept to a minimum.
~	MAT-30006	Logic	O	M	7.5	15
		The purpose of the module is to introduce mathematical concepts for examining philosophical questions about the nature of mathematics as a whole. It attempts to present a sophisticated perspective on mathematics in a way that is accessible to undergraduates. The first half of the module concerns the subject-matter of mathematics. The thesis is developed that all mathematical objects can be understood as sets; set theory is developed in an informal axiomatic spirit, based on G�del&�s notion of transfinite iteration of the $�set of&� operation. The second half examines mathematical reasoning, which is formalised as predicate calculus and studied metamathematically. The basic apparatus of formal semantics is introduced, and issues such as completeness and categoricity are surveyed informally.
~	MAT-30006	Logic	EP	M	7.5	15
		The purpose of the module is to introduce mathematical concepts for examining philosophical questions about the nature of mathematics as a whole. It attempts to present a sophisticated perspective on mathematics in a way that is accessible to undergraduates. The first half of the module concerns the subject-matter of mathematics. The thesis is developed that all mathematical objects can be understood as sets; set theory is developed in an informal axiomatic spirit, based on G�del&�s notion of transfinite iteration of the $�set of&� operation. The second half examines mathematical reasoning, which is formalised as predicate calculus and studied metamathematically. The basic apparatus of formal semantics is introduced, and issues such as completeness and categoricity are surveyed informally.
~	MAT-30013	Group Theory	O	M	7.5	15
		This module builds on the Group Theory introduced in MAT - 20009 to develop some of the mathematics underlying the classification of finite groups. This culminates in a proof of Sylow&�s First Theorem which offers a partial converse to Lagrange&�s Theorem proved in MAT - 20009. The module also develops some applications of Group Theory, the natural setting for which is that of group actions. Several examples of applying group theoretic ideas to counting combinatorial configurations are presented.
~	MAT-30013	Group Theory	EP	M	7.5	15
		This module builds on the Group Theory introduced in MAT - 20009 to develop some of the mathematics underlying the classification of finite groups. This culminates in a proof of Sylow&�s First Theorem which offers a partial converse to Lagrange&�s Theorem proved in MAT - 20009. The module also develops some applications of Group Theory, the natural setting for which is that of group actions. Several examples of applying group theoretic ideas to counting combinatorial configurations are presented.
#	MAT-30016	Project II - ISP	O	C	7.5	15
		This module is an opportunity for students to explore in depth a particular topic or problem in mathematics or statistics chosen from a list prepared by the division. Each student is supervised by a member of staff throughout the execution of the project. This execution is carried out between September and May. There are many different types of project: some comprise primarily of a review of the literature, the student assembling from several sources their own account of a body of mathematics not covered elsewhere in the programme; others involve tackling an applied problem, and these normally require some computational work, using Mathematica for example; and in others specialised statistical techniques and packages are used to analyse a large data set. The work is assessed by means of a dissertation and an oral presentation to the division.
#	MAT-30016	Project II - ISP	EP	C	7.5	15
		This module is an opportunity for students to explore in depth a particular topic or problem in mathematics or statistics chosen from a list prepared by the division. Each student is supervised by a member of staff throughout the execution of the project. This execution is carried out between September and May. There are many different types of project: some comprise primarily of a review of the literature, the student assembling from several sources their own account of a body of mathematics not covered elsewhere in the programme; others involve tackling an applied problem, and these normally require some computational work, using Mathematica for example; and in others specialised statistical techniques and packages are used to analyse a large data set. The work is assessed by means of a dissertation and an oral presentation to the division.
	MAT-30021	Numerical Analysis	O	C	7.5	15
		This module is concerned with the analysis of numerical methods, concentrating on how to select a suitable method and analyse the results produced, rather than on programming the methods themselves. Substantial use is made of technology in terms of existing teaching packages, computer algebra and hand held technologies, all of which provide transferable skills. There are regular computer laboratory sessions at which students will be expected to use any and all of the methods introduced in lectures. The module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought; 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand; 6. The ability to communicate clearly and effectively in written form;
	MAT-30021	Numerical Analysis	EP	C	7.5	15
		This module is concerned with the analysis of numerical methods, concentrating on how to select a suitable method and analyse the results produced, rather than on programming the methods themselves. Substantial use is made of technology in terms of existing teaching packages, computer algebra and hand held technologies, all of which provide transferable skills. There are regular computer laboratory sessions at which students will be expected to use any and all of the methods introduced in lectures. The module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought; 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand; 6. The ability to communicate clearly and effectively in written form;
~	MAT-30022	Number Theory	O	M	7.5	15
		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. 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. This module will build upon the concepts and techniques introduced in the second year module "Analysis and Abstract Algebra", and consider how these can be extended and applied to both ancient and modern problems, from finding whole number solutions to polynomial equations to primality testing and cryptography.
~	MAT-30022	Number Theory	EP	M	7.5	15
		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. 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. This module will build upon the concepts and techniques introduced in the second year module "Analysis and Abstract Algebra", and consider how these can be extended and applied to both ancient and modern problems, from finding whole number solutions to polynomial equations to primality testing and cryptography.
~	MAT-30026	Probability Models	O	M	7.5	15
		Probability models are widely used in natural, social and financial sciences for the purposes of understanding, predicting and controlling random phenomena. The module builds on the earlier study of probability and stochastic processes. It aims to: (a) provide students with knowledge of some more advanced concepts in probability theory, in particular: conditional expectation, convergence of random variables, martingale sequences, and brownian motion; (b) apply probabilistic techniques to a wide range of stochastic models and processes. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
~	MAT-30026	Probability Models	EP	M	7.5	15
		Probability models are widely used in natural, social and financial sciences for the purposes of understanding, predicting and controlling random phenomena. The module builds on the earlier study of probability and stochastic processes. It aims to: (a) provide students with knowledge of some more advanced concepts in probability theory, in particular: conditional expectation, convergence of random variables, martingale sequences, and brownian motion; (b) apply probabilistic techniques to a wide range of stochastic models and processes. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-30029	Professional Mathematics	O	C	7.5	15
		This module gives the opportunity to further develop skills of mathematical problem solving and application of mathematical results to real world scenarios through group work activities. This aims to develop skills that are needed when undertaking employment or research, such as working in depth on a problem over an extended period, writing reports, communicating mathematical results to different audiences and working in collaboration with others. This module will provide employability skills experiences and develop students� ability to articulate their skills, which will be useful to draw on when applying for jobs. The skills developed will form parts of a student�s Distinctive Keele Curriculum experience. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity, independence of thought; 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6. The ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 7. The knowledge, skills, self-confidence and self-awareness actively to pursue your future goals; 9. A professional and reflective approach, including qualities of leadership, responsibility, personal integrity, empathy, care and respect for others, accountability and self-regulation.

