Modules

These are the Level 7 (MSc level) modules in Mathematics and in Statistics offered by the School of Mathematical Sciences in 2011-12. The precise selection of modules offered varies somewhat from year to year. Please note that MSc Mathematics students may also take some Astronomy MSc modules offered by the School of Physics and Astronomy.

Advanced Combinatorics (MTH742P)

  • Semester B in 2011-12
  • Credit value: 15
  • Level: 7
  • Lecturer: Prof T Müller
  • Subject area: Combinatorial Mathematics
  • Website: tbc

This module builds on the combinatorial ideas of the modules Combinatorics and Extremal Combinatorics and introduces some of the more advanced tools for solving combinatorial and graph theoretic problems. The topics covered will depend on the module organiser's expertise but significant emphasis will be on the techniques used as well as the results proved.

Syllabus

  • This module aims to introduce students to some of the more advanced techniques used in combinatorics such as the Regularity Lemma, probabilistic techniques, the discrete Fourier transform, eigenvalue methods and generating functions with the intention that the students will be able to recognise and then apply the appropriate tools in unfamiliar situations.

Learning resources

Main texts:

  • P. J. Cameron, Enumerative and Asymptotic Combinatorics, Lecture notes.
  • P. Flajolet and R. Sedgwick, Analytic Combinatorics, Cambridge University Press.

Applied Statistics (MTHM002)

Syllabus

The semester will be divided into three four-week 'months'. In each month there is a genuine piece of applied statistics, led by a different lecturer. The lecturer will set it up with at most two lectures. At the end of the month the student will hand in a report of 10-15 pages. Statistical techniques and statistical computing packages from previous statistics courses will be needed. The three topics will be chosen from the following list:

  • Designed experiments
  • Medical statistics
  • Time series analysis of spacecraft data
  • Multivariate data from crop research
  • Agricultural statistics
  • Economic statistics
  • Industrial statistics

Bayesian statistics (MTHM042)

The module aims to introduce you to the Bayesian paradigm. The module will show you some of the problems with frequentist statistical methods, show you that the Bayesian paradigm provides a unified approach to problems of statistical inference and prediction, enable you to make Bayesian inferences in a variety of problems, and illustrate the use of Bayesian methods in real-life examples.

Syllabus

  • The Bayesian paradigm: likelihood principle, sufficiency and the exponential family, conjugate priors, examples of prior to posterior analysis, mixtures of conjugate priors, non-informative priors, two sample problems, predictive distributions, constraints on parameters, point and interval estimation,hypothesis tests, nuisance parameters.
  • Linear models: use of non-informative priors, normal priors, two and three stage hierarchical models, examples of one way model, exchangeability between regressions, growth curves, outliers and influential observations.
  • Approximate methods: normal approximations to posterior distributions, Laplace’s method for calculating ratios of integrals, Gibbs sampling, finding full conditionals, constrained parameter and missing data problems, graphical models. Advantages and disadvantages of Bayesian methods.
  • Examples: appropriate examples will be discussed throughout the course. Possibilities include epidemiological data, randomized clinical trials, radiocarbon dating.

Learning resources

Main text:

  • P.M. Lee, Bayesian statistics: An Introduction, (3rd Edition), Edward Arnold.

Complex Systems (MTH734P)

Complex systems can be defined as systems involving many coupled units whose collective behaviour is more than the sum of the behaviour of each unit. Examples of such systems include coupled dynamical systems, fluids, transport or biological networks, interacting particle systems, etc. The aim of this module is to introduce students with a number of mathematical tools and models used to study complex systems and to explain the mathematical meaning of key concepts of complexity science, such as self-similarity, emergence, and self-organisation. The exact topics covered will depend on the module organiser's expertise with a view to cover practical applications using analytical and numerical tools drawn from other applied modules.

