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

Classical Mechanics

15 credits
Autumn teaching, Year 1

An introduction to mechanics and its applications, covering: Newton's Laws; particle dynamics; work and kinetic energy; potential energy and energy conservation; momentum, impulse and collisions; rocket propulsion; rigid-body rotation; torque and angular momentum; gyroscopes and precession; statics and equilibrium; fluid statics and dynamics; gravitation, satellite motion and Kepler's laws.

Classical Physics

15 credits
Spring teaching, Year 1

This module is focused around three main areas:

Electromagnetism:
-Electric forces and fields in systems with static discrete electric charges and static observers.
-Continuous charge distributions, Gauss's law. Electric potential energy and electric potential. 
-Energy stored by the electric field. Motion of charged particles in static electric fields.
-Conductors and insulators in electric fields. Capacitance and capacitors. Energy storage in capacitors. Dielectrics. Drude's model of conduction. 
-Creation of magnetic fields from linear motion of charges (ie, a current) electron spin and orbital motion; motion perpendicular to an electric field. Force on a charged particle moving perpendicular to magnetic field.

Relativity:
-Historical perspective. 
-Inertial frames and transformations. Newton's laws in inertial frames. 
-Michelson-Morley experiment - observed constancy of speed of light. Einstein's assumptions.
-Lorentz-Einstein transformations; Minkowski diagrams; Lorentz contraction; time dilation. 
-Transformation of velocities - stellar aberration. Variation of mass, mass-energy equivalence. 
-Lorentz transformations for momentum and energy.

Thermodynamics:
-Phases of matter; the zeroth law of thermodynamics; temperature and temperature scales
-Thermal expansion coefficients
-The ideal-gas law
-The kinetic theory of gases; the Maxwell speed distribution; mean free paths; transport properties of gases; the equipartition theorem
-Heat capacity; latent heat
-The first law of thermodynamics; internal energy of gases
-PV diagrams; work
-Adiabatic processes

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 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.

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.

Electrodynamics

15 credits
Autumn teaching, Year 2

This module covers electro/magnetostatics and electrodynamics in differential form with key applications. Topics covered include: mathematical revision.
Electrostatics: equations for the E-field, potential, energy, basic boundary-value setups. Electrostatics: dielectrics, displacement and free charge. Magnetostatics: forces, equations for the B-field, vector potential, Biot-Savart, dipole field of current loops. Magnetostatics: diamagnetism and paramagnetism, auxiliary field H, ferromagnetism. Electrodynamics: Faraday's law, inductance and back emf, circuit applications, Maxwell-Ampere law, energy and Poynting's theorem. Electromagnetic waves: wave equation, plane waves, polarization, waves in dielectrics, reflection at an interface, wave velocity/group velocity/dispersion. Potentials and dipole radiation.

Introduction to Probability and Applied Analysis

15 credits
Autumn teaching, Year 2

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

Quantum Mechanics 1

15 credits
Spring teaching, Year 2

Module topics include

• Introduction to quantum mechanics, wave functions and the Schroedinger equation in 1D.
• Statistical interpretation of quantum mechanics, probability density, expectation values, normalisation of the wave function.
• Position and momentum, Heisenberg uncertainty relation.
• Time-independent Schroedinger equation, stationary states, eigenstates and eigenvalues.
• Bound states in a potential, infinite square well.
• Completeness and orthogonality of eigenstates.
• Free particle, probability current, wave packets, group and phase velocities, dispersion.
• General potentials, bound and continuum states, continuity of the wave function and its first
• derivative.
• Bound states in a finite square well.
• Left- and right-incident scattering of a finite square well, reflection and transmission probabilities.
• Reflection and transmission at a finite square well.
• Reflection and transmission at a square barrier, over-the-barrier reflection, tunnelling, resonant
• tunnelling through multiple barriers.
• Harmonic oscillator (analytic approach).
• Quantum mechanics in 3D, degeneracy in the 3D isotropic harmonic oscillator.
• Orbital angular momentum, commutators and simultaneous measurement.
• Motion in a central potential, Schroedinger equation in spherical polar coordinates.
• Schroedinger equation in a Coulomb potential.
• H atom.
• Spin, identical particles, spin-statistics theorem.
• Helium, basics of atomic structure. 
• Time-independent perturbation theory for non-degenerate bound states.
• Applications of perturbation theory, fine structure in the H atom.
• Schroedinger equation for a particle coupled to an electromagnetic field.
• Summary and revision

Thermal and Statistical Physics

15 credits
Spring teaching, Year 2

Topics covered include:

• Review of kinetic theory of gases and first law of thermodynamics.
• Basics of statistical mechanics. Microstates, entropy, second law.
• Classical thermodynamics. Engines and refrigerators.
• More statistical mechanics. Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac distributions.
• Blackbody radiation.
• Elements of phase transitions.

