Coding Theory

15 credits
Spring teaching, year 1

Topics covered include: 

• Introduction to error-correcting codes. The main coding theory problem. Finite fields.
• Vector spaces over finite fields. Linear codes. Encoding and decoding with a linear code.
• The dual code and the parity check matrix. Hamming codes. Constructions of codes.
• Weight enumerators. Cyclic codes. MDS codes.

Continuum Mechanics

15 credits
Spring teaching, year 1

Topics include: 

• Kinematics: Eulerian and Lagrangian descriptions, velocity, acceleration, rate of change of physical quantities, material derivatives, streamlines.
• Deformation: stress and strain tensors, Hooke's law, equilibrium equations.
• Conservation laws for mass, momentum and energy.
• Phase/group velocities of travelling wave solutions.
• Models of fluid and solid mechanics.

Cryptography

15 credits
Spring teaching, year 1

You will cover the following areas: 

• Symmetric-key cryptosystems.
• Hash functions and message authentication codes.
• Public-key cryptosystems.
• Complexity theory and one-way functions.
• Primality and randomised algorithms.
• Random number generation.
• Elliptic curve cryptography.
• Attacks on cryptosystems.
• Quantum cryptography.
• Cryptographic standards.

Differential Geometry

15 credits
Autumn teaching, year 1

This module covers: Manifolds and differentiable structures, Lie derivatives, Parallel transport, Riemannian metrics and affine connections, Curvature tensor, Sectional curvature, Scalar curvature, Ricci curvature, Bianchi identities, Schur's lemma, Complete manifolds, Hopf-Rinow theorem, Hadmard's theorem, Geodescis and Jacobi fields, Bonnet-Meyer and Synge theorems, Laplace-Beltrami operator, Heat kernels and index theorem.

Financial Mathematics

15 credits
Autumn teaching, year 1

You will study generalized cash flows, time value of money, real and money interest rates, compound interest functions, equations of value, loan repayment schemes, investment project evaluation and comparison, bonds, term structure of interest rates, some simple stochastic interest rate models, and project writing.

Functional Analysis

15 credits
Spring teaching, year 1

Topics include: Banach spaces (Banach fixed point theorem); Baire's theorem; Bounded linear operators and on Banach spaces; continuous linear functionals; Banach-Steinhaus Uniform Boundedness Principle; open mapping and closed graph theorems; Hahn-Banach theorem; Hilbert spaces; orthogonal expansions; and Riesz-Fischer theorem.

Harmonic Analysis and Wavelets

15 credits
Autumn teaching, year 1

This module introduces you to the concepts of harmonic analysis and the basics of wavelet theory. We will discuss the concepts of normed linear spaces and Hilbert spaces, with a focus on sequence spaces and spaces of functions, most notably the space of square-integrable functions on an interval or on the real line. You will be intoroduces to the ideas of best approximation, orthogonal projection, orthogonal sums, orthonormal bases and Fourier series in a separable Hilbert space, and apply these to the concrete case of classical trigonometric Fourier series. You will also use these strategies to prove both Fejer's theorem and the Weierstrass approximation theorem. Finally you will apply the concepts for Hilbert spaces to discuss wavelet analysis using the example of the Haar wavelet and the Haar scaling function. The concepts of an orthogonal wavelet and a multiresolution analysis (with a scaling function) will initially be introduced for the case of the Haar wavelet, but will also be defined in general.

Introduction to Mathematical Biology

15 credits
Autumn teaching, year 1

The module will introduce you to the concepts of mathematical modelling with applications to biological, ecological and medical phenomena. The main topics will include:

• Continuous populations models for single species;
• Discrete population models for single species;
• Phase plane analysis;
• Interacting populations (continuous models);
• Enzyme kinetics;
• Dynamics of infectious diseases and epidemics.