Syllabus

  • Introduction to the field of complex systems via a number of representative examples and models of these systems (e.g., coupled dynamical systems, time-delayed systems, stochastic processes, networks, time series, fractals, multifractals, particle models).
  • Introduction to basic tools and quantities used in the study of complex systems (e.g., bifurcation diagram, symbolic dynamics, dimensions, Lyapunov exponents, complexity measures, entropies).
  • Introduction to the concepts of emergence and self-organisation in the context of basic models of complex models.
  • Introduction to basic computational and numerical methods used to study complex systems.

Computational Statistics (MTHM731)

This module introduces modern methods of statistical inference for small samples, which use computational methods of analysis, rather than asymptotic theory. Some of these methods such as permutation tests and bootstrapping, are now used regularly in modern business, finance and science.

Syllabus

The techniques developed will be applied to a range of problems arising in business, economics, industry and science. Data analysis will be carried out using the user-friendly, but comprehensive, statistics package R.

  • Probability density functions: the empirical cdf; q-q plots; histogram estimation; kernel density estimation.
  • Nonparametric tests: permutation tests; randomization tests; link to standard methods; rank tests.
  • Data splitting: the jackknife; bias estimation; cross-validation; model selection.
  • Bootstrapping: the parametric bootstrap; the simple bootstrap; the smoothed bootstrap; the balanced bootstrap; bias estimation; bootstrap confidence intervals; the bivariate bootstrap; bootstrapping linear models.

Learning resources

  • J. Gentle, Elements of Computational Statistics, Springer.

Dynamical Systems (MTH744P)

A dynamical system is any system which evolves over time according to some pre-determined rule. The goal of dynamical systems theory is to understand this evolution. This module develops the theory of dynamical systems systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations and chaos. Much emphasis is placed on applications.

Syllabus

  • First-order differential equations (one-dimensional flows): linear and nonlinear equations, graphical solutions, bifurcations.
  • Two-dimensional flows: phase plane, stability of fixed points, periodic solutions, and limit cycles. Introduction to bifurcation theory, local and global bifurcations. Tools for studying global behavior of flows: Lyapunov functions, Poincare-Bendixson Theorem, gradient flows.
  • Three-dimensional flows: Lyapunov exponents, Poincare sections, strange attractors, chaos.

Learning resources

  • Steven H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Studies in Nonlinearity)

Extremal Combinatorics (MTHM044)

An extremal result is one which determines the extreme value of some parameter over a class of combinatorial structures. A classical example is Mantel's theorem which answers the question: how many edges can a graph have without it containing a triangle. Another instance is the Erdos-Ko-Rado theorem which answers the question: how large can a family of r-subsets of an n-set be if any two of them have non-empty intersection. This module will cover results of this flavour on both graphs and set systems. There will be an emphasis on techniques as well as results, including the use of linear algebra, probabilistic methods and compressions.

Syllabus

  • Extremal Graph Theory: introduction (what is an extremal problem/result, some simple examples). Cycles (Dirac's theorem). Complete Graphs (Turan's theorem). Zarankiewicz problem (bipartite analogue of Turan). Erdos-Stone theorem.
  • The Discrete Cube: Sperner's theorem. Shadows and isoperimetric inequalities (LYM inequality, the Kruskal-Katona theorem, Harper's theorem, edge isoperimetric inequality).
  • Intersecting Families: Erdos-Ko-Rado theorem. Katona's t-intersecting theorem. Brief discussion of uniform t-intersecting problem (with statement but not proof of Ahlswede and Khachatrian's complete intersection theorem). Modular intersections (Frankl-Wilson theorem and some extensions and applications).
  • Other Topics: other topics of a similar flavour chosen according to class interest and time.

Learning resources

The lecture notes will be self contained. Examples of books giving background material and further reading are:

  • B Bollobás, Combinatorics, Cambridge University Press, 1986
  • B Bollobás, Modern Graph Theory, Springer-Verlag, 1998
  • S Jukna, Extremal Combinatorics: With Applications in Computer Science, Springer-Verlag, 2001

Further Topics in Algebra (MTH745P)

  • Semester B in 2011-12
  • Credit value: 15
  • Level: 7
  • Lecturer: Prof A McIntyre
  • Subject area: Algebra
  • Website: tbc

This module provides exposure to advanced techniques in algebra at an MSc or MSci level. Algebra encompasses familiar objects such as integers, fields, polynomial rings and matrices and has applications throughout mathematics including to geometry, number theory and topology. The module will complement the algebra module offered in Semester A and will cover topics either in commutative or noncommutative algebra. Included will be basic definitions and theorems in either case, normally with rings or fields as a starting point.

