## 4000-level modules

MT4003 Groups

MT4004 Real Analysis

MT4005 Linear and Nonlinear Waves

MT4111 Symbolic Computation

MT4112 Computing in Mathematics

MT4113 Computing in Statistics

MT4501 Topics in the History of Mathematics

MT4507 Classical Mechanics

MT4508 Dynamical Systems

MT4509 Fluid Dynamics

MT4510 Solar Theory

MT4511 Asymptotic Methods

MT4513 Fractal Geometry

MT4514 Graph Theory

MT4515 Functional Analysis

MT4516 Finite Mathematics

MT4519 Number Theory

MT4526 Topology

MT4527 Time Series Analysis

MT4528 Markov Chains and Processes

MT4530 Population Genetics

MT4531 Bayesian inference

MT4537 Spatial Processes

MT4539 Quantitative Risk Management

MT4551 Financial Mathematics

MT4552 Mathematical Biology 1

MT4553 Theory of Electric and Magnetic Fields

MT4599 Project in Mathematics/Statistics (final year of BSc/MA degree only)

MT4606 Statistical inference

MT4607 Generalized Linear Models and Data Analysis

MT4608 Sampling Theory

MT4609 Multivariate Analysis

MT4614 Design of Experiments

#### MT4003 Groups

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2016/7 |

Timetable | 9.00 am Mon (even weeks), Tue and Thu |

Description | This module introduces students to group theory, which is one of the central fields of the 20th century mathematics. The main theme of the module is classifying groups with various additional properties, and the development of tools necessary in this classification. In particular, the students will meet the standard algebraic notions, such as substructures, homomorphisms, quotients and products, and also various concepts peculiar to groups, such as normality, conjugation and Sylow theory. The importance of groups in mathematics, arising from the fact that groups may be used to describe symmetries of any mathematical object, will be emphasised throughout the module. |

Prerequisites | MT3600 or (MT2002 and MT3501) or MT2505 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10), 1 tutorial and 1 examples class (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | Prof N Ruskuc |

Lecturer | Prof N Ruskuc |

#### Syllabus

- Groups: definitions, examples;
- Subgroups, generating sets, cosets, Lagrange's Theorem;
- Normal subgroups, quotient groups, homomorphisms, kernels and images, the Isomorphism Theorems, the Correspondence Theorem;
- Direct products, the Classification of Finite Simple Groups;
- Simplicity of the alternating groups
*A*_{n}for*n*≥ 5; - The centre, conjugation, commutators, centralizers, normalizers;
- Sylow's Theorem;
- Classification of groups of small order.

#### Assumed knowledge

- It will be assumed that students are familiar with calculating with permutations (as presented in MT2002, MT2505 or MT1003).
- It will be assumed that students have met the definition of a group and some of the basic ideas, though these will be reviewed briefly at the beginning of the course.

#### Reading list

- R.B.J.T. Allenby,
*Rings, Fields and Groups: An Introduction to Abstract Algebra*, Butterworth-Heinemann, 1991: Chapters 5 and 6. - W. Ledermann,
*Introduction to Group Theory*, Oliver and Boyd. - John S. Rose,
*A Course on Group Theory*, Dover 1994: up to Chapter 6. - T.S. Blyth & E.F. Robertson,
*Algebra Through Practice: A Collection of Problems in Algebra with Solutions, Book 5: Groups*, CUP, 1995. - Thomas W. Hungerford,
*Algebra*, Holt, Rinehart and Winston, 1974: Chapters 1 and 2. - I.N. Herstein,
*Topics in Algebra, Second Edition*, Wiley, 1975: Chapter 2. - Derek J.S. Robinson,
*A Course in the Theory of Groups, Second Edition*, Springer, 1996: some parts of the early chapters. - Joseph J. Rotman,
*An Introduction to the Theory of Groups, Fourth Edition*, Springer, 1995: some parts of the early chapters.

#### MT4004 Real and Abstract Analysis

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2016/7 |

Timetable | 11.00 am Mon (even weeks), Tue and Thu |

Description | This module continues the development of real analysis that was begun in MT2502 and continued through MT3502. Topics covered will include limits and continuity in metric spaces, differentiation in higher dimensions and the theoretical underpinning of Fourier series. This module will present some of the highlights of the study of analysis, such as Baire’s Category Theorem, the Contraction Mapping Theorem, the Weierstrass Approximation Theorem, and the Inverse Function Theorem. |

Prerequisites | MT3502 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10), 1 tutorial and 1 examples class (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | Prof L Olsen |

Lecturer | Prof L Olsen |

#### Syllabus

- Metric spaces, continuity and completeness: Recall the basic definitions of metric spaces and continuity; complete metric spaces; examples.
- Applications of complete metric spaces: (1) Contraction Mapping Theorem; examples including, for example, solutions to integral equations. (2) Baire’s Category Theorem; examples including, for example, (i) Existence of continuous and nowhere differentiable functions; (ii) The rationals is not a G
_{δ}set. (3) Weierstrass approximation Theorem - Continuity in metric spaces: (1) Functions of bounded variation; (2) Absolutely continuous functions.
- Differentiation in higher dimensions: (1) Definition of differentiation in higher dimensions and examples; (2) The Mean Value Theorem in
**R**^{n} - Applications of differentiation in higher dimensions: (1) Inverse Function Theorem; (2) The Implicit Function Theorem; (3) Sard’s Theorem.
- Differentiation in one dimension: Lebesgue’s theorem on differentiability almost everywhere
- Fourier Series: Definitions; summability and pointwise convergence of Fourier series and uniform convergence of Fourier series.