Linear Statistical Models

15 credits
Autumn teaching, year 1

Topics include: full-rank model (multiple and polynomial regression), estimation of parameters, analysis of variance and covariance; model checking; comparing models, model selection; transformation of response and regressor variables; models of less than full rank (experimental design), analysis of variance, hypothesis testing, contrasts; simple examples of experimental designs, introduction to factorial experiments; and use of a computer statistical package to analyse real data sets.

Mathematical Models in Finance and Industry

15 credits
Spring teaching, year 1

Topics include: partial differential equations (and methods for their solution) and how they arise in real-world problems in industry and finance. For example: advection/diffusion of pollutants, pricing of financial options.

Measure and Integration

15 credits
Autumn teaching, year 1

Topics include: 

• Countably additive measures, sigma-algebras, Borel sets, measure spaces.
• Outer measures and Caratheodory's construction of measures.
• Construction and properties of Lebesgue measure in Euclidean spaces.
• Measurable and integrable functions, Lebesgue integration theory on measure spaces, L^p spaces and their properties.
• Convergence theorems - monotone convergence, dominated convergence, Fatou's lemma.
• Application of limit theorems to continuity and differentiability of integrals depending on a parameter.
• Properties of finite measure spaces and probability theory.

Medical Statistics

15 credits
Spring teaching, year 1

Topics include: logistic regression, fitting and interpretation. Survival times; Kaplan-Meier estimate, log-rank test, Cox proportional hazard model. Designing medical research. Clinical trials; phases I-IV, randomised double-blind controlled trial, ethical issues, sample size, early stopping. Observational studies: prospective/retrospective, longitudinal/cross-sectional. Analysis of categorical data; relative risk, odds ratio; McNemar's test, meta-analysis (Mantel-Haenszel method). Diagnostic tests; sensitivity and specificity; receiver operating characteristic. Standardised mortality rates.

Numerical Linear Algebra

15 credits
Autumn teaching, year 1

Topics covered include:

• matrix analysis, vector norms, canonical forms and spectral radius
• floating-point arithmetic, stability, conditioning
• direct methods for linear systems, back substitution, Gaussian elimination, pivoting, Cholesky factorisation
• iterative methods, Jacobi, Gauss-Seidel, conjugate gradients
• eigenvalues, basic properties, reduction to Hessenberg form, power methods.

Numerical Solution of ODEs

15 credits
Spring teaching, year 1

Topics that you will study include: linear multistep methods and Runge-Kutta methods; consistency, stability and convergence theory; absolute stability; initial boundary value problems; finite difference method; finite element method; modes of application; and error analysis.

Numerical Solution of Partial Differential Equations

15 credits
Spring teaching, year 1

Topics covered include: variational formulation of boundary value problems; function spaces; abstract variational problems; Lax-Milgram Theorem; Galerkin method; finite element method; examples of finite elements; and error analysis.

Object Oriented Programming

15 credits
Autumn teaching, year 1

You will be introduced to object-oriented programming, and in particular to understanding, writing, modifying, debugging and assessing the design quality of simple Java applications.

You do not need any previous programming experience to take this module, as it is suitable for absolute beginners.

Partial Differential Equations

15 credits
Autumn teaching, year 1

Topics include: Second-order Partial Differential Equations: wave equation, heat equation, Laplace equation. D'Alembert's solution, separation of variables, Duhamel's principle, energy method, Maximum principle, Green's identities.

Perturbation theory and calculus of variations

15 credits
Spring teaching, year 1

The aim of this module is to introduce you to a variety of techniques primarily involving ordinary differential equations, that have applications in various branches of applied mathematics. No particular application is emphasised.

Topics covered include

• Dimensional analysis and scaling:
• physical quantities and their measurement;
• dimensions;
• change of units;
• physical laws;
• Buckingham Pi Theorem;
• scaling.

• Regular perturbation methods:
• direct method applied to algebraic equations and initial value problems (IVP);
• Poincar method for periodic solutions;
• validity of approximations.

