Analysis 1

15 credits
Spring teaching, Year 1

Topics include: 

• Sequences: convergence, Cauchy sequences, subsequences
• Series: proof and application of convergence/divergence criteria
• Limits of functions: definitions, examples and properties
• Continuity: intermediate value theorem, uniform continuity
• Differentiability: definition, proofs of mean value theorems

Calculus

15 credits
Autumn teaching, Year 1

Topics include: functions of one real variable: graphical representation, inverse functions, composition of functions, polynomial, trigonometric, exponential and hyperbolic functions. Limits, continuity and differentiation: one-sided limits, infinite limits, algebra of limits, continuity and the intermediate value theorem, differentiation from first principles, product rule and chain rule, Rolle’s theorem, the mean value theorem and Taylor’s theorem, stationary points of a function. Integration: indefinite and definite integrals, fundamental theorem of calculus, integration by parts and integration by substitution. Solutions to first order ODEs. Manipulations with absolute values. Quadratic forms

Geometry

15 credits
Autumn teaching, Year 1

Topics include: vectors in two and three dimensions. Vector algebra: addition, scalar product, vector product, including triple products. Applications in two- and three-dimensional geometry: points, lines, planes, geometrical theorems. Area and volume. Linear dependence and determinants. Polar co-ordinates in two and three dimensions. Definitions of a group and a field. Polynomials. Complex numbers, Argand plane, De Moivre's theorem. Matrices: addition, multiplication, inverses. Transformations in R^2 and R^3: isometries. Analytical geometry: classification and properties of conics.

Introduction to Economics

15 credits
Autumn teaching, Year 1

This course provides an introduction to the fundamental principles of economics. The first half of the course deals with microeconomic issues including the behaviour of individuals and firms, their interaction in markets and the role of government. The second half of the course is devoted to macroeconomics and examines the determinants of aggregate economic variables, such as national income, inflation, and the balance of payments, and the relationships between them. This course also provides students with a basic introduction to mathematical economics, covering solving linear equations, differential calculus, and discounting.

 

Introduction to Pure Mathematics

15 credits
Autumn teaching, Year 1

Topics covered include:

• Numbers: introduction of mathematical symbols, natural numbers, integers, rationals, real numbers, basic number algebra. Ordering, inequalities, absolute value (modulus), homogeneity, triangle inequality. Concept of algebraic structure, groups. Sequences, Induction Principle, Well Ordering Principle, sums, products, factorials, Fibonacci numbers, fractions.
• Irrational roots of integers, divisibility, prime numbers, Euclidean Division, highest common factor, Euclidean Algorithm, Number Theory, Atomic Property of Primes, Coprime Factorisation, Fundamental Theorem of Arithmetic, square-free numbers.
• Logic: concept of proof, logical argument, direct proof, propositional manipulation, basic logic, and, or, not, implication, contraposition, contradiction, logical equivalence, quantifiers. 
• Axiomatic set theory: Extension Axiom, equality of sets, Specification Axiom, intersection (meet), difference (take), subset, Existence Axiom, empty set, Pairing Axiom, singletons, pairs, ordered pairs, Union Axiom, cartesian products, Power Axiom, power set. 
• Counting: maps and functions, distinguished functions, injections, surjections, bijections, one-to-one correspondences, Pigeon Hole Principle, counting the power set, counting subsets of the power set, Cherry Picking, binomial coefficients, binomial formula, combinatorics, Inclusion-Exclusion formula, permutations, counting maps.
• Functions and maps: formal definition, finite and infinite sets, Peano's Axiom (Infinity Axiom/Induction Axiom), Pigeon Hole Principle revisited, counterimage, inverse functions, partial inverses, Axiom of Choice. 
• Relations: relations, equivalence relations, modular arithmetic and quotient sets, order relations, partial ordering, total ordering, linear ordering. Rigorous extension of N to Z and Q. Rings, fields. Examples.
• Real numbers: ordering and Archimedean Property of Reals, countable vs. uncountable sets, Cantor's "Diagonal".

 

 

 

 

 

 

Linear Algebra

15 credits
Spring teaching, Year 1

You will cover:

Matrices, Elementary row and column operations, Vector spaces, Linear independence, Basis and dimension, Inner products and orthogonality, Gram-Schmidt orthonormalisation process, Linear transformations, Determinants, Eigen-values and Eigen-vectors, Polynomials, Cayley-Hamilton 
theorem, Quadratic forms.