#### Reading list

- S Krantz & H Parks, Geometric Integration Theory, Birkhauser, 2008.
- W Rudin, Principles of Mathematical Analysis, Third Edition, McGraw-‐‑Hill, 1976.
- M Spivak, Calculus on Manifolds, 1968.
- K Stromberg, An Introduction to Classical Real Analysis, Wadsworth & Brooks, 1982.

#### MT4005 Linear and Nonlinear Waves

Credits | 15.0 |
---|---|

Semester | 1 |

Academic year | 2016/7 |

Timetable | 11.00 am Mon (even weeks), Tue and Thu |

Description | This module gives an introduction to wave motion and its importance in many areas of applied mathematics. It begins with a discussion of the linear approximation for small amplitude waves and discusses properties of these such as dispersion relations, phase and group velocities, dissipation and dispersion. Some nonlinear effects such as wave steepening are then treated and an introduction given to some of the equations, for example Burger's and Korteweg de Vries, which are used to model nonlinear wave propagation. |

Prerequisites | (MT2003 or MT2506 or PH3081) and (MT3503 or MT3504) |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | Dr A N Wright |

Lecturer | Dr A N Wright |

#### MT4111 Symbolic Computation

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2016/7 |

Timetable | 9.00 am Mon (odd weeks), Wed and Fri |

Description | This module aims to enable students to use a computer as a tool in their other modules and to turn naturally to a computer when solving mathematical problems. The module aims to illustrate the following points: computation allows one to conduct mathematical experiments; computation allows one to collect data about a problem being studied. This is similar to the way other scientists work. It is easier to try several different approaches to a problem and see which works. The computer is not intelligent; intelligence comes from the user. The user thinks, the user interprets, the computer calculates. |

Prerequisites | Any of MT3501-MT3506 |

Antirequisites | MT5611 |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 practical session (weeks 2 - 11) |

Assessment | 2-hour Written Examination = 70%, Coursework = 30% |

Module coordinator | Dr J D Mitchell |

Lecturer | Dr J D Mitchell, Dr C M Roney-Dougal, Dr L Theran |

MT4111 runs in alternate years.

#### MT4112 Computing in Mathematics

Credits | 15.0 |
---|---|

Semester | 1 |

Academic year | 2017/8 |

Timetable | 9.00 am Mon (even weeks), Tue and Thu |

Description | This module is intended to introduce students to FORTRAN and the writing of computer codes to implement mathematical algorithms. The module includes a basic introduction to FORTRAN, and the implementation of mathematical algorithms in a well-documented FORTRAN program. Students are required to complete a project in addition to sitting the examination. |

Prerequisites | either pre- or co-requisites MT3501, MT3503 or MT3504 |

Antirequisites | MT5612, Honours or Joint Honours Programme in Computer Science. |

Lectures and tutorials | 2.5 lectures. |

Assessment | 2-hour Written Examination = 70%, Coursework: Project = 30% |

Module coordinator | TBC |

Lecturer | TBC (2015/6 - Dr D H Mackay, Dr V Archontis) |

MT4112 runs in alternate years.

#### MT4113 Computing in Statistics

Credits | 15.0 |
---|---|

Semester | 1 |

Academic year | 2017/8 |

Timetable | 12.00 noon Mon (odd weeks) and Wed, 12.00 noon–2.00 pm Fri |

Description | The aim of this module is to teach computer programming skills, including principles of good programming practice, with an emphasis on statistical computing. Practical work focusses on the widely-used statistical language R. Practical skills are developed through a series of computing exercises that include 1) modular programming, 2) manipulating data, 3) simulating data with specific statistical properties, and 4) investigating behaviour of statistical procedures under failure of statistical assumptions. |

Prerequisites | pre- or co-requisite MT2508 or MT2004 |

Antirequisites | MT3607 |

Lectures and tutorials | 1.5-hour lectures (x 10 weeks), 2-hour pracitcal classes (x 10 weeks) |

Assessment | 2-hour Written Examination = 40% , Coursework = 60% |

Module coordinator | Dr L J Thomas |

Lecturer | Dr L J Thomas, Dr E Rexstad |

#### Syllabus

- Computer arithmetic; algorithms; introduction to R.
- Control structures.
- Modular programming; programming practice; code testing.
- Computer intensive statistics: bootstrapping; randomisation tests.
- Optimisation.
- Reproducible analysis.

