| 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&İs theorems, complex power series, singularities, the residue theorem, contour integration, differentiation of vectors, differential operators, integration of vectors, the divergence theorem and Stokes&İ 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&İs 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-20010 |
Linear and Metric Spaces |
O |
M
|
7.5 |
15 |
|
|
This module introduces the concept of an abstract vector (linear) space, and the concept of a metric space. The module builds on the knowledge of analysis and matrix algebra gained from MAT-10029, 10031 and 20009. The module includes: vector spaces, subspaces, bases and dimension, linear transformations, metric spaces, convergence, topological ideas, completeness and contraction mappings, and compactness. |
|
|
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. |
| 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&İs 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&İs 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 Gödel&İs notion of transfinite iteration of the $ùset of&İ 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 Gödel&İs notion of transfinite iteration of the $ùset of&İ 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-30008 |
Mathematical Programming |
O |
M
|
7.5 |
15 |
|
|
Mathematical programming involves maximizing or minimizing a real function of several variables subject to a list of constraints. It is a major branch of Operational Research, where the formulation is typically a model of an organizational problem; although there are many applications in the social and natural sciences and also in other areas of mathematics. The module studies the mathematical underpinnings of various algorithms for solving different classes of problem, with a strong emphasis on linearity. Some specific applications are mentioned, but the module deals primarily with theory and methods. The work builds upon the account of the Simplex method given in MAT - 10005; this material will be assumed. |
|
|
MAT-30008 |
Mathematical Programming |
EP |
M
|
7.5 |
15 |
|
|
Mathematical programming involves maximizing or minimizing a real function of several variables subject to a list of constraints. It is a major branch of Operational Research, where the formulation is typically a model of an organizational problem; although there are many applications in the social and natural sciences and also in other areas of mathematics. The module studies the mathematical underpinnings of various algorithms for solving different classes of problem, with a strong emphasis on linearity. Some specific applications are mentioned, but the module deals primarily with theory and methods. The work builds upon the account of the Simplex method given in MAT - 10005; this material will be assumed. |
|
|
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&İs First Theorem which offers a partial converse to Lagrange&İs 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&İs First Theorem which offers a partial converse to Lagrange&İs 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. |
|
|
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. |
|
|
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. |
| 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-30009 |
Discrete Mathematics |
O |
M
|
7.5 |
15 |
|
|
This module contains a selection of topics in Discrete Mathematics. The motivation is to show how real projective geometry may be generalised to geometries over any field, and then applied, in the case of finite fields, to solve problems in the construction of combinatorial designs and error-correcting codes. Approximately equal emphasis is placed on theory and applications. |
|
|
MAT-30009 |
Discrete Mathematics |
EP |
M
|
7.5 |
15 |
|
|
This module contains a selection of topics in Discrete Mathematics. The motivation is to show how real projective geometry may be generalised to geometries over any field, and then applied, in the case of finite fields, to solve problems in the construction of combinatorial designs and error-correcting codes. Approximately equal emphasis is placed on theory and applications. |
|
|
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. |
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. |
|
|
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. |
| 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. |
|
|
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. |
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-20009 |
Analysis and Abstract Algebra |
C |
M
|
7.5 |
15 |
|
|
This module builds upon the first year analysis module in order to develop a rigorous theory of functions of a real variable, and introduces abstraction in mathematics through the study of abstract algebra. The module includes: limits of functions of a real variable, continuity, differentiation and Riemann integration; equivalence relations, elementary group theory, applications to number theory, modulo arithmetic, and an introduction to rings and fields. |
|
|
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. |
| 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&İs theorems, complex power series, singularities, the residue theorem, contour integration, differentiation of vectors, differential operators, integration of vectors, the divergence theorem and Stokes&İ 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&İs 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-20010 |
Linear and Metric Spaces |
O |
M
|
7.5 |
15 |
|
|