Syllabus

The aim of this module is to expose students to advanced techniques in algebra, which complement those presented in the module Group Theory. The module is also seen as a way to prepare students to study more advanced algebra subjects at PhD level.

The topics covered will depend on the expertise of the lecturer. These could be drawn from the theory of rings and their modules, Galois theory or elements of algebraic geometry. Commutative algebra or noncommutative algebra could also be covered.

Group Theory (MTHM024)

This module provides an introduction to advanced group theory. The aim is to explore the theory of finite groups by studying important examples in detail, such as simple groups. In particular, the projective special linear groups over small fields provide a rich vein of interesting cases on which to hang the general theory.

Syllabus

  • Revision of basic group theory, isomorphism theorems, Jordan-Holder theorem, Sylow's theorems, the structure theorem for finite abelian groups.
  • Permutation groups: transitivity, primitivity, symmetric and alternating groups. Maximal subgroups, wreath products, Iwasawa's Lemma. The outer automorphism of S6.
  • Linear groups: finite fields, general linear groups, projective special linear groups. Projective lines and isomorphisms of some projective special linear groups with alternating groups. Simplicity of PSL n(q).

Mathematical Statistics (MTHM736)

Syllabus

This module covers the classical theory of statistical inference and probability theory which are required for more advanced study in statistics. It is aimed at mathematicians who have done little statistics in their undergraduate studies. It will cover material approximately equivalent to two undergraduate modules, at a fast pace and taking a mathematical approach.

Topics covered include: conditional probability; hypothesis testing; distribution theory; estimation; multivariate normal distribution; Laws of Large Numbers and the Central Limit Theorem; confidence intervals; general theory of testing; matrix algebra; least squares; Gauss-Markov Theorem.

Learning resources

  • Rice, Mathematical Statistics and Data Analysis, Duxbury, 1994.
  • Wackerley, Mendenhall and Schaeffer, Mathematical Statistics with Applications, Duxbury, 2002.

Measure Theory and Probability (MTHM007)

This is an introductory module on the Lebesgue theory of measure and integral with application to probability. You are expected to know the theory of Riemann integration. Measure in the line and plane, outer measure, measurable sets, Lebesgue measure, non-measurable sets. Sigma-algebras, measures, probability measures, measurable functions, random variables. Simple functions, Lebesgue integration, integration with respect to general measures. Expectation of random variables. Monotone and dominated convergence theorems, and applications. Absolute continuity and singularity, Radon-Nikodym theorem, probability densities. Possible further topics: product spaces, Fubini's theorem.

Syllabus

  • Measure in the line and plane, outer measure, measurable sets, Lebesgue measure, non-measurable sets.
  • Sigma-algebras, measures, probability measures, measurable functions, random variables.
  • Simple functions, Lebesgue integration, integration with respect to general measures. Expectation of random variables. Monotone and dominated convergence theorems, and applications.
  • Absolute continuity and singularity, Radon-Nikodym theorem, probability densities.
  • Possible further topics: product spaces, Fubini's theorem.

Relativity and Gravitation (MTHM033)

Introduction to general relativity. Derivation from basic principles of the Schwarzschild, Reisner-Nordstrom, Kerr and Kerr-Neuman solutions of Einstein's field equations. Physical aspects of strong gravitational fields around black holes. Generation, propagation and detection of gravitational waves. Weak general relativistic effects in the solar system and binary pulsars. Alternative theories of gravity and experimental tests of general relativity.