#### MT4501 Topics in the History of Mathematics

Credits | 15.0 |
---|---|

Semester | 1 |

Academic year | 2017/8 |

Timetable | 12.00 noon Mon (odd weeks), Wed and Fri |

Description | The aim of this module is to give students an insight into the historical development of mathematics. Topics to be covered may include some of: the development of algebra, the origins of the calculus, the history of logarithms, the work of some individual mathematicians. |

Prerequisites | either pre- or co-requisites: Any of MT3501 - MT3508 or MT3606 |

Antirequisites | MT5613 |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | Written Examination = 50% (2 x 1-hour class tests), Coursework: Project = 50% |

Module coordinator | TBC |

Lecturer | TBC ( 2015/6 - Dr C P Bleak, Dr C M Roney-Dougal, TBC) |

MT4501 runs in alternate years.

#### MT4507 Classical Mechanics

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2016/7 |

Timetable | 10.00 am Mon (even weeks), Tue and Thu |

Description | The object of this module is to introduce students to some of the ideas and mathematical techniques used in understanding the behaviour of dynamical systems that obey Newton's Laws. These notions are arguably the foundations of physics and applied mathematics. The module will include: Newton's laws of motion; conservative forces; central forces; non-inertial/accelerating frames of reference; dynamics of a system of particles; mechanics of a rigid body; Euler's equations; Lagrange's equations; Hamilton's equations. |

Prerequisites | (MT2003 or MT2503 or PH3081) and MT3504 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | Prof T Neukirch |

Lecturer | Prof T Neukirch |

MT4507 runs in alternate years.

#### MT4508 Dynamical Systems

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2017/8 |

Timetable | 10.00 am Mon (even weeks), Tue and Thu |

Description | This module aims to introduce students to the basic ideas of the modern theory of dynamical systems and to the concepts of chaos and strange attractors. The module will include: period doubling; intermittency and chaos; geometrical approach to differential equations; homoclinic and heteroclinic orbits; Poincaré sections; the Smale horseshoe mapping; centre manifold theory. |

Prerequisites | MT3504 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | TBC |

Lecturer | TBC ( 2015/6- Dr A P Naughton, Prof T Neukirch) |

MT4508 runs in alternate years.

#### MT4509 Fluid Dynamics

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2016/7 |

Timetable | 11.00 am Mon (even weeks), Tue and Thu |

Description | This module provides an introduction to the theory of incompressible fluid dynamics, which describes the motion of liquids and gases at speeds small compared to the sound speed. Special attention is paid to a precise foundation of the various conservation laws that govern fluid dynamics, as this provides a convenient framework in which to study specific examples as well as extensions of the basic theory. |

Prerequisites | (MT2506 and MT3504) or MT3601 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | Written Examination = 100% (2-hour final exam = 90%, class test = 10%) |

Module coordinator | Dr M Carr |

Lecturer | Dr M Carr |

#### MT4510 Solar Theory

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2016/7 |

Timetable | 11.00 am Mon (odd weeks), Wed and Fri |

Description | The object of this module is to describe the basic dynamic processes at work in the Sun, a subject which is being enlivened by dramatic new results from space missions. |

Prerequisites | (MT2506 and MT3504) or MT3601 |

Antirequisites | MT4504, MT5804 |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | Prof I De Moortel |

Lecturer | Prof I De Moortel |

#### MT4511 Asymptotic Methods

Credits | 15.0 |
---|---|

Semester | 1 |

Academic year | 2016/7 |

Timetable | 9.00 am Mon (even weeks), Tue and Thu |

Description | This module is designed to introduce students to asymptotic methods used in the construction of analytical approximations to integrals and solutions of differential equations. |

Prerequisites | MT3504 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | Dr A Wilmot-Smith |

Lecturer | Dr A Wilmot-Smith |

MT4511 runs in alternate years.

#### Syllabus

- Introduction to ideas of expansion using a small parameter, review of basic expansions, definition of order symbols;
- Expansions of integrals, Watson's lemma, the Laplace method and method of stationary phase;
- Asymptotic methods for differential equations, multiple scales;
- Matched asymptotic expansions;
- The WKB method.

#### Assumed knowledge

- A basic understanding of limits and convergence.
- A good knowledge of basic expansions is essential - for example Taylor series, Binomial series.
- Students must also be able to solve simple o.d.e.s and be able to do basic integration.

#### Reading list

- Perturbation Methods - E.J. Hinch
- Asymptotic Analysis - J.D. Murray
- Pertubation Methods for Engineers and Scientists - C.M. Bender & S.A. Orszag
- Perturbation Methods for Engineers and Scientists - A.W. Bush
- Pertubation Methods - A.H. Nayfeh

#### MT4513 Fractal Geometry

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2017/8 |

Timetable | 12.00 noon Mon (even weeks), Tue and Thu |

Description | The aim of this module is to introduce the mathematics used to describe and analyse fractals and to show how the theory may be applied to examples drawn from across mathematics and science. The module discusses the philosophy and scope of fractal geometry; and may include topics such as dimension, representation of fractals by iterated function systems, fractals in other areas of mathematics such as dynamical systems and number theory, Julia sets and the Mandelbrot set. |

Prerequisites | (MT2503 or MT2001) and any one of MT3501 - MT3504 |

Antirequisites | MT5813 |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | TBC |

Lecturer | TBC ( 2015/6 - Prof K J Falconer) |

MT4513 runs in alternate years.