• Singular perturbation methods:
• finding approximate solutions to algebraic solutions;
• finding approximate solutions to boundary value problems (BVP) including boundary layers and matching.

• Calculus of Variations:
• necessary conditions for a function to be an extremal of a fixed or free end point problem involving a functional of integral form;
• isoperimetric problems.

Probability Models

15 credits
Autumn teaching, year 1

Topics include: probability spaces as models of chance experiments; axioms, conditional probability; random variables, distributions, densities, mass functions; random vectors, joint and marginal distributions, conditioning; expectation, indicator variables, laws of large numbers, moment generating functions, central limit theorem; ideas of convergence of random variables; Markov chains, including random walk; Poisson processes; and The Wiener process.

Programming in C++

15 credits
Autumn teaching, year 1

After a review of the basic concepts of the C++ language, you are introduced to object oriented programming in C++ and its application to scientific computing. This includes writing and using classes and templates, operator overloading, inheritance, exceptions and error handling. In addition, Eigen, a powerful library for linear algebra is introduced. The results of programs are displayed using the graphics interface dislin.

Random processes

15 credits
Spring teaching, year 1

The aim of this module is to present a systematic introductory account of several principal areas in stochastic processes. You cover basic principles of model building and analysis with applications that are drawn from mainly biology and engineering.

Topics include:

• Poisson processes:
• Definition and assumptions.
• Density and distribution of inter-event time.
• Pooled Poisson process.
• Breaking down a Poisson process.

• Birth processes, birth- and death- processes:
• The simple birth process.
• The pure death process.
• The Kolmogorov equations.
• The simple birth-death process.
• Simple birth-death: extinction.
• An embedded process.
• The immigration-death model.

• Queues:
• The simple M/M/1 queue.
• Queue size.
• The M/M/n queue.
• The M/M/ queue.
• The M/D/1 queue.
• The M/G/1 queue.
• Equilibrium theory.
• Other queues.

• Renewal processes:
• Discrete-time renewal processes.
• The ordinary renewal process.
• The equilibrium renewal process.

• Epidemic models:
• The simple epidemic.
• General epidemic.
• The threshold in epidemic models.

Ring Theory

15 credits
Autumn teaching, year 1

In this module we will explore how to construct fields such as the complex numbers and investigate other properties and applications of rings.

Topics covered include

• Rings and types of rings: examples.
• Special rings and special elements: unit, zero, divisor, integral domain, fraction field, irreducible element, prime element.
• Factorising polynomials: roots and multiple roots, differentiation, roots of unity, polynomials in Q[x] and Z[x], Gauss' lemma, Eisenstein's criterion.
• Manipulating roots and symmetry: coefficients of polynomials and roots, Newton's theorem.
• Euclidean domains: Gaussian integers, Euclidean algorithm, gcd's and lcm's.
• Homomorphisms and ideals: quotient rings, principal, maximal and prime ideals. 
• Finite fields.
• Unique factorisation domains: generalising Gauss' lemma.
• Special topics: Quaternions, valuations, Hurwitz ring, the four squares theorem.

Special Topics in Mathematics

15 credits
Autumn teaching, year 1

The module will give Masters students with an especially strong mathematical background the opportunity to study an advanced area of Mathematics in-depth with support of an expert in that area. There will be no fixed syllabus, but a menu of options available to you. The major options available initially will be in the areas on numerical analysis, partial differential equations and mathematical biology.

Topology and Advanced Analysis

15 credits
Spring teaching, year 1

Topics that will be covered in this module include:

• Topological spaces
• Base and sub-base
• Separation axioms
• Continuity
• Metrisability
• Completeness
• Compactness and Coverings
• Total Boundedness
• Lebesgue numbers and Epsilon-nets
• Sequential Compactness
• Arzela-Ascoli Theorem
• Montel's theorem
• Infinite Products
• Box and Product Topologies
• Tychonov Theorem.