Semester 2			C/O	TYP	ECTS	CATS
	MAT-30001	Graph Theory	O	M	7.5	15
		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 via worked examples and coursework.
	MAT-30001	Graph Theory	EP	M	7.5	15
		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 via worked examples and coursework.
	MAT-30004	Fluid Mechanics	O	M	7.5	15
		The module is a first course in fluid mechanics, designed for students with no previous knowledge of the subject. Although previous acquaintance with vectors and differential equations would be helpful, the lecturer will offer revision of the required parts of these subjects as the module progresses.
	MAT-30004	Fluid Mechanics	EP	M	7.5	15
		The module is a first course in fluid mechanics, designed for students with no previous knowledge of the subject. Although previous acquaintance with vectors and differential equations would be helpful, the lecturer will offer revision of the required parts of these subjects as the module progresses.
~	MAT-30010	Complex Variable II	O	M	7.5	15
		This module revises and consolidates the material of MAT - 20004: Complex Variable I, before studying further applications of contour integration and the Residue Theorem. It then provides an introduction to conformal mappings, together with some of their applications, in particular their application in determining harmonic functions in two-dimensional regions. The module also covers certain analytical aspects of complex functions.
~	MAT-30010	Complex Variable II	EP	M	7.5	15
		This module revises and consolidates the material of MAT - 20004: Complex Variable I, before studying further applications of contour integration and the Residue Theorem. It then provides an introduction to conformal mappings, together with some of their applications, in particular their application in determining harmonic functions in two-dimensional regions. The module also covers certain analytical aspects of complex functions.
	MAT-30011	Waves	O	M	7.5	15
		The module aims to give an account of the underlying mathematical theory that describes the behaviour of waves. The mathematical development of the subject is combined with a discussion of applications, for example musical instruments. The module material is illuminated by small demonstrations and by computer-generated animations of wave processes. The topics include: the wave equation, waves on stretched strings, waves on membranes, waves on beams, sound waves, and waves in liquids with a free surface.
	MAT-30011	Waves	EP	M	7.5	15
		The module aims to give an account of the underlying mathematical theory that describes the behaviour of waves. The mathematical development of the subject is combined with a discussion of applications, for example musical instruments. The module material is illuminated by small demonstrations and by computer-generated animations of wave processes. The topics include: the wave equation, waves on stretched strings, waves on membranes, waves on beams, sound waves, and waves in liquids with a free surface.
	MAT-30014	Medical Statistics	O	M	7.5	15
		This module illustrates the application of statistical techniques to health related research. Methods are applied using data from real-life studies, with 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.
	MAT-30014	Medical Statistics	EP	M	7.5	15
		This module illustrates the application of statistical techniques to health related research. Methods are applied using data from real-life studies, with 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.
#	MAT-30016	Project II - ISP	O	C	7.5	15
		This module is an opportunity for students to explore in depth a particular topic or problem in mathematics or statistics chosen from a list prepared by the division. Each student is supervised by a member of staff throughout the execution of the project. This execution is carried out between September and May. There are many different types of project: some comprise primarily of a review of the literature, the student assembling from several sources their own account of a body of mathematics not covered elsewhere in the programme; others involve tackling an applied problem, and these normally require some computational work, using Mathematica for example; and in others specialised statistical techniques and packages are used to analyse a large data set. The work is assessed by means of a dissertation and an oral presentation to the division.
#	MAT-30016	Project II - ISP	EP	C	7.5	15
		This module is an opportunity for students to explore in depth a particular topic or problem in mathematics or statistics chosen from a list prepared by the division. Each student is supervised by a member of staff throughout the execution of the project. This execution is carried out between September and May. There are many different types of project: some comprise primarily of a review of the literature, the student assembling from several sources their own account of a body of mathematics not covered elsewhere in the programme; others involve tackling an applied problem, and these normally require some computational work, using Mathematica for example; and in others specialised statistical techniques and packages are used to analyse a large data set. The work is assessed by means of a dissertation and an oral presentation to the division.
	MAT-30023	Mathematical Biology	O	M	7.5	15
		This module will show you how mathematics is an interdisciplinary subject, with particular attention to biology. Applications of mathematics to biological situations is one of the fastest growing areas where mathematics can explain and predict behaviour. These predictions are not just theoretical: every day people's lives are saved due to the predictions possible. We shall investigate a diverse set of applications. Game theory has provided new mathematical tools to study the evolution of animal behaviour. The biology of population growth and disease transmission, in particular, recent advances in our mathematical understanding of biology has provided new insight into the spread of MRSA. In the last few years there have been advances in the application of mathematics to the study of animal gaits (the different method of locomotion).
	MAT-30023	Mathematical Biology	EP	M	7.5	15
		This module will show you how mathematics is an interdisciplinary subject, with particular attention to biology. Applications of mathematics to biological situations is one of the fastest growing areas where mathematics can explain and predict behaviour. These predictions are not just theoretical: every day people's lives are saved due to the predictions possible. We shall investigate a diverse set of applications. Game theory has provided new mathematical tools to study the evolution of animal behaviour. The biology of population growth and disease transmission, in particular, recent advances in our mathematical understanding of biology has provided new insight into the spread of MRSA. In the last few years there have been advances in the application of mathematics to the study of animal gaits (the different method of locomotion).
~	MAT-30024	Ring and Field Theory	EP	M	7.5	15
		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 Pure Mathematics where field theory and group theory come together to answer some of the oldest questions in mathematics 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 also be given. This will include the application to classical proofs concerning geometrical constructions using ruler and compasses.
~	MAT-30024	Ring and Field Theory	O	M	7.5	15
		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 Pure Mathematics where field theory and group theory come together to answer some of the oldest questions in mathematics 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 also be given. This will include the application to classical proofs concerning geometrical constructions using ruler and compasses.
~	MAT-30027	Codes and Cryptography	O	M	7.5	15
		This module will illustrate how abstract ideas from group theory, number theory and linear 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. This module will develop the following graduate attributes: 1) An open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds. 2) An appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge. 4) The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6) The ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences 9) A professional and reflective approach, including qualities of leadership, responsibility, personal integrity, empathy, care and respect for others, accountability and self-regulation
~	MAT-30027	Codes and Cryptography	EP	M	7.5	15
		This module will illustrate how abstract ideas from group theory, number theory and linear 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. This module will develop the following graduate attributes: 1) An open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds. 2) An appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge. 4) The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6) The ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences 9) A professional and reflective approach, including qualities of leadership, responsibility, personal integrity, empathy, care and respect for others, accountability and self-regulation
	MAT-30028	Perturbation Methods	O	M	7.5	15
		This module introduces students to the asymptotic techniques which are indispensible 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.
	MAT-30028	Perturbation Methods	EP	M	7.5	15
		This module introduces students to the asymptotic techniques which are indispensible 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.

Mathematics Major - Level 1 Modules

Semester 1			C/O	TYP	ECTS	CATS
	MAT-10038	Algebra I	C	M	7.5	15
		This module is intended to help students with the transition from the methods based approach of A level mathematics to the higher levels of understanding and rigour expected at degree level. It begins by discussing mathematical statements and the meaning and basic strategies of proof. This is followed by a short exposition of naive set theory and by a careful treatment of the notion of a function. The remainder of the module covers the algebraic development of number systems and their properties. The module ends with a brief discussion of the properties of polynomials, including the Fundamental Theorem of Algebra. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.
	MAT-10039	Calculus I	C	M	7.5	15
		This module is intended to help students with the transition from the methods based approach of A level Mathematics to the higher levels of understanding and rigour expected at degree level. The module will develop students' skills in mathematical techniques, mainly in differentiation and integration. It begins by revising the standard elementary functions and their properties and continues with the revision of basic results and techniques in differentiation. It then revises basic results and techniques in integration. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.

Semester 2			C/O	TYP	ECTS	CATS
	MAT-10040	Algebra II	C	M	7.5	15
		This module continues from Algebra I and provides an introduction to matrix and vector algebra. Topics will include matrix algebra, elementary row operations, solving linear equations, determinants, eigenvalues and eigenvectors, diagonalisation of matrices, three dimensional vector algebra and geometry, lines and planes, general vector algebra, linear independence and bases. Some of this material will be applied in a treatment of linear programming, a widely-used optimization technique. In particular the Simplex method and an introduction to duality theory are covered. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.
	MAT-10041	Calculus II	C	M	7.5	15
		Many physical problems are governed by ordinary or partial differential equations, the solution of which can help us understand their properties and characteristics. For instance, the oscillation frequency of a pendulum, the transfer time for sending a spaceship from the Earth to Mars, and the population evolution of a fish species in a lake can all be determined by solving ordinary differential equations. This module, which is a prerequisite for a number of other modules in the second and third years, will introduce some of the basic techniques for solving ordinary differential equations, familiarize students with partial differentiations, and explain how double integrals can be evaluated and used to compute areas and volumes. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.

Mathematics Major - Level 2 Modules

Semester 1			C/O	TYP	ECTS	CATS
	MAT-20008	Differential Equations	C	M	7.5	15
		This module focuses on methods for solving ordinary differential equations. The topics include: solutions to first-order equations, higher order linear equations, power series methods, graphical aspects of differential equations, Fourier series and Laplace transforms.
	MAT-20011	Mathematics - Study Abroad I	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20012	Mathematics - Study Abroad II	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20015	Numerical Methods	O	M	7.5	15
		This module studies numerical methods as applied in linear algebra and to differential equations. Topics include: the solution of algebraic equations, numerical quadrature, the numerical solution of differential equations including the Euler and Runge-Kutta methods, the solution of linear equations by direct and iterative methods, and numerical methods for determining eigenvalues and eigenvectors.
	MAT-20015	Numerical Methods	EP	M	7.5	15
		This module studies numerical methods as applied in linear algebra and to differential equations. Topics include: the solution of algebraic equations, numerical quadrature, the numerical solution of differential equations including the Euler and Runge-Kutta methods, the solution of linear equations by direct and iterative methods, and numerical methods for determining eigenvalues and eigenvectors.
	MAT-20023	Probability	O	M	7.5	15
		Probability is the mathematics of uncertainty and randomness. The module begins with classical notions of probability associated with the analysis of games of chance using cards, dice, etc. It then moves to the axiomatic treatment of the probability of random events. This leads to the definitions of statistical independence and conditional probability. The remainder of the module is concerned with a systematic study of discrete and continuous, univariate and bivariate, random variables, covering expectation, variance, covariance and the moment generating function. The theory is applied to a wide range of theoretical and applied problems. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20023	Probability	EP	M	7.5	15
		Probability is the mathematics of uncertainty and randomness. The module begins with classical notions of probability associated with the analysis of games of chance using cards, dice, etc. It then moves to the axiomatic treatment of the probability of random events. This leads to the definitions of statistical independence and conditional probability. The remainder of the module is concerned with a systematic study of discrete and continuous, univariate and bivariate, random variables, covering expectation, variance, covariance and the moment generating function. The theory is applied to a wide range of theoretical and applied problems. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20024	Operational Research	O	M	7.5	15
		This module presents a variety of topics, techniques and models in operational research. 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. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20024	Operational Research	EP	M	7.5	15
		This module presents a variety of topics, techniques and models in operational research. 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. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20025	Abstract Algebra	O	M	7.5	15
		This module introduces and studies the abstract algebraic structure known as a group. Starting 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, Lagrange'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. The module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 2. An appreciation of the development and value of Mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.
	MAT-20025	Abstract Algebra	EP	M	7.5	15
		This module introduces and studies the abstract algebraic structure known as a group. Starting 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, Lagrange'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. The module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 2. An appreciation of the development and value of Mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.
	MAT-20026	Linear Algebra	O	M	7.5	15
		This module introduces the concept of an abstract vector space The module builds on the knowledge of vectors and matrices gained from MAT-10040. 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 just different specialisations of the same general concept. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity, and independence of thought. 2. An appreciation of the development and value of mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.
	MAT-20026	Linear Algebra	EP	M	7.5	15
		This module introduces the concept of an abstract vector space The module builds on the knowledge of vectors and matrices gained from MAT-10040. 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 just different specialisations of the same general concept. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity, and independence of thought. 2. An appreciation of the development and value of mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.