The module will introduce the mathematics used to describe and analyse fractals, and to show how the theory may be applied to examples drawn from across mathematics and science. The approach will have a geometric emphasis and the treatment will precise though not over-formal.

#### Syllabus

- The philosophy and scope of fractal geometry.
- The concept of dimension.
- Box-counting and Hausdorff dimensions, their properties and calculation in simple cases.
- Representation of fractals by iterated function schemes, relation to dimensions and applications
- Fractals in dynamical systems, number theory, and other areas of mathematics.
- Iteration of complex functions, Julia sets and the Mandelbrot set.

#### Reading list

Books providing an overview of fractal geometry:

- Fractals - A Very Short Introduction, Kenneth Falconer (Oxford U.P., 2013)
- Introducing Fractals - A Graphic Guide, N. Lesmoir-Gordon, W. Rood & R. Edny (Icon Books, 2009)
- The Fractal Geometry of Nature, Benoit Mandelbrot (W.H. Freeman, 1982)

Books at the level of the course

- Fractal Geometry: Mathematical Foundations and Applications, Kenneth Falconer (John Wiley, 3rd Ed, 2014)
- Chaos and Fractals, H.-O. Peitgen, H. Jurgens & D. Saupe (Springer-Verlag, 1992)
- Fractals Everywhere, Michael Barnsley (Academic Press, 2nd Ed, 1993)
- Measure, Topology and Fractal Geometry, G.A. Edgar (Springer-Verlag, 1990)

#### MT4514 Graph Theory

Credits | 15.0 |
---|---|

Semester | 1 |

Academic year | 2016/7 |

Timetable | 10.00 am Mon (even weeks), Tue and Thu |

Description | The aim of this module is to introduce students to the study of graph theory as a tool for representing connections between data. Topics to be covered may include: basic theory and applications, Eulerian graphs, Hamiltonian graphs, planar graphs, spanning trees and applications, networks, matching problems. |

Prerequisites | MT1003 or MT2504 or MT2005 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | Prof N Ruskuc |

Lecturer | Prof N Ruskuc |

MT4514 runs in alternate years.

#### Syllabus

- graphs: definition of simple, directed, labelled, and weighted graphs
- subgraphs, graph homomorphisms, graph homeomorphisms
- complete graphs, bipartite graphs, trees
- shortest paths
- Eulerian cycles, Hamiltonian cycles, planarity, colourability
- spanning trees
- Hall's marriage theorem
- NP complete problems

#### Assumed knowledge

- It will be assumed that students are familiar with abstract definitions of mathematical objects and basic formal proofs.

#### MT4515 Functional Analysis

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2016/7 |

Timetable | 12.00 noon Mon (even weeks), Tue and Thu |

Description | This object of this module is to familiarise students with the basic notions of functional analysis, that is analysis on normed spaces and Hilbert space. The module will cover normed spaces, convergence and completeness, operators, Hilbert spaces and may include topics such as spectral theory and the Hahn-Banach theorem. |

Prerequisites | MT2002 or (MT2501 and MT2502) |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | Prof K J Falconer |

Lecturer | Prof K J Falconer |

MT4515 runs in alternate years.

#### Syllabus

- Normed spaces and examples.
- Bounded linear operators on norm spaces.
- The operator norm.
- Completeness.
- The contraction mapping theorem.
- Finite dimensional spaces.
- Spaces of continuous functions.
- Banach spaces.
- Inner product spaces and Hilbert spaces.
- The spectrum of a bounded linear operator.
- Compact operators.
- The spectral theorem for compact self-adjoint operators.

#### Reading list

- Rynne, B.P. & Youngson, M.A., Linear Functional Analysis, Springer, 2nd Ed 2007.
- Young, N., An Introduction to Hilbert Space, Cambridge University Press, 1988.
- Griffel, D.H., Applied Functional Analysis, Dover, 2nd Ed 2002.
- Bollobas, B, Linear Analysis: An Introductory Course, Cambridge University Press, 2nd Ed 1999.

#### MT4516 Finite Mathematics

Credits | 15.0 |
---|---|

Semester | 1 |

Academic year | 2017/8 |

Timetable | 10.00 am Mon (even weeks), Tue and Thu |

Description | The aim of this module is to introduce students to some topics in the mathematics of combinatorial structures. This theory has wide applications, both in classical mathematics and in theoretical computer science. Topics to be covered may include: coding theory, finite geometries, Latin squares, designs. |

Prerequisites | MT2504 or MT2505 or MT2002 or MT2005 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | TBC |

Lecturer | TBC (2015/6 - Dr C M Roney-Dougal, Prof I Rivin) |

MT4516 runs in alternate years.