This module introduces the concept of an abstract vector (linear) space, and the concept of a metric space. The module builds on the knowledge of analysis and matrix algebra gained from MAT-10029, 10031 and 20009. The module includes: vector spaces, subspaces, bases and dimension, linear transformations, metric spaces, convergence, topological ideas, completeness and contraction mappings, and compactness. |
|
|
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. |
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 |
C |
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&İs 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 Gödel&İs notion of transfinite iteration of the $ùset of&İ 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 Gödel&İs notion of transfinite iteration of the $ùset of&İ 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-30008 |
Mathematical Programming |
O |
M
|
7.5 |
15 |
|
|
Mathematical programming involves maximizing or minimizing a real function of several variables subject to a list of constraints. It is a major branch of Operational Research, where the formulation is typically a model of an organizational problem; although there are many applications in the social and natural sciences and also in other areas of mathematics. The module studies the mathematical underpinnings of various algorithms for solving different classes of problem, with a strong emphasis on linearity. Some specific applications are mentioned, but the module deals primarily with theory and methods. The work builds upon the account of the Simplex method given in MAT - 10005; this material will be assumed. |
|
|
MAT-30008 |
Mathematical Programming |
EP |
M
|
7.5 |
15 |
|
|
Mathematical programming involves maximizing or minimizing a real function of several variables subject to a list of constraints. It is a major branch of Operational Research, where the formulation is typically a model of an organizational problem; although there are many applications in the social and natural sciences and also in other areas of mathematics. The module studies the mathematical underpinnings of various algorithms for solving different classes of problem, with a strong emphasis on linearity. Some specific applications are mentioned, but the module deals primarily with theory and methods. The work builds upon the account of the Simplex method given in MAT - 10005; this material will be assumed. |
|
|
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&İs First Theorem which offers a partial converse to Lagrange&İs 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&İs First Theorem which offers a partial converse to Lagrange&İs 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. |
|
|
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. |
|
|
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. |
| 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-30009 |
Discrete Mathematics |
O |
M
|
7.5 |
15 |
|
|
This module contains a selection of topics in Discrete Mathematics. The motivation is to show how real projective geometry may be generalised to geometries over any field, and then applied, in the case of finite fields, to solve problems in the construction of combinatorial designs and error-correcting codes. Approximately equal emphasis is placed on theory and applications. |
|
|
MAT-30009 |
Discrete Mathematics |
EP |
M
|
7.5 |
15 |
|
|
This module contains a selection of topics in Discrete Mathematics. The motivation is to show how real projective geometry may be generalised to geometries over any field, and then applied, in the case of finite fields, to solve problems in the construction of combinatorial designs and error-correcting codes. Approximately equal emphasis is placed on theory and applications. |
|
|
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-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. |
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. |
|
|
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. |
| 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. |
|
|
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. |
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-20009 |
Analysis and Abstract Algebra |
C |
M
|
7.5 |
15 |
|
|
This module builds upon the first year analysis module in order to develop a rigorous theory of functions of a real variable, and introduces abstraction in mathematics through the study of abstract algebra. The module includes: limits of functions of a real variable, continuity, differentiation and Riemann integration; equivalence relations, elementary group theory, applications to number theory, modulo arithmetic, and an introduction to rings and fields. |
|
|
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. |
| 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&İs theorems, complex power series, singularities, the residue theorem, contour integration, differentiation of vectors, differential operators, integration of vectors, the divergence theorem and Stokes&İ 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&İs 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-20010 |
Linear and Metric Spaces |
O |
M
|
7.5 |
15 |
|
|
This module introduces the concept of an abstract vector (linear) space, and the concept of a metric space. The module builds on the knowledge of analysis and matrix algebra gained from MAT-10029, 10031 and 20009. The module includes: vector spaces, subspaces, bases and dimension, linear transformations, metric spaces, convergence, topological ideas, completeness and contraction mappings, and compactness. |
|
|
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. |
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&İs 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. |