Semester 2			C/O	TYP	ECTS	CATS
	MAT-20004	Complex Variable I and Vector Calculus	C	M	7.5	15
		This module contains a first course on vector calculus and a first course in functions of a complex variable. Complex variable leads to elegant results in pure mathematics and both complex variable and vector calculus provide a framework for solving physical and geometrical problems. The topics covered include: complex functions, analytic functions, Cauchy&�s theorems, complex power series, singularities, the residue theorem, contour integration, differentiation of vectors, differential operators, integration of vectors, the divergence theorem and Stokes&� theorem.
	MAT-20005	Dynamics	O	M	7.5	15
		The module is an introduction to dynamics with applications mainly to systems which can be modelled by particle dynamics. The topics investigated include: Newton&�s laws, momentum, kinetic and potential energy, projectiles, simple harmonic motion, springs, the pendulum, rocket motion, planetary and satellite orbits, linear theory of oscillations, and normal modes. 2 hours lectures & 1 hour examples class
	MAT-20005	Dynamics	EP	M	7.5	15
		The module is an introduction to dynamics with applications mainly to systems which can be modelled by particle dynamics. The topics investigated include: Newton&�s laws, momentum, kinetic and potential energy, projectiles, simple harmonic motion, springs, the pendulum, rocket motion, planetary and satellite orbits, linear theory of oscillations, and normal modes. 2 hours lectures & 1 hour examples class
	MAT-20006	Stochastic Processes	O	M	7.5	15
		This module examines the mathematics of random processes with particular reference to biological and social sciences. The course will include: revision of probability and distribution theory, difference equations, Markov chains, branching and Poisson processes, birth and death processes, queues, reliability and lifetime distributions. 2 hours lectures & 1 hour examples class
	MAT-20006	Stochastic Processes	EP	M	7.5	15
		This module examines the mathematics of random processes with particular reference to biological and social sciences. The course will include: revision of probability and distribution theory, difference equations, Markov chains, branching and Poisson processes, birth and death processes, queues, reliability and lifetime distributions. 2 hours lectures & 1 hour examples class
	MAT-20016	Mathematical Modelling	O	C	7.5	15
		The aim of the module is to demonstrate how real-world problems can be modelled mathematically. The mathematical modelling process will be introduced through a six-step problem-solving approach: identifying a suitable problem from the particular real-world scenario, making assumptions to simplify the problem, classifying variables influencing the problem, constructing a mathematical model to determine interrelationships among the variables, solving and interpreting the model and finally validating the model with real-world data. Mathematical tools that will be used in the model construction and solution process include: Ordinary Differential Equations and their solution methods, such as 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.
	MAT-20016	Mathematical Modelling	EP	C	7.5	15
		The aim of the module is to demonstrate how real-world problems can be modelled mathematically. The mathematical modelling process will be introduced through a six-step problem-solving approach: identifying a suitable problem from the particular real-world scenario, making assumptions to simplify the problem, classifying variables influencing the problem, constructing a mathematical model to determine interrelationships among the variables, solving and interpreting the model and finally validating the model with real-world data. Mathematical tools that will be used in the model construction and solution process include: Ordinary Differential Equations and their solution methods, such as 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.
	MAT-20017	Mathematics - Study Abroad Vi	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20018	Mathematics - Study Abroad VII	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20021	Real Analysis	O	M	7.5	15
		This module is intended as an introduction to Mathematical Analysis, which can be described as the study of the infinitely small and infinitely large. The module studies the real number system, sequences, series, limits of functions, continuity and Taylor series. Such a study is essential for understanding subsequent modules. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal forms.
	MAT-20021	Real Analysis	EP	M	7.5	15
		This module is intended as an introduction to Mathematical Analysis, which can be described as the study of the infinitely small and infinitely large. The module studies the real number system, sequences, series, limits of functions, continuity and Taylor series. Such a study is essential for understanding subsequent modules. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal forms.
	MAT-20027	Linear Statistical Models	O	M	7.5	15
		This module presents an introduction to statistical inference and illustrates the theory with practical applications to real life data sets. The first half of the module will encorporate hypothesis testing, confidence interval and parameter estimation. The second half of the module considers regression and anova methods, and demonstrates how statistical models can be used to draw conclusions from observations and experimental data collected in the physical and social sciences. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20027	Linear Statistical Models	EP	M	7.5	15
		This module presents an introduction to statistical inference and illustrates the theory with practical applications to real life data sets. The first half of the module will encorporate hypothesis testing, confidence interval and parameter estimation. The second half of the module considers regression and anova methods, and demonstrates how statistical models can be used to draw conclusions from observations and experimental data collected in the physical and social sciences. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.