#### MT4519 Number Theory

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2017/8 |

Timetable | 10.00 am Mon (even weeks), Tue and Thu |

Description | The aim of this module is to introduce students to some important topics in number theory. Topics to be covered may include: prime numbers, cryptography, continued fractions, Pell's equation, the Gaussian integers and writing numbers as sums of squares. |

Prerequisites | (MT2505 or MT2002) and one of MT3501 - MT3505 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | TBC |

Lecturer | TBC - (2015/6 - Dr C P Bleak) |

MT4519 runs in alternate years.

#### MT4526 Topology

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2017/8 |

Timetable | 10.00 am Mon (odd weeks), Wed and Fri |

Description | This module introduces the ideas of metric and topological spaces. A metric space is simply a set together with a 'distance' between any two points. This idea is pervasive in mathematics: from situations such as the usual distance in n-dimensional space, to the Hamming distance between words in an error-correcting code and the distance between functions approximating a given function. Metric spaces can be thought of as particular instances of topological spaces, where the fundamental concept is that of points being 'close' to each other rather than the precise distance between points. Topological spaces are a powerful generalisation of metric spaces, and have had a profound influence in the development of mathematics. Many examples of metric spaces and topological spaces will be introduced and fundamental ideas within topology will be discussed, including separation axioms, compactness and connectedness. |

Prerequisites | MT2002 or MT2502 or MT3600 or MT4004 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | TBC |

Lecturer | TBC (2015/6 - Prof P J Cameron, Dr J D Mitchell) |

MT4526 runs in alternate years.

#### Syllabus

- Metric spaces, convergence and continuity in metric spaces, complete metric spaces;
- Topological spaces, continuity in topological spaces, separation axioms;
- Subspaces, product spaces and quotient spaces;
- Compactness, the Heine-Borel Theorem, and Tychonoff's Theorem;
- Connectedness.

Additional topics may also be covered, for instance, the Contraction Mapping Theorem, and Baire's Category Theorem.

#### Reading list

- Sutherland, Introduction to Metric and Topological Spaces
- Munkres, Topology: a first course
- Willard, General Topology

#### MT4527 Time Series Analysis

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2016/7 |

Timetable | 10.00 am Mon (even weeks), Tue and Thu |

Description | This module provides an introduction to univariate linear times series models (ARIMA processes) and univariate non-linear times-series models (ARCH and GARCH). The syllabus includes: forecasting methods for constant mean and trend models, the ARIMA class of models (including seasonal ARIMA models), fitting and forecasting ARIMA models, ARCH and GARCH processes. |

Prerequisites | MT2004 or MT2508 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 0.5 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | Dr V M Popov |

Lecturer | Dr V M Popov |

MT4527 runs in alternate years.

#### Syllabus

- Introduction
- Basic time series models
- Models with trend and seasonality
- ARMA models
- ARIMA models
- ARCH and GARCH models

#### Reading list

- P. Brockwell & P. Davis:
*Introduction to Time Series and Forecasting*, Springer Texts in Statistics, 2002 - R. Tsay:
*Analysis of Financial Time Series*, 3rd ed., Wiley, 2010 - D.C. Montgomery, , C.L. Jennings & Kulahci, M.:
*Introduction to Time Series Analysis and Forecasting*, Wiley-Interscience, 2008 - Box, G.E.P., Jenkins, G.M. & Reinsel, G.:
*Time Series Analysis -**Forecasting and Control*, 4th ed., Wiley Series in Probability and Statistics, 2008

#### MT4528 Markov Chains and Processes

Credits | 15.0 |
---|---|

Semester | 1 |

Academic year | 2017/8 |

Timetable | 11.00 noon Mon (even weeks), Tue and Thu |

Description | This module provides an introduction to the theory of stochastic processes and to their use as models, including applications to population processes and queues. The syllabus includes the Markov property, Chapman-Kolmogorov equations, classification of states of Markov chains, decomposition of chains, stationary distributions, random walks, branching processes, the Poisson process, birth-and-death processes and their transient behaviour, embedded chains, Markovian queues and hidden Markov models. |

Prerequisites | MT2504 or MT2004 |

Antirequisites | MT3706 |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 8 tutorials over the semester. |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | TBC |

Lecturer | TBC (2015/6 Dr V Popov) |

MT4528 runs in alternate years.

#### Syllabus

- Preliminaries: Introduction to the ideas of stochastic processes and models (including index sets and state space); brief review of probability theory; probability generating functions.
- Markov chains: Definition; directed graphs; transition probability matrices; Chapman-Kolmogorov equations; classification of states; decomposition of states; periodicity; limiting distributions; stationary distributions.
- Random walks: gambler's ruin; probability of ruin; absorption probabilities.
- Branching processes: Galton-Watson process; extinction probabilities; criticality theorem.
- Markov processes: continuous-time Markov chains; Poisson process; birth/death process; Kolmogorov forward equation.
- Queueing systems: Kendall notation; special queues.
- Hidden Markov models: model formulation; basic properties; applications.