Microeconomics 1

15 credits
Spring teaching, Year 1

This module develops consumer and producer theory, examining such topics as consumer surplus, labour supply, production and costs of the firm, alternative market structures and factor markets. It explores the application of these concepts to public policy, making use of real-world examples to illustrate the usefulness of the theory.

Numerical Analysis 1

15 credits
Spring teaching, Year 1

This module covers topics such as:

Introduction to Computing with MATLAB 

• Basic arithmetic and vectors, M-File Functions, For Loops, If and else, While statements

Introduction to Numerical Analysis

• Operating with floating point numbers, round-off error, cancellation error
• Polynomial interpolation, Basic idea of interpolation, Order of approximation, Lagrangian interpolation, Runge's example, Piecewise linear interpolation
• Numerical differentiation, finite difference quotients, order of approximation
• Numerical integration, Derive standard numerical integration scheme's and analyse, using polynomial interpolation (midpoint formula, trapezoidal rule, Simpson's formula)
• Nonlinear equations, bisection method, fixed point iteration method, Newton's method/ Secant method

Analysis 2

15 credits
Autumn teaching, Year 2

Topics covered: power series, radius of convergence; Taylor series and Taylor's formula; applications and examples; upper and lower sums; the Riemann integral; basic properties of the Riemann integral; primitive; fundamental theorem of calculus; integration by parts and change of variable; applications and examples. Pointwise and uniform convergence of sequences and series of functions: interchange of differentiation or integration and limit for sequences and series; differentiation and integration of power series term by term; applications and examples. Metric spaces and normed linear spaces: inner products; Cauchy sequences, convergence and completeness; the Euclidean space R^n; introduction to general topology; applications and examples.

Calculus of Several Variables

15 credits
Autumn teaching, Year 2

Differential Equations

15 credits
Spring teaching, Year 2

Topics include:

Ordinary differential equations:

• Solution methods: Variation of the constant formula, separation of variables
• Solution of linear ODE with constant coefficients
• Lipschitz continuity
• Existence and uniqueness (Picard-Lindeloef), maximal solutions, Gronwall
• Higher order equations into system of first order
• Boundary value problems.

Partial differential equations:

• Partial and total derivatives
• First order PDEs: Method of characteristics for semilinear and quasilinear equations, initial boundary value problems.

Introduction to Probability and Applied Analysis

15 credits
Autumn teaching, Year 2

Macroeconomics 1

15 credits
Spring teaching, Year 2

This module introduces core short-run and medium-run macroeconomics.

First you will consider what determines demand for goods and services in the short run. You will be introduced to financial markets, and outline the links between financial markets and demand for goods. The Keynesian ISLM model encapsulates these linkages. Second, you will turn to medium-term supply. You will bring together the market for labour and the price-setting decisions of firms in order to build an understanding of how inflation and unemployment are determined. Finally, you will look at supply and the ISLM together to produce a full medium-term macroeconomic model.

Microeconomics 2

15 credits
Autumn teaching, Year 2

This module develops the economics principles learned in Microeconomics 1. Alternative market structures such as oligopoly and monopolistic competition are studied and comparisons drawn with perfect competition and monopoly. Decision-making under uncertainty and over multiple time periods is introduced, relaxing some of the restrictive assumptions made in the level 1 module. The knowledge gained is applied to such issues as investment in human capital (eg education), saving and investment decisions, insurance and criminal deterrence.

Numerical Analysis 2

15 credits
Spring teaching, Year 2

Topics covered include:

• Linear systems (conditioning, LU factorization, basic iterative methods, convergence analysis)
• Nonlinear systems: Newton's method
• Numerical solution of differential equations:
    - finite difference methods for first and second order initial value problems
    - finite difference/element methods for one-dimensional boundary value problems

Probability and Statistics

15 credits
Spring teaching, Year 2

Topics include: 

• Descriptive Statistics: types of data, histograms, sample mean, variance, standard deviation, quantiles;
• Statistical Inference: estimation, maximum likelihood, standard distributions, central limit theorem, model validation;
• Distribution theory: Chebychev's inequality, weak law of large numbers, distribution of sums of random variables, t,\chi^2 and F distributions;
• Confidence intervals;
• Statistical tests including z- and t-tests, \chi^2 tests;
• Linear regression;
• Nonparametric methods;
• Random number generation;
• Introduction to stochastic processes.