Mathematics Major - Level 3 Modules

Semester 1			C/O	TYP	ECTS	CATS
	MAT-30002	Non-linear Differential Equations	O	M	7.5	15
		Systems that evolve in time can often be modelled by differential equations. There are countless examples of such systems from the physical world including the weather, climate change, stock markets, the economy, population dynamics, mechanical systems, etc. The great variety of behaviours exhibited by these systems is reflected in the solutions to the corresponding differential equations. This module introduces a number of methods for identifying and classifying various types of behaviour in various types of differential equation. While linear differential equations model some processes, the majority are described by nonlinear equations, and it is these that display the greatest diversity of behaviour. However, very few nonlinear differential equations have exact solutions. Nevertheless, a great deal of insight can be obtained from qualitative methods. This module focuses on geometric methods for constructing phase plane representations of dynamics and perturbation methods for obtaining approximate solutions. With these tools it is then possible to examine the changes in behaviour that can occur when a parameter is varied, and bifurcation theory is introduced to describe this. The relation between the evolution of differential equations and the evolution of maps is explained, and more exotic behaviour, like period doubling and chaos, are then studied in terms of the dynamics of maps.
	MAT-30002	Non-linear Differential Equations	EP	M	7.5	15
		Systems that evolve in time can often be modelled by differential equations. There are countless examples of such systems from the physical world including the weather, climate change, stock markets, the economy, population dynamics, mechanical systems, etc. The great variety of behaviours exhibited by these systems is reflected in the solutions to the corresponding differential equations. This module introduces a number of methods for identifying and classifying various types of behaviour in various types of differential equation. While linear differential equations model some processes, the majority are described by nonlinear equations, and it is these that display the greatest diversity of behaviour. However, very few nonlinear differential equations have exact solutions. Nevertheless, a great deal of insight can be obtained from qualitative methods. This module focuses on geometric methods for constructing phase plane representations of dynamics and perturbation methods for obtaining approximate solutions. With these tools it is then possible to examine the changes in behaviour that can occur when a parameter is varied, and bifurcation theory is introduced to describe this. The relation between the evolution of differential equations and the evolution of maps is explained, and more exotic behaviour, like period doubling and chaos, are then studied in terms of the dynamics of maps.
	MAT-30003	Partial Differential Equations	O	M	7.5	15
		This module provides an introduction to analytical methods for solving partial differential equations (PDEs). Throughout the module focuses on PDEs in two independent variables, although generalisation to three, or more, independent variables is briefly discussed. The module begins by introducing the method of characteristics for solving first order linear and quasi-linear PDEs. This is followed by the classification of linear second order PDEs into hyperbolic, parabolic or elliptic type, with a detailed treatment of the reduction to canonical form in each case. Thereafter, the module introduces the separation of variables technique and transform methods to solve the wave, diffusion, Laplace and Poisson equations. These PDEs are frequently encountered in many branches of applied mathematics, including fluid dynamics, mathematical biology, financial mathematics and electromagnetism. Finally the module introduces the technique of eigenfunction expansions for solving inhomogeneous PDEs, and the powerful technique of Green&�s functions.
	MAT-30003	Partial Differential Equations	EP	M	7.5	15
		This module provides an introduction to analytical methods for solving partial differential equations (PDEs). Throughout the module focuses on PDEs in two independent variables, although generalisation to three, or more, independent variables is briefly discussed. The module begins by introducing the method of characteristics for solving first order linear and quasi-linear PDEs. This is followed by the classification of linear second order PDEs into hyperbolic, parabolic or elliptic type, with a detailed treatment of the reduction to canonical form in each case. Thereafter, the module introduces the separation of variables technique and transform methods to solve the wave, diffusion, Laplace and Poisson equations. These PDEs are frequently encountered in many branches of applied mathematics, including fluid dynamics, mathematical biology, financial mathematics and electromagnetism. Finally the module introduces the technique of eigenfunction expansions for solving inhomogeneous PDEs, and the powerful technique of Green&�s functions.
	MAT-30005	Relativity	O	M	7.5	15
		Relativity is a theory of space and time. It consists of two fundamentally different areas, that is, Special Relativity and General Relativity. Special Relativity was first proposed by Einstein in 1905 and is concerned with the connection between observations made in frames of reference that move with constant velocity relative to each other. It has many practical applications in situations which involve extremely high speeds. General Relativity was first proposed by Einstein in 1916 and incorporates non-uniform motion. Its main applications lie in gravitation theory, including astrophysics and cosmology. This module gives an introduction to both Special and General Relativity from a mainly mathematical point of view. The standard applications to physical problems are given, but experimental details are kept to a minimum.
	MAT-30005	Relativity	EP	M	7.5	15
		Relativity is a theory of space and time. It consists of two fundamentally different areas, that is, Special Relativity and General Relativity. Special Relativity was first proposed by Einstein in 1905 and is concerned with the connection between observations made in frames of reference that move with constant velocity relative to each other. It has many practical applications in situations which involve extremely high speeds. General Relativity was first proposed by Einstein in 1916 and incorporates non-uniform motion. Its main applications lie in gravitation theory, including astrophysics and cosmology. This module gives an introduction to both Special and General Relativity from a mainly mathematical point of view. The standard applications to physical problems are given, but experimental details are kept to a minimum.
~	MAT-30006	Logic	O	M	7.5	15
		The purpose of the module is to introduce mathematical concepts for examining philosophical questions about the nature of mathematics as a whole. It attempts to present a sophisticated perspective on mathematics in a way that is accessible to undergraduates. The first half of the module concerns the subject-matter of mathematics. The thesis is developed that all mathematical objects can be understood as sets; set theory is developed in an informal axiomatic spirit, based on G�del&�s notion of transfinite iteration of the $�set of&� operation. The second half examines mathematical reasoning, which is formalised as predicate calculus and studied metamathematically. The basic apparatus of formal semantics is introduced, and issues such as completeness and categoricity are surveyed informally.
~	MAT-30006	Logic	EP	M	7.5	15
		The purpose of the module is to introduce mathematical concepts for examining philosophical questions about the nature of mathematics as a whole. It attempts to present a sophisticated perspective on mathematics in a way that is accessible to undergraduates. The first half of the module concerns the subject-matter of mathematics. The thesis is developed that all mathematical objects can be understood as sets; set theory is developed in an informal axiomatic spirit, based on G�del&�s notion of transfinite iteration of the $�set of&� operation. The second half examines mathematical reasoning, which is formalised as predicate calculus and studied metamathematically. The basic apparatus of formal semantics is introduced, and issues such as completeness and categoricity are surveyed informally.
~	MAT-30013	Group Theory	O	M	7.5	15
		This module builds on the Group Theory introduced in MAT - 20009 to develop some of the mathematics underlying the classification of finite groups. This culminates in a proof of Sylow&�s First Theorem which offers a partial converse to Lagrange&�s Theorem proved in MAT - 20009. The module also develops some applications of Group Theory, the natural setting for which is that of group actions. Several examples of applying group theoretic ideas to counting combinatorial configurations are presented.
~	MAT-30013	Group Theory	EP	M	7.5	15
		This module builds on the Group Theory introduced in MAT - 20009 to develop some of the mathematics underlying the classification of finite groups. This culminates in a proof of Sylow&�s First Theorem which offers a partial converse to Lagrange&�s Theorem proved in MAT - 20009. The module also develops some applications of Group Theory, the natural setting for which is that of group actions. Several examples of applying group theoretic ideas to counting combinatorial configurations are presented.
#	MAT-30016	Project II - ISP	O	C	7.5	15
		This module is an opportunity for students to explore in depth a particular topic or problem in mathematics or statistics chosen from a list prepared by the division. Each student is supervised by a member of staff throughout the execution of the project. This execution is carried out between September and May. There are many different types of project: some comprise primarily of a review of the literature, the student assembling from several sources their own account of a body of mathematics not covered elsewhere in the programme; others involve tackling an applied problem, and these normally require some computational work, using Mathematica for example; and in others specialised statistical techniques and packages are used to analyse a large data set. The work is assessed by means of a dissertation and an oral presentation to the division.
#	MAT-30016	Project II - ISP	EP	C	7.5	15
		This module is an opportunity for students to explore in depth a particular topic or problem in mathematics or statistics chosen from a list prepared by the division. Each student is supervised by a member of staff throughout the execution of the project. This execution is carried out between September and May. There are many different types of project: some comprise primarily of a review of the literature, the student assembling from several sources their own account of a body of mathematics not covered elsewhere in the programme; others involve tackling an applied problem, and these normally require some computational work, using Mathematica for example; and in others specialised statistical techniques and packages are used to analyse a large data set. The work is assessed by means of a dissertation and an oral presentation to the division.
	MAT-30021	Numerical Analysis	O	C	7.5	15
		This module is concerned with the analysis of numerical methods, concentrating on how to select a suitable method and analyse the results produced, rather than on programming the methods themselves. Substantial use is made of technology in terms of existing teaching packages, computer algebra and hand held technologies, all of which provide transferable skills. There are regular computer laboratory sessions at which students will be expected to use any and all of the methods introduced in lectures. The module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought; 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand; 6. The ability to communicate clearly and effectively in written form;
	MAT-30021	Numerical Analysis	EP	C	7.5	15
		This module is concerned with the analysis of numerical methods, concentrating on how to select a suitable method and analyse the results produced, rather than on programming the methods themselves. Substantial use is made of technology in terms of existing teaching packages, computer algebra and hand held technologies, all of which provide transferable skills. There are regular computer laboratory sessions at which students will be expected to use any and all of the methods introduced in lectures. The module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought; 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand; 6. The ability to communicate clearly and effectively in written form;
~	MAT-30022	Number Theory	O	M	7.5	15
		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. 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. This module will build upon the concepts and techniques introduced in the second year module "Analysis and Abstract Algebra", and consider how these can be extended and applied to both ancient and modern problems, from finding whole number solutions to polynomial equations to primality testing and cryptography.
~	MAT-30022	Number Theory	EP	M	7.5	15
		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. 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. This module will build upon the concepts and techniques introduced in the second year module "Analysis and Abstract Algebra", and consider how these can be extended and applied to both ancient and modern problems, from finding whole number solutions to polynomial equations to primality testing and cryptography.
~	MAT-30026	Probability Models	O	M	7.5	15
		Probability models are widely used in natural, social and financial sciences for the purposes of understanding, predicting and controlling random phenomena. The module builds on the earlier study of probability and stochastic processes. It aims to: (a) provide students with knowledge of some more advanced concepts in probability theory, in particular: conditional expectation, convergence of random variables, martingale sequences, and brownian motion; (b) apply probabilistic techniques to a wide range of stochastic models and processes. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
~	MAT-30026	Probability Models	EP	M	7.5	15
		Probability models are widely used in natural, social and financial sciences for the purposes of understanding, predicting and controlling random phenomena. The module builds on the earlier study of probability and stochastic processes. It aims to: (a) provide students with knowledge of some more advanced concepts in probability theory, in particular: conditional expectation, convergence of random variables, martingale sequences, and brownian motion; (b) apply probabilistic techniques to a wide range of stochastic models and processes. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-30029	Professional Mathematics	O	C	7.5	15
		This module gives the opportunity to further develop skills of mathematical problem solving and application of mathematical results to real world scenarios through group work activities. This aims to develop skills that are needed when undertaking employment or research, such as working in depth on a problem over an extended period, writing reports, communicating mathematical results to different audiences and working in collaboration with others. This module will provide employability skills experiences and develop students� ability to articulate their skills, which will be useful to draw on when applying for jobs. The skills developed will form parts of a student�s Distinctive Keele Curriculum experience. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity, independence of thought; 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6. The ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 7. The knowledge, skills, self-confidence and self-awareness actively to pursue your future goals; 9. A professional and reflective approach, including qualities of leadership, responsibility, personal integrity, empathy, care and respect for others, accountability and self-regulation.