#### Reading list

- Jones, P. W. and Smith, P.
*Stochastic processes : an introduction*, Arnold/OUP. - Stirzaker, D. R.
*Stochastic processes and models*, OUP. - Ross, S. M.
*Introduction to probability models, 9th ed*., Academic Press. - Norris, J.R.
*Markov chains*, CUP. - Grimmett, G. R. and Stirzaker, D. R.
*Probability and random processes, 3rd ed*., OUP. - Taylor, H. M. and Karlin, S.
*An introduction to stochastic modeling, 3rd ed*., Academic Press. - Zucchini, W. and MacDonald, I. L.
*Hidden Markov models for time series: an introduction using R*. Chapman & Hall.

#### MT4530 Population Genetics

Credits | 15.0 |
---|---|

Semester | 1 |

Academic year | 2017/8 |

Timetable | 9.00 am Mon (even weeks), Tue and Thu |

Description | This module aims to show how the frequencies of characteristics in large natural populations can be explained using mathematical models and how statistical techniques may be used to investigate model validity. The syllabus includes: Mendel's First and Second Laws, random mating and random union of gametes, Hardy-Weinberg equilibrium, linkage, inbreeding, assortative mating, X-linked loci, selection and mutation. |

Prerequisites | MT2004 or MT2508 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 0.5 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | TBC |

Lecturer | TBC (2015/6 - Dr I B J Goudie) |

MT4530 runs in alternate years.

#### MT4531 Bayesian Inference

Credits | 15.0 |
---|---|

Semester | 1 |

Academic year | 2016/7 |

Timetable | 10.00 am Mon (even weeks), Tue and Thu |

Description | This module is intended to offer a re-examination of standard statistical problems from a Bayesian viewpoint and an introduction to recently developed computational Bayes methods. The syllabus includes Bayes' theorem, inference for Normal samples; univariate Normal linear regression; principles of Bayesian computational, Markov chain Monte Carlo - theory and applications. |

Prerequisites | MT3507 or MT3606 |

Antirequisites | MT5831 |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 8 tutorials/practical classes over the semester. |

Assessment | 2-hour Written Examination = 80%, Coursework = 20% |

Module coordinator | Dr L Thomas |

Lecturer | Dr L Thomas |

#### MT4537 Spatial Processes

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2017/8 |

Timetable | 10.00 am Mon (even weeks), Tue and Thu |

Description | This module will study probabilistic and inferential problems for spatial processes. It commences with a discussion on different types of spatial data. In the context of spatial point processes functional and non-functional summary characteristics for point patterns are considered. Spatial point process models, including homogeneous and inhomogeneous Poisson processes as well as Gibbs processes and Cox processes along with the approaches to parameter estimation and model evaluation, are introduced. Models in geostatistics based on empirical variograms and kirging approaches and spatial models for lattice data (CAR model, Gauss Markov random fields) are also discussed. |

Prerequisites | MT3507 or MT3606 |

Antirequisites | MT4536 |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 4 tutorials over the semester. |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | TBC |

Lecturer | TBC (2015/6 - Dr J B Illian, Dr V M Popov) |

MT4537 runs in alternate years.

#### Syllabus

- Types of spatial data
- Spatial point processes, exploratory analysis (intensity estimation, indices as summary characteristics, K-function, pair correlation function)
- Spatial point processes, modelling (homogeneous and inhomogeneous Poisson process, cluster models, Gibbs processes, Cox processes)
- Geostatistics (semivarigrams, exponential, Gaussian, power model, anisotropy)
- Spatial models for lattice data (CAR model, Gauss Markov random fields)

#### MT4539 Quantitative Risk Management

Credits | 15 |
---|---|

Semester | 2 |

Academic year | 2017/8 |

Timetable | 12.00 noon Mon (odd weeks), Wed, Fri and 2.00 pm Fri |

Description | The module introduces the concept of financial risk and discusses the importance of its regulation. The emphasis is laid on the popular risk measure Value at Risk (VaR). After a brief discussion on asset returns, various modelling techniques - ranging from the simple Historical Simulation to the more advanced ARMA and GARCH models - are presented and applied for the calculation of VaR using real financial data. The aim of this module is to provide a solid basis in risk management for those students considering a career in finance. |

Prerequisites | MT2504, MT2508 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (x 10 weeks), 5 tutorials and 5 practical sessions. |

Assessment | 2-hour Written Examination = 80%, Coursework = 20% |

Module coordinator | Dr V Popov |

Lecturer | Dr V Popov |

#### MT4551 Financial Mathematics

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2017/8 |

Timetable | 10.00 am Mon (odd weeks), Wed and Fri |

Description | Students are introduced to the application of mathematical models to financial instruments. The course will include an overview of financial markets and the terminology in common usage but the emphasis will be on the mathematical description of risk and return as a means of pricing contracts and options. |

Prerequisites | (MT2001 or MT2503) and (MT1007 or MT2004 or MT2504 or EC2003) and MT3504 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | TBC |

Lecturer | TBC (2015/6 - Dr D H Mackay) |

MT4551 runs in alternate years.