Atomic Physics

15 credits
Autumn teaching, Year 3

Topics covered include: physics of the hydrogen atom; relativistic hydrogen atom (fine structure, antimatter); hyperfine structure of hydrogen and the 21-cm line; interaction with external fields (Zeeman Effect, Stark Effect); helium atom; multi-electron atoms and the periodic system; molecules and chemical binding; molecular structure: vibration and rotation; radiative processes, emission and absorption spectra.

Complex Analysis

15 credits
Spring teaching, Year 3

Topics covered include:

• Holomorphic functions, Cauchy's theorem and its consequences.
• Power series, integration, differentiation and analysis of convergence.
• Taylor expansions and circle of convergence.
• Laurent expansions and classification of isolated singularities.
• Residue theorem and evaluation of integrals.
• Rouche's theorem and the fundamental theorem of algebra.

Condensed State Physics

15 credits
Autumn teaching, Year 3

Classification of Solids
1. Types of solids; Classification of elements and compounds by physical properties. 2. Types of bonding. 3. Basic band theory of metals, electrical insulators and semiconductors. 

Crystal Structures 
4. Crystals; Unit cells and lattice parameters. 5. Bravais lattices; Crystallographic basis; Crystal axes and planes. 6. Cubic and hexagonal structures. 7. Reciprocal lattice.

Diffraction by Crystals
8. Physical Processes; Braggs law; Atomic and geometrical scattering factors. 9. Diffraction crystallography. 

Lattice Vibrations
10. Thermal properties of electrical insulators: Specific Heat and Thermal Conductivity. 11. Vibrations of monatomic and diatomic 1-D crystals; Acoustic and Optical modes. 12. Quantisation of Lattice Vibrations; Phonons. 13. Einstein and Debye Models for Lattice Specific Heat.

The Free Electron Model
14. Classical Free Electron Gas. 15. Quantised Free Electron Model. 16. Specific Heat of the Conduction Electrons. 17. Electrical and Thermal Conductivity of metals. 18. AC conductivity and Optical Properties of metals.

Dielectric and Optical Properties of Insulators
19. Dielectric Constant and Polarizability. 20. Sources of Polarizability; Dipolar Dispersion.

Nuclear and Particle Physics

15 credits
Autumn teaching, Year 3

This module on nuclear and particle physics covers:

• Chronology of discoveries.
• Basic nuclear properties.
• Nuclear forces.
• Models of nuclear structure.
• Magic numbers.
• Nuclear reactions, nuclear decay and radioactivity, including their roles in nature.
• The weak force.
• Existence and properties of neutrinos.
• Qualitative introduction to neutrino oscillations.
• C, P and T symmetries.
• Classification of elementary particles, and their reactions and decays.
• Particle structure.
• Qualitative introduction to Feynman diagrams.

Quantum Mechanics 2

15 credits
Spring teaching, Year 3

This module on quantum mechanics employing Dirac notation and algebraic methods. Topics covered include:

• Dirac's formulation of quantum mechanics - bras&kets, observables, algebraic treatment of harmonic oscillator, x&p representation, compatibility, uncertainty
• Symmetries and conservation laws - generators of translations&rotations, parity, time evolution, Heisenberg picture
• Angular momentum - algebraic treatment, spin, "addition" of angular momenta, explicit form of rotation operators
• Approximation methods - time-independent perturbation theory: first and second orders, degeneracies; WKB approximation & tunneling
• Interaction picture and time-dependent perturbation theory
• Basics of field quantisation - creation and annihilation operators, EM transitions
• Basic scattering theory
• Mixed states and quantum measurement - density matrix, Bell's inequality
• Elements of relativistic QM and antiparticles

Advanced Condensed State Physics

15 credits
Spring teaching, Year 3

This module covers the following topics:

• Electronic Energy bands in Solids. Electrons in periodic potentials; Brillouin Zones; Bloch states. Nearly Free Electron (NFE) model. Tight-Binding Approximation (TBA) model. Band structure of selected metals, insulators and semiconductors. Optical Properties.
• Electron Dynamics. Electrons and holes. Effective Mass. Mobilities. Magneto-transport.
• Semiconductors. Classification; Energy Gaps. Donor and Acceptor doping. Equilibrium carrier statistics in intrinsic and doped materials. Temperature dependence of electrical and optical properties.
• Semiconductor Devices. p-n junctions. Diodes, LEDs, Lasers, Transistors. Superlattices and 2DEG devices. 
• Lattice Defects. Types of defects. Electronic and optical effects of defects in semiconductors and insulators.