Semester 2			C/O	TYP	ECTS	CATS
	MAT-30001	Graph Theory	O	M	7.5	15
		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 via worked examples and coursework.
	MAT-30001	Graph Theory	EP	M	7.5	15
		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 via worked examples and coursework.
	MAT-30004	Fluid Mechanics	O	M	7.5	15
		The module is a first course in fluid mechanics, designed for students with no previous knowledge of the subject. Although previous acquaintance with vectors and differential equations would be helpful, the lecturer will offer revision of the required parts of these subjects as the module progresses.
	MAT-30004	Fluid Mechanics	EP	M	7.5	15
		The module is a first course in fluid mechanics, designed for students with no previous knowledge of the subject. Although previous acquaintance with vectors and differential equations would be helpful, the lecturer will offer revision of the required parts of these subjects as the module progresses.
~	MAT-30010	Complex Variable II	O	M	7.5	15
		This module revises and consolidates the material of MAT - 20004: Complex Variable I, before studying further applications of contour integration and the Residue Theorem. It then provides an introduction to conformal mappings, together with some of their applications, in particular their application in determining harmonic functions in two-dimensional regions. The module also covers certain analytical aspects of complex functions.
~	MAT-30010	Complex Variable II	EP	M	7.5	15
		This module revises and consolidates the material of MAT - 20004: Complex Variable I, before studying further applications of contour integration and the Residue Theorem. It then provides an introduction to conformal mappings, together with some of their applications, in particular their application in determining harmonic functions in two-dimensional regions. The module also covers certain analytical aspects of complex functions.
	MAT-30011	Waves	O	M	7.5	15
		The module aims to give an account of the underlying mathematical theory that describes the behaviour of waves. The mathematical development of the subject is combined with a discussion of applications, for example musical instruments. The module material is illuminated by small demonstrations and by computer-generated animations of wave processes. The topics include: the wave equation, waves on stretched strings, waves on membranes, waves on beams, sound waves, and waves in liquids with a free surface.
	MAT-30011	Waves	EP	M	7.5	15
		The module aims to give an account of the underlying mathematical theory that describes the behaviour of waves. The mathematical development of the subject is combined with a discussion of applications, for example musical instruments. The module material is illuminated by small demonstrations and by computer-generated animations of wave processes. The topics include: the wave equation, waves on stretched strings, waves on membranes, waves on beams, sound waves, and waves in liquids with a free surface.
	MAT-30014	Medical Statistics	O	M	7.5	15
		This module illustrates the application of statistical techniques to health related research. Methods are applied using data from real-life studies, with 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.
	MAT-30014	Medical Statistics	EP	M	7.5	15
		This module illustrates the application of statistical techniques to health related research. Methods are applied using data from real-life studies, with 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.
#	MAT-30016	Project II - ISP	O	C	7.5	15
		This module is an opportunity for students to explore in depth a particular topic or problem in mathematics or statistics chosen from a list prepared by the division. Each student is supervised by a member of staff throughout the execution of the project. This execution is carried out between September and May. There are many different types of project: some comprise primarily of a review of the literature, the student assembling from several sources their own account of a body of mathematics not covered elsewhere in the programme; others involve tackling an applied problem, and these normally require some computational work, using Mathematica for example; and in others specialised statistical techniques and packages are used to analyse a large data set. The work is assessed by means of a dissertation and an oral presentation to the division.
#	MAT-30016	Project II - ISP	EP	C	7.5	15
		This module is an opportunity for students to explore in depth a particular topic or problem in mathematics or statistics chosen from a list prepared by the division. Each student is supervised by a member of staff throughout the execution of the project. This execution is carried out between September and May. There are many different types of project: some comprise primarily of a review of the literature, the student assembling from several sources their own account of a body of mathematics not covered elsewhere in the programme; others involve tackling an applied problem, and these normally require some computational work, using Mathematica for example; and in others specialised statistical techniques and packages are used to analyse a large data set. The work is assessed by means of a dissertation and an oral presentation to the division.
	MAT-30023	Mathematical Biology	O	M	7.5	15
		This module will show you how mathematics is an interdisciplinary subject, with particular attention to biology. Applications of mathematics to biological situations is one of the fastest growing areas where mathematics can explain and predict behaviour. These predictions are not just theoretical: every day people's lives are saved due to the predictions possible. We shall investigate a diverse set of applications. Game theory has provided new mathematical tools to study the evolution of animal behaviour. The biology of population growth and disease transmission, in particular, recent advances in our mathematical understanding of biology has provided new insight into the spread of MRSA. In the last few years there have been advances in the application of mathematics to the study of animal gaits (the different method of locomotion).
	MAT-30023	Mathematical Biology	EP	M	7.5	15
		This module will show you how mathematics is an interdisciplinary subject, with particular attention to biology. Applications of mathematics to biological situations is one of the fastest growing areas where mathematics can explain and predict behaviour. These predictions are not just theoretical: every day people's lives are saved due to the predictions possible. We shall investigate a diverse set of applications. Game theory has provided new mathematical tools to study the evolution of animal behaviour. The biology of population growth and disease transmission, in particular, recent advances in our mathematical understanding of biology has provided new insight into the spread of MRSA. In the last few years there have been advances in the application of mathematics to the study of animal gaits (the different method of locomotion).
~	MAT-30024	Ring and Field Theory	EP	M	7.5	15
		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 Pure Mathematics where field theory and group theory come together to answer some of the oldest questions in mathematics 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 also be given. This will include the application to classical proofs concerning geometrical constructions using ruler and compasses.
~	MAT-30024	Ring and Field Theory	O	M	7.5	15
		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 Pure Mathematics where field theory and group theory come together to answer some of the oldest questions in mathematics 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 also be given. This will include the application to classical proofs concerning geometrical constructions using ruler and compasses.
~	MAT-30027	Codes and Cryptography	O	M	7.5	15
		This module will illustrate how abstract ideas from group theory, number theory and linear 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. This module will develop the following graduate attributes: 1) An open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds. 2) An appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge. 4) The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6) The ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences 9) A professional and reflective approach, including qualities of leadership, responsibility, personal integrity, empathy, care and respect for others, accountability and self-regulation
~	MAT-30027	Codes and Cryptography	EP	M	7.5	15
		This module will illustrate how abstract ideas from group theory, number theory and linear 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. This module will develop the following graduate attributes: 1) An open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds. 2) An appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge. 4) The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6) The ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences 9) A professional and reflective approach, including qualities of leadership, responsibility, personal integrity, empathy, care and respect for others, accountability and self-regulation
	MAT-30028	Perturbation Methods	O	M	7.5	15
		This module introduces students to the asymptotic techniques which are indispensible 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.
	MAT-30028	Perturbation Methods	EP	M	7.5	15
		This module introduces students to the asymptotic techniques which are indispensible 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.

Mathematics Minor - Level 1 Modules

Semester 1			C/O	TYP	ECTS	CATS
	MAT-10038	Algebra I	C	M	7.5	15
		This module is intended to help students with the transition from the methods based approach of A level mathematics to the higher levels of understanding and rigour expected at degree level. It begins by discussing mathematical statements and the meaning and basic strategies of proof. This is followed by a short exposition of naive set theory and by a careful treatment of the notion of a function. The remainder of the module covers the algebraic development of number systems and their properties. The module ends with a brief discussion of the properties of polynomials, including the Fundamental Theorem of Algebra. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.
	MAT-10039	Calculus I	C	M	7.5	15
		This module is intended to help students with the transition from the methods based approach of A level Mathematics to the higher levels of understanding and rigour expected at degree level. The module will develop students' skills in mathematical techniques, mainly in differentiation and integration. It begins by revising the standard elementary functions and their properties and continues with the revision of basic results and techniques in differentiation. It then revises basic results and techniques in integration. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.

Semester 2			C/O	TYP	ECTS	CATS
	MAT-10040	Algebra II	C	M	7.5	15
		This module continues from Algebra I and provides an introduction to matrix and vector algebra. Topics will include matrix algebra, elementary row operations, solving linear equations, determinants, eigenvalues and eigenvectors, diagonalisation of matrices, three dimensional vector algebra and geometry, lines and planes, general vector algebra, linear independence and bases. Some of this material will be applied in a treatment of linear programming, a widely-used optimization technique. In particular the Simplex method and an introduction to duality theory are covered. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.
	MAT-10041	Calculus II	C	M	7.5	15
		Many physical problems are governed by ordinary or partial differential equations, the solution of which can help us understand their properties and characteristics. For instance, the oscillation frequency of a pendulum, the transfer time for sending a spaceship from the Earth to Mars, and the population evolution of a fish species in a lake can all be determined by solving ordinary differential equations. This module, which is a prerequisite for a number of other modules in the second and third years, will introduce some of the basic techniques for solving ordinary differential equations, familiarize students with partial differentiations, and explain how double integrals can be evaluated and used to compute areas and volumes. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.

Mathematics Minor - Level 2 Modules

Semester 1			C/O	TYP	ECTS	CATS
	MAT-20008	Differential Equations	C	M	7.5	15
		This module focuses on methods for solving ordinary differential equations. The topics include: solutions to first-order equations, higher order linear equations, power series methods, graphical aspects of differential equations, Fourier series and Laplace transforms.
	MAT-20011	Mathematics - Study Abroad I	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20012	Mathematics - Study Abroad II	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20015	Numerical Methods	O	M	7.5	15
		This module studies numerical methods as applied in linear algebra and to differential equations. Topics include: the solution of algebraic equations, numerical quadrature, the numerical solution of differential equations including the Euler and Runge-Kutta methods, the solution of linear equations by direct and iterative methods, and numerical methods for determining eigenvalues and eigenvectors.
	MAT-20015	Numerical Methods	EP	M	7.5	15
		This module studies numerical methods as applied in linear algebra and to differential equations. Topics include: the solution of algebraic equations, numerical quadrature, the numerical solution of differential equations including the Euler and Runge-Kutta methods, the solution of linear equations by direct and iterative methods, and numerical methods for determining eigenvalues and eigenvectors.
	MAT-20023	Probability	O	M	7.5	15
		Probability is the mathematics of uncertainty and randomness. The module begins with classical notions of probability associated with the analysis of games of chance using cards, dice, etc. It then moves to the axiomatic treatment of the probability of random events. This leads to the definitions of statistical independence and conditional probability. The remainder of the module is concerned with a systematic study of discrete and continuous, univariate and bivariate, random variables, covering expectation, variance, covariance and the moment generating function. The theory is applied to a wide range of theoretical and applied problems. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20023	Probability	EP	M	7.5	15
		Probability is the mathematics of uncertainty and randomness. The module begins with classical notions of probability associated with the analysis of games of chance using cards, dice, etc. It then moves to the axiomatic treatment of the probability of random events. This leads to the definitions of statistical independence and conditional probability. The remainder of the module is concerned with a systematic study of discrete and continuous, univariate and bivariate, random variables, covering expectation, variance, covariance and the moment generating function. The theory is applied to a wide range of theoretical and applied problems. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20024	Operational Research	O	M	7.5	15
		This module presents a variety of topics, techniques and models in operational research. 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. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20024	Operational Research	EP	M	7.5	15
		This module presents a variety of topics, techniques and models in operational research. 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. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20025	Abstract Algebra	O	M	7.5	15
		This module introduces and studies the abstract algebraic structure known as a group. Starting 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, Lagrange'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. The module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 2. An appreciation of the development and value of Mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.
	MAT-20025	Abstract Algebra	EP	M	7.5	15
		This module introduces and studies the abstract algebraic structure known as a group. Starting 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, Lagrange'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. The module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 2. An appreciation of the development and value of Mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.
	MAT-20026	Linear Algebra	O	M	7.5	15
		This module introduces the concept of an abstract vector space The module builds on the knowledge of vectors and matrices gained from MAT-10040. 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 just different specialisations of the same general concept. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity, and independence of thought. 2. An appreciation of the development and value of mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.
	MAT-20026	Linear Algebra	EP	M	7.5	15
		This module introduces the concept of an abstract vector space The module builds on the knowledge of vectors and matrices gained from MAT-10040. 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 just different specialisations of the same general concept. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity, and independence of thought. 2. An appreciation of the development and value of mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.

