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.