#### MT4552 Mathematical Biology 1

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2016/7 |

Timetable | 9.00 am Mon (even weeks), Tue and Thu |

Description | This module will explore real world applications of mathematics to biological problems e.g. harvesting of fish stocks, host-parasitoid systems, predator-prey dynamics, molecular interactions. The mathematical techniques used in the modelling will be nonlinear difference equations and ordinary differential equations. The module will be useful to students who wish to specialise in Applied Mathematics in their degree programme. |

Prerequisites | MT3504 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 1 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 90%, Coursework (Class Test) = 10% |

Module coordinator | Dr C Venkataraman |

Lecturer | Dr C Venkataraman |

#### Continuous assessment

50-minute class test: 10% of final mark

#### Syllabus

- Difference equation models of single species (e.g. insect populations; difference equation difference equation models of interacting species models (e.g. host-parasitoid systems, plant-herbivore systems); delay-difference equation models.
- Differential equation models of single species (including harvesting); differential equation models of interacting species (e.g. Lotka-Volterra system; predator-prey systems); Limit cycles and the Hopf bifurcation theorem.
- Introductory Systems Biology - intracellular signalling modelling and biochemical reactions; Law of Mass Action; Michaelis-Menten kinetics; Quasi-Steady State Assumption; Matched asymptotics; Metabolic pathways; Autocatalysis and activator-inhibitor systems.

#### Reading list

- N.F. Britton,
*Essential Mathematical Biology*, (Springer 2003). - J.D. Murray,
*Mathematical Biology I: An Introduction*, (Springer, 3rd ed. 2003). - J.D. Murray,
*Mathematical Biology II: Spatial Models and Biomedical Applications*, (Springer, 3rd ed. 2003). - L. Edelstein-Keshet,
*Mathematical Models in Biology*, (SIAM Classics in Applied Mathematics, SIAM Publishing 2005).

#### MT4553 Theory of Electric and Magnetic Fields

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2016/7 |

Timetable | 10.00 am Mon (odd weeks), Wed, Fri |

Description | The module will consider the mathematical and physical principles that describe the theory of electric and magnetic fields. It will first describe the basic principles of electrostatics and magneto-statics and following this electrodynamics. Next Maxwell's equations are described along with the properties of electro-magnetic waves in a variety of media. Finally an application to the area of plasma physics is carried out through considering the orbits of charged particles in a variety of spatially and time varying magnetic fields. |

Prerequisites | MT2503, MT2506 and MT3504 |

Antirequisites | PH3007 |

Lectures and tutorials | 2.5 lectures (x 10 weeks) and 1 tutorial (x 10 weeks). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | Dr D Mackay |

Lecturer | Dr D Mackay |

#### MT4599 Project in Mathematics / Statistics

Credits | 15.0 |
---|---|

Semester | Whole Year |

Academic year | 2016/7 |

Timetable | none |

Description | The student will choose a project from a list published annually although a topic outwith the list may be approved. Students will be required to report regularly to their supervisor and a report of no more than 5,000 words must be submitted by the end of the April. |

Prerequisites | |

Antirequisites | |

Lectures and tutorials | Typically and on average, 20 mins of project supervisions per week over whole year. |

Assessment | Coursework = 100%: Project = 80%, Presentation = 20% |

Module coordinator | Prof C E Parnell |

Lecturer |

Booklet for MT4599 BSc/MA Honours projects 2017-2018 (PDF, 478 KB)

and the Project allocation form 2017-2018 (PDF, 70 KB)

Booklet for MT4599 BSc/MA Honours projects 2016-2017 (PDF, 470 KB)

and the Project allocation form 2016-2017 (PDF, 68 KB)

#### MT4606 Statistical Inference

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2017/8 |

Timetable | 10.00 am Mon (odd weeks), Wed and Fri |

Description | This module aims to show how the methods of estimation and hypothesis testing met in MT2004 and MT3606 can be justified and derived; to extend those methods to a wider variety of situations. The syllabus includes: comparison of point estimators; the Rao-Blackwell Theorem; distribution theory; Fisher information and the Cramer-Rao lower bound; maximum likelihood estimation; hypothesis-testing; confidence sets. |

Prerequisites | MT3507 or MT3606 |

Antirequisites | MT3701, MT5701 |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 0.5 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | TBC |

Lecturer | TBC (2015/6 - Dr I B J Goudie) |

MT4606 runs in alternate years.