Semester 2			C/O	TYP	ECTS	CATS
	MAT-20004	Complex Variable I and Vector Calculus	C	M	7.5	15
		This module contains a first course on vector calculus and a first course in functions of a complex variable. Complex variable leads to elegant results in pure mathematics and both complex variable and vector calculus provide a framework for solving physical and geometrical problems. The topics covered include: complex functions, analytic functions, Cauchy&�s theorems, complex power series, singularities, the residue theorem, contour integration, differentiation of vectors, differential operators, integration of vectors, the divergence theorem and Stokes&� theorem.
	MAT-20005	Dynamics	O	M	7.5	15
		The module is an introduction to dynamics with applications mainly to systems which can be modelled by particle dynamics. The topics investigated include: Newton&�s laws, momentum, kinetic and potential energy, projectiles, simple harmonic motion, springs, the pendulum, rocket motion, planetary and satellite orbits, linear theory of oscillations, and normal modes. 2 hours lectures & 1 hour examples class
	MAT-20005	Dynamics	EP	M	7.5	15
		The module is an introduction to dynamics with applications mainly to systems which can be modelled by particle dynamics. The topics investigated include: Newton&�s laws, momentum, kinetic and potential energy, projectiles, simple harmonic motion, springs, the pendulum, rocket motion, planetary and satellite orbits, linear theory of oscillations, and normal modes. 2 hours lectures & 1 hour examples class
	MAT-20006	Stochastic Processes	O	M	7.5	15
		This module examines the mathematics of random processes with particular reference to biological and social sciences. The course will include: revision of probability and distribution theory, difference equations, Markov chains, branching and Poisson processes, birth and death processes, queues, reliability and lifetime distributions. 2 hours lectures & 1 hour examples class
	MAT-20006	Stochastic Processes	EP	M	7.5	15
		This module examines the mathematics of random processes with particular reference to biological and social sciences. The course will include: revision of probability and distribution theory, difference equations, Markov chains, branching and Poisson processes, birth and death processes, queues, reliability and lifetime distributions. 2 hours lectures & 1 hour examples class
	MAT-20016	Mathematical Modelling	O	C	7.5	15
		The aim of the module is to demonstrate how real-world problems can be modelled mathematically. The mathematical modelling process will be introduced through a six-step problem-solving approach: identifying a suitable problem from the particular real-world scenario, making assumptions to simplify the problem, classifying variables influencing the problem, constructing a mathematical model to determine interrelationships among the variables, solving and interpreting the model and finally validating the model with real-world data. Mathematical tools that will be used in the model construction and solution process include: Ordinary Differential Equations and their solution methods, such as 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.
	MAT-20016	Mathematical Modelling	EP	C	7.5	15
		The aim of the module is to demonstrate how real-world problems can be modelled mathematically. The mathematical modelling process will be introduced through a six-step problem-solving approach: identifying a suitable problem from the particular real-world scenario, making assumptions to simplify the problem, classifying variables influencing the problem, constructing a mathematical model to determine interrelationships among the variables, solving and interpreting the model and finally validating the model with real-world data. Mathematical tools that will be used in the model construction and solution process include: Ordinary Differential Equations and their solution methods, such as 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.
	MAT-20017	Mathematics - Study Abroad Vi	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20018	Mathematics - Study Abroad VII	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20021	Real Analysis	O	M	7.5	15
		This module is intended as an introduction to Mathematical Analysis, which can be described as the study of the infinitely small and infinitely large. The module studies the real number system, sequences, series, limits of functions, continuity and Taylor series. Such a study is essential for understanding subsequent modules. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal forms.
	MAT-20021	Real Analysis	EP	M	7.5	15
		This module is intended as an introduction to Mathematical Analysis, which can be described as the study of the infinitely small and infinitely large. The module studies the real number system, sequences, series, limits of functions, continuity and Taylor series. Such a study is essential for understanding subsequent modules. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal forms.
	MAT-20027	Linear Statistical Models	O	M	7.5	15
		This module presents an introduction to statistical inference and illustrates the theory with practical applications to real life data sets. The first half of the module will encorporate hypothesis testing, confidence interval and parameter estimation. The second half of the module considers regression and anova methods, and demonstrates how statistical models can be used to draw conclusions from observations and experimental data collected in the physical and social sciences. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20027	Linear Statistical Models	EP	M	7.5	15
		This module presents an introduction to statistical inference and illustrates the theory with practical applications to real life data sets. The first half of the module will encorporate hypothesis testing, confidence interval and parameter estimation. The second half of the module considers regression and anova methods, and demonstrates how statistical models can be used to draw conclusions from observations and experimental data collected in the physical and social sciences. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.

Mathematics Minor - Level 3 Modules

Semester 1			C/O	TYP	ECTS	CATS
	MAT-30003	Partial Differential Equations	EP	M	7.5	15
		This module provides an introduction to analytical methods for solving partial differential equations (PDEs). Throughout the module focuses on PDEs in two independent variables, although generalisation to three, or more, independent variables is briefly discussed. The module begins by introducing the method of characteristics for solving first order linear and quasi-linear PDEs. This is followed by the classification of linear second order PDEs into hyperbolic, parabolic or elliptic type, with a detailed treatment of the reduction to canonical form in each case. Thereafter, the module introduces the separation of variables technique and transform methods to solve the wave, diffusion, Laplace and Poisson equations. These PDEs are frequently encountered in many branches of applied mathematics, including fluid dynamics, mathematical biology, financial mathematics and electromagnetism. Finally the module introduces the technique of eigenfunction expansions for solving inhomogeneous PDEs, and the powerful technique of Green&�s functions.
	MAT-30021	Numerical Analysis	EP	C	7.5	15
		This module is concerned with the analysis of numerical methods, concentrating on how to select a suitable method and analyse the results produced, rather than on programming the methods themselves. Substantial use is made of technology in terms of existing teaching packages, computer algebra and hand held technologies, all of which provide transferable skills. There are regular computer laboratory sessions at which students will be expected to use any and all of the methods introduced in lectures. The module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought; 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand; 6. The ability to communicate clearly and effectively in written form;

Semester 2			C/O	TYP	ECTS	CATS
	MAT-30001	Graph Theory	EP	M	7.5	15
		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 via worked examples and coursework.
	MAT-30023	Mathematical Biology	EP	M	7.5	15
		This module will show you how mathematics is an interdisciplinary subject, with particular attention to biology. Applications of mathematics to biological situations is one of the fastest growing areas where mathematics can explain and predict behaviour. These predictions are not just theoretical: every day people's lives are saved due to the predictions possible. We shall investigate a diverse set of applications. Game theory has provided new mathematical tools to study the evolution of animal behaviour. The biology of population growth and disease transmission, in particular, recent advances in our mathematical understanding of biology has provided new insight into the spread of MRSA. In the last few years there have been advances in the application of mathematics to the study of animal gaits (the different method of locomotion).

Mathematics Single Honours - Level 1 Modules

Semester 1			C/O	TYP	ECTS	CATS
	MAT-10038	Algebra I	C	M	7.5	15
		This module is intended to help students with the transition from the methods based approach of A level mathematics to the higher levels of understanding and rigour expected at degree level. It begins by discussing mathematical statements and the meaning and basic strategies of proof. This is followed by a short exposition of naive set theory and by a careful treatment of the notion of a function. The remainder of the module covers the algebraic development of number systems and their properties. The module ends with a brief discussion of the properties of polynomials, including the Fundamental Theorem of Algebra. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.
	MAT-10039	Calculus I	C	M	7.5	15
		This module is intended to help students with the transition from the methods based approach of A level Mathematics to the higher levels of understanding and rigour expected at degree level. The module will develop students' skills in mathematical techniques, mainly in differentiation and integration. It begins by revising the standard elementary functions and their properties and continues with the revision of basic results and techniques in differentiation. It then revises basic results and techniques in integration. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.
	MAT-10044	Investigations and Problem Solving	C	C	7.5	15
		This module is designed to aid the transition from A level mathematics to degree level mathematics. At A level, students are presented with highly structured questions having been taught the specific processes required to solve them. In this module students will be presented with unfamiliar and less structured problems which may be open to several different approaches. The module enhances students' employability skills through team-work on extended projects, presentations, and the opportunity to reflect and articulate their strengths and weaknesses. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity, independence of thought. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences. 8. the ability and motivation to participate responsibly and collaboratively as an active citizen in the communities in which you live and work. 9. A professional and reflective approach, including qualities of leadership, responsibility, personal integrity, empathy, care and respect for others, accountability and self-regulation. 10. the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.

Semester 2			C/O	TYP	ECTS	CATS
	MAT-10035	Analysis	C	M	7.5	15
		This module is intended as an introduction to Mathematical Analysis, which can be described as the study of the infinitely small and infinitely large. The module studies the real number system, sequences, series, limits of functions, continuity and Taylor series. Such a study is essential for understanding subsequent modules. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.
	MAT-10036	Geometry	O	C	7.5	15
		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 plane, as first explored by the mathematicians of ancient Greece. It then progresses to examine straightedge and compass constructions, constructible numbers, isometries of the plane, together with some non-Euclidian 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. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.
	MAT-10040	Algebra II	C	M	7.5	15
		This module continues from Algebra I and provides an introduction to matrix and vector algebra. Topics will include matrix algebra, elementary row operations, solving linear equations, determinants, eigenvalues and eigenvectors, diagonalisation of matrices, three dimensional vector algebra and geometry, lines and planes, general vector algebra, linear independence and bases. Some of this material will be applied in a treatment of linear programming, a widely-used optimization technique. In particular the Simplex method and an introduction to duality theory are covered. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.
	MAT-10041	Calculus II	C	M	7.5	15
		Many physical problems are governed by ordinary or partial differential equations, the solution of which can help us understand their properties and characteristics. For instance, the oscillation frequency of a pendulum, the transfer time for sending a spaceship from the Earth to Mars, and the population evolution of a fish species in a lake can all be determined by solving ordinary differential equations. This module, which is a prerequisite for a number of other modules in the second and third years, will introduce some of the basic techniques for solving ordinary differential equations, familiarize students with partial differentiations, and explain how double integrals can be evaluated and used to compute areas and volumes. This module will develop the following graduate attributes: 1) an open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds; 2) an appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge; 4) the ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6) the ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 10) the flexibility to thrive in rapidly changing and uncertain external environments and to update skills and knowledge as circumstances require.
	MAT-10042	Applicable Mathematics	O	C	7.5	15
		This module is designed to help students appreciate mathematics as a method for describing and solving real-world problems. It begins by introducing mathematical problem solving, providing students with a transition from the example-based problem solving encountered at A-level to higher levels of problem solving expected at degree level and in employment. Physical or computer experiments will be used to motivate the study of phenomena such as symmetry, period doubling bifurcations and chaos. Using the theory of discrete dynamical systems the final part of the module will examine the mathematics of, for example, population interactions. The mathematical and problem solving ideas will be developed through a number of group projects.