Macroeconomics 2

15 credits
Autumn teaching, Year 3

This module is concerned with two main topics. 'The long run' is an introduction to how economies grow, gradually raising the standard of living, decade by decade. Once we have the basic analysis in place, we can begin to explain why there are such huge disparities in living standards around the world. 'Expectations' is a deepening of the behavioural background to modelling, saving and investment decisions, emphasising the intrinsically forward-looking nature of saving and investment decisions and analysing the financial markets which coordinate these decisions.

Advanced Macroeconomics

15 credits
Spring teaching, Year 3

The module completes the macroeconomics sequence, starting with a consideration of the policy implications of rational expectations. The macroeconomy is then opened up to international trade and capital movements: the operation of monetary and fiscal policies and the international transmission of disturbances under fixed and flexible exchange rates are contrasted, and the issues bearing on the choice of exchange-rate regime are explored. The major macroeconomic problems of hyperinflation, persistent unemployment and exchange-rate crises are examined. The module concludes by drawing together the implications of the analysis for the design and operation of macroeconomic policy.

 

Advanced Microeconomics

15 credits
Spring teaching, Year 3

This module covers the topics of general equilibrium and welfare economics, including the important issue of market failure. General equilibrium is illustrated using Sen's entitlement approach to famines and also international trade. Welfare economics covers concepts of efficiency and their relationship to the market mechanism. Market failure includes issues such as adverse selection and moral hazard, and applications are drawn from health insurance, environmental economics and the second-hand car market.

 

Advanced Numerical Analysis

15 credits
Autumn teaching, Year 3

Coding Theory

15 credits
Spring teaching, Year 3

Topics 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.

Continuum Mechanics

15 credits
Spring teaching, Year 3

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
Autumn teaching, Year 3

Topics covered include:

• Symmetric-key cryptosystems.
• Hash functions and message authentication codes.
• Public-key cryptosystems.
• Complexity theory and one-way functions.
• Random number generation.
• Attacks on cryptosystems.
• Cryptographic standards.

Dynamical Systems

15 credits
Spring teaching, Year 3

Topics covered include:

• General dynamical systems: semiflow, stability and attraction, omega-limit set, global attractor. 
• Ordinary differential equations: Linear systems, Lyapunov function, linearized systems around fixed points, periodic orbits, Two-dimensional systems, centre manifold, bifurcation.
• Discrete systems (iterations): Linear systems, linearized systems around fixed points.

E-Business and E-Commerce Systems

15 credits
Autumn teaching, Year 3

Topics for this module include: elementary economic theory and its interaction with e-business; alternative e-business strategies, as theories and as case studies; legal and behavioural issues; marketing, branding, and customer relationship issues; software systems for e-business and e-commerce; and commercial website management.

Financial Mathematics

15 credits
Autumn teaching, Year 3

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, and project writing.

Functional Analysis

15 credits
Spring teaching, Year 3

Topics on this module 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; compactness of sets; Hilbert spaces; orthogonal expansions; Riesz-Fischer Theorem.

Harmonic Analysis and Wavelets

15 credits
Autumn teaching, Year 3

You will be introduced to the concepts of harmonic analysis and the basics of wavelet theory: you 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 introduced to the ideas of best approximation, orthogonal projection, orthogonal sums, orthonormal bases and Fourier series in a separable Hilbert space.

You will then apply these concepts to the concrete case of classical trigonometric Fourier series, and both Fejer's theorem and the Weierstrass approximation theorem will be proved.

Finally, you will apply the introduced concepts for Hilbert to discuss wavelet analysis for the example of the Haar wavelet and the Haar scaling function. You will be introduced to the concepts of an orthogonal wavelet and a multiresolution analysis (with a scaling function for the case of the Haar wavelet), but will also be defined in general. 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 3

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 3

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 Research Project

15 credits
Spring teaching, Year 3

The project aims to introduce you under the guidance of a supervisor into a mathematical topic 

Measure and Integration

15 credits
Autumn teaching, Year 3

Topics for this module 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 3

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.

Partial Differential Equations

15 credits
Autumn teaching, Year 3

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 3

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 3

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.

Random processes

15 credits
Spring teaching, Year 3

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 3

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.

Topology and Advanced Analysis

15 credits
Spring teaching, Year 3

This module will introduce you to some of the basic concepts and properties of topological spaces. The subject of topology has a central role in all of Mathematics and having a proper understanding of its concepts and main theorem is essential as part of an undergraduate mathematics curriculum.

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.