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.

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.

Lasers

15 credits
Spring teaching, Year 3

This module covers:

• Light-matter interaction. 
• Rate equations of lasers. 
• Principles of Gaussian optics and optical cavities. 
• Types of lasers and their applications.

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.

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.

Particle Physics

15 credits
Spring teaching, Year 3

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.

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. 

MMath Project

30 credits
Autumn & spring teaching, Year 4

The work for the project and the writing of the project report will have a major role in bringing together material that you have mastered up to Year 3 and is mastering in Year 4. It will consist of a sustained investigation of a mathematical topic at Masters' level. The project report will be typeset using TeX/LaTeX (mathematical document preparation system). The use of mathematical typesetting, (mathematics-specific) information technology and databases and general research skills such as presentation of mathematical material to an audience, gathering information, usage of (electronic) scientific libraries will be taught and acquired during the project.

Advanced Numerical Analysis

15 credits
Autumn teaching, Year 4

Advanced Particle Physics

15 credits
Spring teaching, Year 4

You will acquire an overview of the current status of modern particle physics and current experimental techniques used in an attempt to answer today's fundamental questions in this field. 

The topics discussed will be: 

• Essential skills for the experimental particle physicist
• Neutrino physics: Neutrino oscillations and reactor neutrinos
• Neutrino physics: SuperNova, geo- and solar- neutrinos and direct neutrino mass measurements
• Cosmic ray physics
• Dark matter
• Introduction to QCD (jets, particles distribution functions, etc)
• Higgs physics
• BSM (including supersymmetry)
• Flavour physics & CP violation
• Electric dipole measurements
• Future particle physics experiments.

Beyond the Standard Model

15 credits
Spring teaching, Year 4

This module covers:

• Basics of global supersymmetry: motivation and algebra, the Wess-Zumino model, superfields and superspace, construction of supersymmetry-invariant Lagrangians.
• Weak scale supersymmetry: the gauge hierarchy problem, the Minimal Supersymmetric Standard Model (MSSM).
• Grand unification: SUS(5), the gauge sector, fermion masses, proton decay.
• Extra dimensions: Kaluza-Klein reduction for scalars, fermions and gauge fields, generation of hierarchies, warped geometry.

Data Analysis Techniques

15 credits
Autumn teaching, Year 4

This module introduces you to the mathematical and statistical techniques used to analyse data. The module is fairly rigorous, and is aimed at students who have, or anticipate having, research data to analyse in a thorough and unbiased way.

Topics include: probability distributions; error propagation; maximum likelihood method and linear least squares fitting; chi-squared testing; subjective probability and Bayes' theorem; monte Carlo techniques; and non-linear least squares fitting.

Differential Geometry

15 credits
Autumn teaching, Year 4

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.

Early Universe

15 credits
Spring teaching, Year 4

An advanced module on cosmology.

Topics include:

• Hot big bang and the FRW model; Redshifts, distances, Hubble law
• Thermal history, decoupling, recombination, nucleosynthesis
• Problems with the hot big bang and inflation with a single scalar field
• Linear cosmological perturbation theory
• Quantum generation of perturbations in inflation
• Scalar and tensor power spectrum predictions from inflation
• Perturbation evolution and growth after reheating; free streaming and Silk damping
• Matter power spectrum and CMB anisotropies.

Financial Mathematics

15 credits
Autumn teaching, Year 4

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 4

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.

General Relativity

15 credits
Autumn teaching, Year 4

This module provides an introduction to the general theory of relativity, including:

• Brief review of special relativity
• Scalars, vectors and tensors
• Principles of equivalence and covariance
• Space-time curvature
• The concept of space-time and its metric
• Tensors and curved space-time; covariant differentiation
• The energy-momentum tensor
• Einstein's equations
• The Schwarzschild solution and black holes
• Tests of general relativity
• Weak field gravity and gravitational waves
• Relativity in cosmology and astrophysics.