Mathematics Single Honours - Level 2 Modules

Semester 1			C/O	TYP	ECTS	CATS
	MAT-20008	Differential Equations	C	M	7.5	15
		This module focuses on methods for solving ordinary differential equations. The topics include: solutions to first-order equations, higher order linear equations, power series methods, graphical aspects of differential equations, Fourier series and Laplace transforms.
	MAT-20011	Mathematics - Study Abroad I	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20012	Mathematics - Study Abroad II	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20013	Mathematics - Study Abroad III	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20014	Mathematics - Study Abroad IV	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20015	Numerical Methods	O	M	7.5	15
		This module studies numerical methods as applied in linear algebra and to differential equations. Topics include: the solution of algebraic equations, numerical quadrature, the numerical solution of differential equations including the Euler and Runge-Kutta methods, the solution of linear equations by direct and iterative methods, and numerical methods for determining eigenvalues and eigenvectors.
	MAT-20015	Numerical Methods	EP	M	7.5	15
		This module studies numerical methods as applied in linear algebra and to differential equations. Topics include: the solution of algebraic equations, numerical quadrature, the numerical solution of differential equations including the Euler and Runge-Kutta methods, the solution of linear equations by direct and iterative methods, and numerical methods for determining eigenvalues and eigenvectors.
	MAT-20023	Probability	C	M	7.5	15
		Probability is the mathematics of uncertainty and randomness. The module begins with classical notions of probability associated with the analysis of games of chance using cards, dice, etc. It then moves to the axiomatic treatment of the probability of random events. This leads to the definitions of statistical independence and conditional probability. The remainder of the module is concerned with a systematic study of discrete and continuous, univariate and bivariate, random variables, covering expectation, variance, covariance and the moment generating function. The theory is applied to a wide range of theoretical and applied problems. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20024	Operational Research	O	M	7.5	15
		This module presents a variety of topics, techniques and models in operational research. 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. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20024	Operational Research	EP	M	7.5	15
		This module presents a variety of topics, techniques and models in operational research. 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. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20025	Abstract Algebra	C	M	7.5	15
		This module introduces and studies the abstract algebraic structure known as a group. Starting 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, Lagrange'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. The module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 2. An appreciation of the development and value of Mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.
	MAT-20026	Linear Algebra	O	M	7.5	15
		This module introduces the concept of an abstract vector space The module builds on the knowledge of vectors and matrices gained from MAT-10040. 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 just different specialisations of the same general concept. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity, and independence of thought. 2. An appreciation of the development and value of mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.
	MAT-20026	Linear Algebra	EP	M	7.5	15
		This module introduces the concept of an abstract vector space The module builds on the knowledge of vectors and matrices gained from MAT-10040. 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 just different specialisations of the same general concept. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity, and independence of thought. 2. An appreciation of the development and value of mathematics and the links between different areas of the subject. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand. 6. The ability to communicate clearly and effectively in written form.

Semester 2			C/O	TYP	ECTS	CATS
	MAT-20004	Complex Variable I and Vector Calculus	C	M	7.5	15
		This module contains a first course on vector calculus and a first course in functions of a complex variable. Complex variable leads to elegant results in pure mathematics and both complex variable and vector calculus provide a framework for solving physical and geometrical problems. The topics covered include: complex functions, analytic functions, Cauchy&�s theorems, complex power series, singularities, the residue theorem, contour integration, differentiation of vectors, differential operators, integration of vectors, the divergence theorem and Stokes&� theorem.
	MAT-20005	Dynamics	O	M	7.5	15
		The module is an introduction to dynamics with applications mainly to systems which can be modelled by particle dynamics. The topics investigated include: Newton&�s laws, momentum, kinetic and potential energy, projectiles, simple harmonic motion, springs, the pendulum, rocket motion, planetary and satellite orbits, linear theory of oscillations, and normal modes. 2 hours lectures & 1 hour examples class
	MAT-20005	Dynamics	EP	M	7.5	15
		The module is an introduction to dynamics with applications mainly to systems which can be modelled by particle dynamics. The topics investigated include: Newton&�s laws, momentum, kinetic and potential energy, projectiles, simple harmonic motion, springs, the pendulum, rocket motion, planetary and satellite orbits, linear theory of oscillations, and normal modes. 2 hours lectures & 1 hour examples class
	MAT-20006	Stochastic Processes	O	M	7.5	15
		This module examines the mathematics of random processes with particular reference to biological and social sciences. The course will include: revision of probability and distribution theory, difference equations, Markov chains, branching and Poisson processes, birth and death processes, queues, reliability and lifetime distributions. 2 hours lectures & 1 hour examples class
	MAT-20006	Stochastic Processes	EP	M	7.5	15
		This module examines the mathematics of random processes with particular reference to biological and social sciences. The course will include: revision of probability and distribution theory, difference equations, Markov chains, branching and Poisson processes, birth and death processes, queues, reliability and lifetime distributions. 2 hours lectures & 1 hour examples class
	MAT-20016	Mathematical Modelling	C	C	7.5	15
		The aim of the module is to demonstrate how real-world problems can be modelled mathematically. The mathematical modelling process will be introduced through a six-step problem-solving approach: identifying a suitable problem from the particular real-world scenario, making assumptions to simplify the problem, classifying variables influencing the problem, constructing a mathematical model to determine interrelationships among the variables, solving and interpreting the model and finally validating the model with real-world data. Mathematical tools that will be used in the model construction and solution process include: Ordinary Differential Equations and their solution methods, such as 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.
	MAT-20017	Mathematics - Study Abroad Vi	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20018	Mathematics - Study Abroad VII	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20019	Mathematics - Study Abroad VIII	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
	MAT-20020	Mathematics - Study Abroad V	EP	C	7.5	15
		This is a module that is automatically allocated to the records our Keele level II students who are going to Study Abroad at a partner University for a semester of their second year and cannot be selected by any other level II students.
~	MAT-20022	Metric Spaces	O	M	7.5	15
		This module studies the concept of an abstract metric space. The module builds upon the knowledge of Level 1 Analysis. The module includes the following topics: metric spaces, convergence, topological ideas, completeness and contraction mappings, and compactness. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal forms.
	MAT-20027	Linear Statistical Models	O	M	7.5	15
		This module presents an introduction to statistical inference and illustrates the theory with practical applications to real life data sets. The first half of the module will encorporate hypothesis testing, confidence interval and parameter estimation. The second half of the module considers regression and anova methods, and demonstrates how statistical models can be used to draw conclusions from observations and experimental data collected in the physical and social sciences. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-20027	Linear Statistical Models	EP	M	7.5	15
		This module presents an introduction to statistical inference and illustrates the theory with practical applications to real life data sets. The first half of the module will encorporate hypothesis testing, confidence interval and parameter estimation. The second half of the module considers regression and anova methods, and demonstrates how statistical models can be used to draw conclusions from observations and experimental data collected in the physical and social sciences. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.