Syllabus

  • Introduction to general relativity.
  • Derivation from the basic principles Scharzschild.
  • Solution of Einstein's field equations.
  • Reisner-Nordstrom, Kerr and Kerr-Neuman solutions and physical aspects of strong gravitational fields around black holes.
  • Generation, propagation and detection of gravitational waves.
  • Weak general relativistic effects in the solar system and binary pulsars.
  • Alternative theories of gravity and experimental tests of general relativity.

Topics in Scientific Computing (MTH739P)

This module focuses on the use of computers for solving applied mathematical problems. Its aim is to provide students with proper computational tools to solve problems they are likely to encounter while doing their MSc or MSci, and to provide them with a sound understanding of a programming language used in applied sciences. The topics covered will depend on the module organiser’s expertise, with a view to emphasize applications rather than theory.

Syllabus

  • Introduction to programming: Loops; If statements; procedures; input/output; recursion; program structure
  • Numerical solution of ordinary differential equations: Euler scheme; higher-order schemes; Runge-Kutta integrator; applications
  • Random number generation: random vs pseudo-random numbers; linear congruent systems; transformation of random variables; normal variates; applications
  • Monte Carlo methods: direct sampling; importance sampling; Markov chains; applications
  • Simulation of stochastic processes: Random walks; Markov chains; continuous-time limit; Brownian motion; stochastic differential equations; applications

Learning resources

  • R.L. Burden, J.D. Faires, Numerical Analysis, Prindle, Weber & Schmidt, 1985. (PDEs)
  • S. Kay, Intuitive Probability and Random Processes using Matlab, Springer, 2005. (Stochastic processes)
  • W. Krauth, Statistical Mechanics: Algorithms and Computations, Oxford University Press, 2006. (Monte Carlo methods)
  • P. E. Kloeden, E. Platen, H. Schurz, Numerical Solution of SDE Through Computer Experiments, Springer, 1993. (Stochastic differential equations)
  • W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, 2007. (PDEs, random numbers)

Topics in Probability and Stochastic Processes (MTHM012)

This module aims to present some advanced probabilistic concepts and demonstrate their application to stochastic modelling of real-world situations. The topics covered vary from year to year but may include, for example, limit theorems, renewal theory, and continuous-time Markov processes. In addition to exposure to proofs and theoretical material, students develop practical skills through a large number of problems and worked examples.

Syllabus

Topics will be chosen from the following list:

  • Borel-Cantelli lemma, Kolmogorov's inequalities, strong law of large numbers.
  • Weak convergence of distributions.
  • The Central Limit Theorem.
  • Recurrent events and renewal theory.
  • Further topics in random walks.
  • General theory of Markov chains.
  • Classification of states and ergodic properties.
  • Continuous time Markov Processes.

Learning resources

  • W. Feller, An Introduction to Probability Theory and Its Applications 1, Wiley.
  • H M Taylor and S Karlin, An Introduction to Stochastic Modeling, 3rd Edition, Academic Press.

Topology (MTHM732)

Topology is the study of properties of shape which remain the same when pulled, pushed or squeezed by a continuous process of deformation. For example, the property of a space being connected or a surface having a hole is a topological property. In this module we start with general point set topology and formal definitions and move on to study powerful algebraic invariants such as the fundamental group. Topology allows access to many exciting areas of modern mathematics.

Syllabus

  • Topological spaces: examples including discrete, indiscrete, metric and co-finite topologies.
  • Continuity and convergence, homeomorphisms, topological and non-topological properties.
  • Paths and path connectedness.
  • Compactness in a topological space, Heine-Borel theorem, compact implies sequentially compact in metric spaces. Statement of converse.
  • New spaces: subspaces, product spaces, identification spaces (especially of a square).
  • Paths and path homotopies. Simply connectedness.
  • The fundamental group, definition and elementary properties. Fundamental group of a circle. Path and homotopy lifting (proofs non-examinable).
  • Brouwer Fixed Point Theorem in two dimensions. Borsuk-Ulam Theorem in two dimensions.
  • Fundamental group of identifications of the square including the torus and Klein bottle.

Learning resources

  • B Mendelson, Introduction to Topology (Dover Publications).
  • WA Sutherland, Introduction to Metric and Topological Spaces (CUP).