#### MT4607 Generalised Linear Models and Data Analysis

Credits | 15.0 |
---|---|

Semester | 1 |

Academic year | 2016/7 |

Timetable | 9.00 am Mon (even weeks), Tue and Thu |

Description | This module aims to demonstrate the power and elegance of unifying a large number of simple statistical models within the general framework of the generalised linear model. It will train students in the interpretation, analysis and reporting of data, when a single response measurement is interpreted in terms of one or a number of other variables. |

Prerequisites | (MT2001 or MT2503), (MT2004 or MT2508) and either pre- or co-requisite MT3501 |

Antirequisites | MT5753 |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 8 tutorials over the semester |

Assessment | 2-hour Written Examination = 80%, Coursework: Project = 20% |

Module coordinator | Dr M Papathomas |

Lecturer | Dr M Papathomas |

MT4607 runs in alternate years.

#### Syllabus

- Revision of ordinary linear models
- Exponential family of distributions
- Formulation of generalized linear models
- Concept of link function
- Iterated weighted least squares algorithm
- Inference
- Model selection
- Diagnostics

#### Reading list

- Dobson, A. (2008). An Introduction to Generalized Linear Models, Third Edition. Chapman & Hall/CRC
- McCullagh, P., Nelder, J. (1989). Generalized Linear Models, Second Edition. Chapman and Hall/CRC

#### MT4608 Sampling Theory

Credits | 15.0 |
---|---|

Semester | 1 |

Academic year | 2016/7 |

Timetable | 10.00 am Mon (odd weeks), Wed and Fri |

Description | The aims of this module are to introduce students to and interest them in the principles and methods of design-based inference, to convince them of the relevance and utility of the methods in a wide variety of real-world problems, and to give them experience in applying the principles and methods themselves. By the end of the module students should be able to recognise good and poor survey design and analysis, to decide upon and implement the main types of survey design in relatively straightforward settings, and analyse the resulting survey data appropriately. The syllabus includes fundamentals of design based vs model-based inference, simple random sampling, sampling with replacement, ratio and regression estimators, stratified sampling, cluster sampling and unequal probability sampling. |

Prerequisites | MT2004 or MT2508 Pre- or co-requisite: One of MT3501, MT3503, MT3504, MT3606 or any 3000-level MN module |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 8 tutorials over the semester. |

Assessment | 2-hour Written Examination = 85%, Coursework: Project = 15% |

Module coordinator | Dr J B Illian |

Lecturer | Dr J B Illian |

MT4608 runs in alternate years.

#### MT4609 Multivariate Analysis

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2016/7 |

Timetable | 10.00 am Mon (odd weeks), Wed and Fri |

Description | This module aims to introduce students to the ideas and techniques of multivariate statistical analysis. The syllabus includes mean vectors, covariance matrices, correlation matrices; basic properties of multivariate normal distributions; checking multivariate normality; the likelihood ratio and union-intersection principles for constructing multivariate tests; the one-sample and two-sample Hotelling's T-squared tests; tests on covariance matrices, tests of independence; linear discriminant analysis; principal components analysis; canonical correlation. |

Prerequisites | MT3507 or MT3606 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and 0.5 tutorial (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 100% |

Module coordinator | Dr I B J Goudie |

Lecturer | Dr I B J Goudie |

MT4609 runs in alternate years.

#### MT4614 Design of Experiments

Credits | 15.0 |
---|---|

Semester | 2 |

Academic year | 2016/7 |

Timetable | 9.00 am Mon (odd weeks), Wed and Fri |

Description | This module introduces a wide range of features that occur in real comparative experiments, such as choice of blocks and replication as well as type of design. It includes enough about the analysis of data from experiments to show what has to be considered at the design stage. It includes consultation with the scientist and interpretation of the results. |

Prerequisites | (MT2004 or MT2508) and MT3501 |

Antirequisites | |

Lectures and tutorials | 2.5 lectures (weeks 1 - 10) and either tutorial or practical (weeks 2 - 11). |

Assessment | 2-hour Written Examination = 80%, Presentation = 10%, Coursework = 10% |

Module coordinator | Prof R A Bailey |

Lecturer | Prof R A Bailey |

#### Syllabus

- Introduction to concepts in the design of real comparative experiments.
- Randomization, replication, power.
- Simple linear model, orthogonal subspaces, analysis of variance.
- Blocking. Fixed effects or random effects. Orthogonal designs.
- Factorial designs. Main effects and interactions. Control treatments.
- Row-column designs. Latin squares.
- Observational units smaller than experimental units. False replication.
- Split-plot designs. Treatment effects in different strata.
- Structures defined by families of orthogonal factors. Eigenspaces of highly structured variance-covariance matrices.
- Showing factors on a Hasse diagram. Using the Hasse diagram to calculate degrees of freedom and allocate treatment effects to strata. Skeleton analysis of variance.

#### Reading list

- R. A. Bailey, Design of Comparative Experiments, CUP, 2008.
- D. R. Cox, Planning of Experiments, Wiley 1992.
- G. W. Cobb, Introduction to Design and Analysis of Experiments, Springer, 1998.