| 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-10037 |
Computational Mathematics |
C |
C
|
7.5 |
15 |
|
|
This module covers certain aspects of computational mathematics and introduces, and embeds, a computer algebra system (CAS). The application of the CAS brings as an important, transferable, employability skill. By demonstrating the capabilities of the CAS to cover many of the skills taught at A level or equivalent and to extend these to parallel undergraduate studies, this module will embed the use of technology to form a cornerstone of students' undergraduate mathematical studies. |
|
|
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. |
|
|
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. |
| 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. |
|
|
MAT-10036 |
Geometry |
O |
M
|
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 projective geometry and hyperbolic geometry. A study of the symmetries of polygons and polyhedra will introduce students, in an intuitive way, to an algebraic structure called a group. The module concludes with a brief treatment of knots. |
|
|
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. |
|
|
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. |
|
|
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-20002 |
Statistical Inference I |
C |
M
|
7.5 |
15 |
|
|
This module builds on MAT-10028 and illustrates how statistical theory is put into practice in a variety of hypothesis testing situations. The topics covered are: moment generating functions, bivariate distributions, distributions of functions of random variables, sampling theory, properties of point estimators, maximum likelihood, method of moments, interval estimation, hypothesis testing and some parametric and nonparametric tests. |
|
|
MAT-20003 |
Operational Research II |
O |
M
|
7.5 |
15 |
|
|
The module presents a wide variety of techniques for solving quantitative problems arising in management and everyday life. The topics include: the transportation problem, dynamic programming, the Travelling Salesman Problem, scheduling, reliability, replacement and inventory models. |
|
|
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-20009 |
Analysis and Abstract Algebra |
C |
M
|
7.5 |
15 |
|
|
This module builds upon the first year analysis module in order to develop a rigorous theory of functions of a real variable, and introduces abstraction in mathematics through the study of abstract algebra. The module includes: limits of functions of a real variable, continuity, differentiation and Riemann integration; equivalence relations, elementary group theory, applications to number theory, modulo arithmetic, and an introduction to rings and fields. |
|
|
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.
|
| 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&İs theorems, complex power series, singularities, the residue theorem, contour integration, differentiation of vectors, differential operators, integration of vectors, the divergence theorem and Stokes&İ theorem.
|
|
|
MAT-20006 |
Stochastic Processes |
C |
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-20010 |
Linear and Metric Spaces |
C |
M
|
7.5 |
15 |
|
|
This module introduces the concept of an abstract vector (linear) space, and the concept of a metric space. The module builds on the knowledge of analysis and matrix algebra gained from MAT-10029, 10031 and 20009. The module includes: vector spaces, subspaces, bases and dimension, linear transformations, metric spaces, convergence, topological ideas, completeness and contraction mappings, and compactness. |
|
|
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. |
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 |
C |
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&İs 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 Gödel&İs notion of transfinite iteration of the $ùset of&İ 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-30008 |
Mathematical Programming |
O |
M
|
7.5 |
15 |
|
|
Mathematical programming involves maximizing or minimizing a real function of several variables subject to a list of constraints. It is a major branch of Operational Research, where the formulation is typically a model of an organizational problem; although there are many applications in the social and natural sciences and also in other areas of mathematics. The module studies the mathematical underpinnings of various algorithms for solving different classes of problem, with a strong emphasis on linearity. Some specific applications are mentioned, but the module deals primarily with theory and methods. The work builds upon the account of the Simplex method given in MAT - 10005; this material will be assumed. |
|
|
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&İs First Theorem which offers a partial converse to Lagrange&İs 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. |
|
|
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. |
| 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-30009 |
Discrete Mathematics |
O |
M
|
7.5 |
15 |
|
|
This module contains a selection of topics in Discrete Mathematics. The motivation is to show how real projective geometry may be generalised to geometries over any field, and then applied, in the case of finite fields, to solve problems in the construction of combinatorial designs and error-correcting codes. Approximately equal emphasis is placed on theory and applications. |
|
|
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-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-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. |