Mathematics Single Honours - Level 3 Modules

Semester 1			C/O	TYP	ECTS	CATS
	MAT-30002	Non-linear Differential Equations	O	M	7.5	15
		Systems that evolve in time can often be modelled by differential equations. There are countless examples of such systems from the physical world including the weather, climate change, stock markets, the economy, population dynamics, mechanical systems, etc. The great variety of behaviours exhibited by these systems is reflected in the solutions to the corresponding differential equations. This module introduces a number of methods for identifying and classifying various types of behaviour in various types of differential equation. While linear differential equations model some processes, the majority are described by nonlinear equations, and it is these that display the greatest diversity of behaviour. However, very few nonlinear differential equations have exact solutions. Nevertheless, a great deal of insight can be obtained from qualitative methods. This module focuses on geometric methods for constructing phase plane representations of dynamics and perturbation methods for obtaining approximate solutions. With these tools it is then possible to examine the changes in behaviour that can occur when a parameter is varied, and bifurcation theory is introduced to describe this. The relation between the evolution of differential equations and the evolution of maps is explained, and more exotic behaviour, like period doubling and chaos, are then studied in terms of the dynamics of maps.
	MAT-30003	Partial Differential Equations	O	M	7.5	15
		This module provides an introduction to analytical methods for solving partial differential equations (PDEs). Throughout the module focuses on PDEs in two independent variables, although generalisation to three, or more, independent variables is briefly discussed. The module begins by introducing the method of characteristics for solving first order linear and quasi-linear PDEs. This is followed by the classification of linear second order PDEs into hyperbolic, parabolic or elliptic type, with a detailed treatment of the reduction to canonical form in each case. Thereafter, the module introduces the separation of variables technique and transform methods to solve the wave, diffusion, Laplace and Poisson equations. These PDEs are frequently encountered in many branches of applied mathematics, including fluid dynamics, mathematical biology, financial mathematics and electromagnetism. Finally the module introduces the technique of eigenfunction expansions for solving inhomogeneous PDEs, and the powerful technique of Green&�s functions.
	MAT-30005	Relativity	O	M	7.5	15
		Relativity is a theory of space and time. It consists of two fundamentally different areas, that is, Special Relativity and General Relativity. Special Relativity was first proposed by Einstein in 1905 and is concerned with the connection between observations made in frames of reference that move with constant velocity relative to each other. It has many practical applications in situations which involve extremely high speeds. General Relativity was first proposed by Einstein in 1916 and incorporates non-uniform motion. Its main applications lie in gravitation theory, including astrophysics and cosmology. This module gives an introduction to both Special and General Relativity from a mainly mathematical point of view. The standard applications to physical problems are given, but experimental details are kept to a minimum.
~	MAT-30006	Logic	O	M	7.5	15
		The purpose of the module is to introduce mathematical concepts for examining philosophical questions about the nature of mathematics as a whole. It attempts to present a sophisticated perspective on mathematics in a way that is accessible to undergraduates. The first half of the module concerns the subject-matter of mathematics. The thesis is developed that all mathematical objects can be understood as sets; set theory is developed in an informal axiomatic spirit, based on G�del&�s notion of transfinite iteration of the $�set of&� operation. The second half examines mathematical reasoning, which is formalised as predicate calculus and studied metamathematically. The basic apparatus of formal semantics is introduced, and issues such as completeness and categoricity are surveyed informally.
~	MAT-30013	Group Theory	O	M	7.5	15
		This module builds on the Group Theory introduced in MAT - 20009 to develop some of the mathematics underlying the classification of finite groups. This culminates in a proof of Sylow&�s First Theorem which offers a partial converse to Lagrange&�s Theorem proved in MAT - 20009. The module also develops some applications of Group Theory, the natural setting for which is that of group actions. Several examples of applying group theoretic ideas to counting combinatorial configurations are presented.
#	MAT-30016	Project II - ISP	O	C	7.5	15
		This module is an opportunity for students to explore in depth a particular topic or problem in mathematics or statistics chosen from a list prepared by the division. Each student is supervised by a member of staff throughout the execution of the project. This execution is carried out between September and May. There are many different types of project: some comprise primarily of a review of the literature, the student assembling from several sources their own account of a body of mathematics not covered elsewhere in the programme; others involve tackling an applied problem, and these normally require some computational work, using Mathematica for example; and in others specialised statistical techniques and packages are used to analyse a large data set. The work is assessed by means of a dissertation and an oral presentation to the division.
	MAT-30021	Numerical Analysis	O	C	7.5	15
		This module is concerned with the analysis of numerical methods, concentrating on how to select a suitable method and analyse the results produced, rather than on programming the methods themselves. Substantial use is made of technology in terms of existing teaching packages, computer algebra and hand held technologies, all of which provide transferable skills. There are regular computer laboratory sessions at which students will be expected to use any and all of the methods introduced in lectures. The module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought; 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand; 6. The ability to communicate clearly and effectively in written form;
~	MAT-30022	Number Theory	O	M	7.5	15
		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. 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. This module will build upon the concepts and techniques introduced in the second year module "Analysis and Abstract Algebra", and consider how these can be extended and applied to both ancient and modern problems, from finding whole number solutions to polynomial equations to primality testing and cryptography.
~	MAT-30026	Probability Models	O	M	7.5	15
		Probability models are widely used in natural, social and financial sciences for the purposes of understanding, predicting and controlling random phenomena. The module builds on the earlier study of probability and stochastic processes. It aims to: (a) provide students with knowledge of some more advanced concepts in probability theory, in particular: conditional expectation, convergence of random variables, martingale sequences, and brownian motion; (b) apply probabilistic techniques to a wide range of stochastic models and processes. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought. 4. The ability to solve problems creatively using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6. The ability to communicate clearly and effectively in written and verbal form.
	MAT-30029	Professional Mathematics	O	C	7.5	15
		This module gives the opportunity to further develop skills of mathematical problem solving and application of mathematical results to real world scenarios through group work activities. This aims to develop skills that are needed when undertaking employment or research, such as working in depth on a problem over an extended period, writing reports, communicating mathematical results to different audiences and working in collaboration with others. This module will provide employability skills experiences and develop students� ability to articulate their skills, which will be useful to draw on when applying for jobs. The skills developed will form parts of a student�s Distinctive Keele Curriculum experience. This module develops the following Keele Graduate attributes: 1. An open and questioning approach to ideas, demonstrating curiosity, independence of thought; 4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand; 6. The ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences; 7. The knowledge, skills, self-confidence and self-awareness actively to pursue your future goals; 9. A professional and reflective approach, including qualities of leadership, responsibility, personal integrity, empathy, care and respect for others, accountability and self-regulation.

Semester 2			C/O	TYP	ECTS	CATS
	MAT-30001	Graph Theory	O	M	7.5	15
		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 via worked examples and coursework.
	MAT-30004	Fluid Mechanics	O	M	7.5	15
		The module is a first course in fluid mechanics, designed for students with no previous knowledge of the subject. Although previous acquaintance with vectors and differential equations would be helpful, the lecturer will offer revision of the required parts of these subjects as the module progresses.
~	MAT-30010	Complex Variable II	O	M	7.5	15
		This module revises and consolidates the material of MAT - 20004: Complex Variable I, before studying further applications of contour integration and the Residue Theorem. It then provides an introduction to conformal mappings, together with some of their applications, in particular their application in determining harmonic functions in two-dimensional regions. The module also covers certain analytical aspects of complex functions.
	MAT-30011	Waves	O	M	7.5	15
		The module aims to give an account of the underlying mathematical theory that describes the behaviour of waves. The mathematical development of the subject is combined with a discussion of applications, for example musical instruments. The module material is illuminated by small demonstrations and by computer-generated animations of wave processes. The topics include: the wave equation, waves on stretched strings, waves on membranes, waves on beams, sound waves, and waves in liquids with a free surface.
	MAT-30014	Medical Statistics	O	M	7.5	15
		This module illustrates the application of statistical techniques to health related research. Methods are applied using data from real-life studies, with 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.
#	MAT-30016	Project II - ISP	O	C	7.5	15
		This module is an opportunity for students to explore in depth a particular topic or problem in mathematics or statistics chosen from a list prepared by the division. Each student is supervised by a member of staff throughout the execution of the project. This execution is carried out between September and May. There are many different types of project: some comprise primarily of a review of the literature, the student assembling from several sources their own account of a body of mathematics not covered elsewhere in the programme; others involve tackling an applied problem, and these normally require some computational work, using Mathematica for example; and in others specialised statistical techniques and packages are used to analyse a large data set. The work is assessed by means of a dissertation and an oral presentation to the division.
	MAT-30023	Mathematical Biology	O	M	7.5	15
		This module will show you how mathematics is an interdisciplinary subject, with particular attention to biology. Applications of mathematics to biological situations is one of the fastest growing areas where mathematics can explain and predict behaviour. These predictions are not just theoretical: every day people's lives are saved due to the predictions possible. We shall investigate a diverse set of applications. Game theory has provided new mathematical tools to study the evolution of animal behaviour. The biology of population growth and disease transmission, in particular, recent advances in our mathematical understanding of biology has provided new insight into the spread of MRSA. In the last few years there have been advances in the application of mathematics to the study of animal gaits (the different method of locomotion).
~	MAT-30024	Ring and Field Theory	O	M	7.5	15
		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 Pure Mathematics where field theory and group theory come together to answer some of the oldest questions in mathematics 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 also be given. This will include the application to classical proofs concerning geometrical constructions using ruler and compasses.
~	MAT-30027	Codes and Cryptography	O	M	7.5	15
		This module will illustrate how abstract ideas from group theory, number theory and linear 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. This module will develop the following graduate attributes: 1) An open and questioning approach to ideas, demonstrating curiosity, independence of thought and the ability to appreciate a range of perspectives on the natural and social worlds. 2) An appreciation of the development and value of your chosen subjects of study, awareness of their contexts, the links between them, and awareness of the provisional and dynamic nature of knowledge. 4) The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the issue at hand. 6) The ability to communicate clearly and effectively in written and verbal forms for different purposes and to a variety of audiences 9) A professional and reflective approach, including qualities of leadership, responsibility, personal integrity, empathy, care and respect for others, accountability and self-regulation
	MAT-30028	Perturbation Methods	O	M	7.5	15
		This module introduces students to the asymptotic techniques which are indispensible 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.

C	Compulsory Core Module
O	Optional Core Module
EP	Programme Elective Module
EA	Approved Elective Module
EF	Free-Standing Elective Module
M	Mixed Assessment e.g. a mixture of essay(s) and examination, with the latter's weighting below 90%.
E	Examination, providing 90% or more of the mark.
C	Continuous Assessment e.g. essay(s) or practical work (as appropriate).
+	Available to qualified non-principal, Erasmus, Exchange and Study Abroad students but there may be a restriction on the number of places available
~	Specific pre-requisite(s) needed by non-principal, Erasmus, Exchange and Study Abroad students wishing to take these modules
#	Not normally available to Erasmus, Exchange and Study Abroad students (except by prior negotiation with Departmental Tutor)
Note: Modules not marked with a # are available to suitably qualified Erasmus, Exchange and Study Abroad students.

Mathematics School of Computing and Mathematics Faculty of Natural Sciences

Principal Course Timetable Blocks 3

Mathematics Dual Honours - Level 1 Modules

Mathematics Dual Honours - Level 2 Modules

Mathematics Dual Honours - Level 3 Modules

Mathematics Major - Level 1 Modules

Mathematics Major - Level 2 Modules

Mathematics Major - Level 3 Modules

Mathematics Minor - Level 1 Modules

Mathematics Minor - Level 2 Modules

Mathematics Minor - Level 3 Modules

Mathematics Single Honours - Level 1 Modules

Mathematics Single Honours - Level 2 Modules

Mathematics Single Honours - Level 3 Modules

Mathematics
School of Computing and Mathematics
Faculty of Natural Sciences