Harmonic Analysis and Wavelets

15 credits
Autumn teaching, Year 4

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 Cosmology

15 credits
Autumn teaching, Year 4

This module covers:

• observational overview: In visible light and other wavebands; the cosmological principle; the expansion of the universe; particles in the universe.
• Newtonian gravity: the Friedmann equation; the fluid equation; the acceleration equation.
• geometry: flat, spherical and hyperbolic; infinite vs. observable universes; introduction to topology
• cosmological models: solving equations for matter and radiation dominated expansions and for mixtures (assuming flat geometry and zero cosmological constant); variation of particle number density with scale factor; variation of scale factor with time and geometry.
• observational parameters: hubble, density, deceleration.
• cosmological constant: fluid description; models with a cosmological constant.
• the age of the universe: tests; model dependence; consequences
• dark matter: observational evidence; properties; potential candidates (including MACHOS, neutrinos and WIMPS)
• the cosmic microwave background: properties; derivation of photo to baryon ratio; origin of CMB (including decoupling and recombination).
• the early universe: the epoch of matter-radiation equality; the relation between temperature and time; an overview of physical properties and particle behaviour.
• nucleosynthesis: basics of light element formation; derivation of percentage, by mass, of helium; introduction to observational tests; contrasting decoupling and nucleosynthesis.
• inflation: definition; three problems (what they are and how they can be solved); estimation of expansion during inflation; contrasting early time and current inflationary epochs; introduction to cosmological constant problem and quintessence.
• initial singularity: definition and implications.
• connection to general relativity: brief introduction to Einstein equations and their relation to Friedmann equation.
• cosmological distance scales: proper, luminosity, angular distances; connection to observables.
• structures in the universe: CMB anisotropies; galaxy clustering
• constraining cosmology: connection to CMB, large scale structure (inc BAO and weak lensing) and supernovae.

Mathematical Models in Finance and Industry

15 credits
Spring teaching, Year 4

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 4

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.

Numerical Solution of Partial Differential Equations

15 credits
Spring teaching, Year 4

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 4

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.

Programming in C++

15 credits
Autumn teaching, Year 4

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.

Quantum Field Theory 1

15 credits
Autumn teaching, Year 4

This module is an introduction into quantum field theory, covering

• Action principle and Lagrangean formulation of mechanics
• Lagrangean formulation of field theory and relativistic invariance
• Symmetry, invariance and Noether's theorem
• Canonical quantization of the scalar field
• Canonical quantization of the electromagnetic field
• Canonical quantization of the Dirac spinor field
• Interactions, the S matrix, and perturbative expansions
• Feynman rules and radiative corrections.

Quantum Field Theory 2

15 credits
Spring teaching, Year 4

Module topics include:

• Path integrals: Path integrals in quantum mechanics; Functionals; Path integral quantisation of scalar field; Gaussian integration; Free particle Green's functions ; Vacuum-vacuum transition function Z[J]. 
• Interacting field theory in path integral formulation. Generating functional W[J]; Momentum space Greens functions; S-matrix and LSZ reduction formula; Grassmann variables; Fermionic path integral. 
• Gauge field theory: Internal symmetries; Gauge symmetry 1: Abelian; The electromagnetic field; Gauge symmetry 2: non-Abelian. 
• Renormalisation of scalar field theory; Quantum gauge theory; Path integral quantisation of non-Abelian gauge theories; Faddeev-Popov procedure, ghosts; Feynman rules in covariant gauge; Renormalisation.

Quantum Optics and Quantum Information

15 credits
Autumn teaching, Year 4

The module will introduce you to quantum optics and quantum information, covering:

• Quantum systems and the qubit
• Non-locality in quantum mechanics
• Methods of quantum optics
• The density matrix
• The process of measurement
• Introduction of irreversibility
• Decoherence and quantum information
• Quantum and classical communication
• Measures of entanglement and distance between states
• Logic operations and quantum algorithms
• Requirements for quantum computers
• Physical systems for quantum information processing.

Random processes

15 credits
Spring teaching, Year 4

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.

Symmetry in Particle Physics

15 credits
Autumn teaching, Year 4

The module provides an introduction into group theory and aspects of symmetry in particle physics, covering:

• Groups and representations
• Lie groups and Lie algebras
• Space-time symmetries and Poincare group
• Symmetry and conservation laws
• Global, local, and discrete symmetry
• Symmetry breaking and the origin of mass
• Symmetry of the standard model, CKM matrix, neutrino masses, tree-level interactions.

Topology and Advanced Analysis

15 credits
Spring teaching, Year 4

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.