MT5420 Advanced Quantum Theory (Term 2: Dr F Mota-Furtado)Prerequisite: An undergraduate course in quantum theoryTeaching: 33hr lectures, 167hr private study, including problem sheetsAssessment: 2hr written examinationAims To derive methods, such as the Rayleigh-Ritz variational principle and perturbation

theory, in order to obtain approximate solutions of the Schrödinger equation. To introduce spin and the Pauli exclusion principle and hence explain the mathematical

basis of the Periodic table of elements. To introduce the quantum theory of the interaction of electromagnetic radiation with

matter using time dependent perturbation theory. To show how scattering theory is used to probe interactions between particles and hence

to show how the probability or cross section for a scattering event to occur can be derived from quantum theory.

Learning outcomesOn completion of the course students should be able to: use various methods to obtain approximate eigenvalues and eigenfunctions of any given

Schrödinger equation, to understand the importance of spin in quantum theory, to appreciate how the Periodic Table of elements follows from quantum theory, to write down the Schrödinger equation for the interaction of electromagnetic radiation

with the hydrogen atom and to work out photoabsorption cross sections for hydrogen, to define the scattering cross section and to work it out for some simple systems.ContentVariational principles in quantum mechanics: the Rayleigh-Ritz variational principle. Bounds on energy levels for quantum systems.Perturbation theory: Rayleigh-Schrödinger time-independent perturbation theory. Perturbations of energy levels due to external electromagnetic fields.The electron’s spin: the eigenfunctions and eigenvalues of the spin operator. The Pauli exclusion principle. The periodic table of elements. Spin precession in an external magnetic field.Radiative transitions: the absorption and emission of electromagnetic radiation by matter. Photoabsorption cross-sections for the hydrogen atom.Scattering theory: definition of the scattering cross-section and the scattering amplitude. Decomposition of the scattering amplitude into partial waves. Phase shifts and the S-matrix. Integral representations of the scattering amplitude. The Born approximation. Potential scattering.Indicative texts Quantum Physics – S Gasiorowicz (Wiley 1974) Library reference 530.12 GASQuantum Mechanics – P C W Davies (Chapman and Hall 1984)Library reference 530.12 DAV

MT5421 Aerodynamics and Geophysical fluid dynamics (Term 2, but not given in 2007/08)Prerequisite: MT322 or other undergraduate course in fluid mechanics.

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Teaching: 33hr lectures, 167hr private study, including problem sheetsAssessment: 2hr written examination

AimsThis course aims to show how the mathematical models of MT222 and MT322 are successful in describing how aircraft are able to fly, and how the motions of the atmosphere and the oceans are caused. It also gives insight into the effect that individual terms in the mathematical model may have on the behaviour of the whole system.

Learning outcomesAt the end of the course the students should be able to derive the freezing-in of vortex lines for incompressible fluids; use complex variable theory to derive the formula for lift on an infinite cylinder; explain in broad terms how an aircraft is able to fly; understand the role of Coriolis and centrifugal forces in a rotating fluid; describe how rotation causes various phenomena in fluids; solve the simple equations for motion in an Ekman layer.

ContentVortex dynamics: freezing-in of vortex lines, why vorticity can be treated as a pollutant. Examples.Flow past wing sections: two-dimensional flow, flow at sharp corners, generation of lift. Blasius’ formula. Three-dimensional flows, trailing vortices, induced drag. Supersonic flow past wing sections.Rotating fluid systems: equation of motion of a rotating fluid. Geostrophic flow and simple properties. Secondary flow and examples (e.g. meanders, tea leaves in a cup). Inertial waves.Viscosity-rotation interactions: Ekman layers and boundary fluxes.The atmosphere and oceans: large-scale motions and the role of Coriolis forces. Tornado generation. Effects of the earth’s curvature and induced waves.

Indicative textFluid Mechanics – P K Kundu and I M Cohen (Academic Press 2002) Library ref. 532 KUN

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MT5422 Advanced Electromagnetism and Special Relativity (Term 1, but not given in 2007/08)Prerequisite: An undergraduate course in Electromagnetism (MT324 may be taken at the same time)Teaching: 33hr lectures, 167hr private study, including problem sheetsAssessment: 2hr written examination

Aims To show how Maxwell’s equations lead to electromagnetic waves and indirectly to the

special theory of relativity; To show how electromagnetic fields propagate with the speed of light; To derive the laws of optics from Maxwell’s equations; To show how the laws of special relativity lead to time dilation and length contraction.

Learning outcomesOn completion of the course students should be able to use Maxwell’s equations to demonstrate the polarization, reflection and refraction of

electromagnetic waves; understand the fundamental ideas of electromagnetic radiation; demonstrate the Galilean non-invariance and Lorentz invariance of Maxwell’s equations; derive the fundamental properties of relativistic optics.

ContentElectromagnetic theory: electromagnetic waves, reflection and refraction with both normal and oblique incidence, total internal reflection, waves in conducting media, wave guides. Radiation: the Hertz vector and related field strengths, fields of moving charges, Lienhard-Wiechart potentials, motion of charged particles.Special relativity: the Lorentz transformation. Relativistic invariance, the Fitzgerald contraction, time dilation. Relativistic electromagnetic theory: Lorentz invariance of Maxwell’s equations, the transformation of and . Relativistic mechanics: mass, momentum, energy. Relativistic optics: aberration, the Doppler effect.

Indicative textFoundations of Electromagnetic Theory (Fourth Edition) – J R Reitz, F J Milford and R W Christy (Addison-Wesley 1993) Library reference 538.141 REI.

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MT5423 Magnetohydrodynamics (Term 2)Prerequisite: An undergraduate course in fluid dynamics.Teaching: 33hr lectures, 167 hours private studyAssessment: 2hr written examinationAimsThis course aims to introduce the study of the motion of conducting fluids in the presence of a magnetic field. Practical applications and a discussion of the structure of sunspots and the origin of the Earth’s magnetic field will be given.Learning outcomesOn completion of the course the student should be able to: demonstrate an understanding of the basic principles of MHD; apply appropriate mathematical techniques to solve a wide variety of

problems in MHD.ContentFoundations of Magnetohydrodynamics (MHD): Consideration of the electrodynamics of moving media and MHD approximations, leading to the induction equation - an equation central to MHD. Alfvén's theorem for a medium of infinite electrical conductivity - its proof and physical importance. The necessity for an additional term in the equation of motion - the electromagnetic body force. Alternative description in terms of electromagnetic stresses. MHD waves: Alfvén waves in a medium of infinite electrical conductivity, reflection and transmission at a discontinuity in density, effect of finite electrical conductivity and/or viscosity, waves in a compressible medium. MHD shock waves.Steady flow problems: including Hartmann flow.Magnetohydrostatics: Pressure balanced configurations. Force-free fields.Indicative TextsAn Introduction to Magneto-fluid Mechanics V C A Ferraro & C Plumpton (2nd edition) (OUP 1966). Library Ref. 538.6 FERAn Introduction to Magnetohydrodynamics – P A Davidson (CUP 2001). Library Ref. 538.6 DA

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MT5441 Channels (Term 1)Prerequisite: Undergraduate courses on coding theory and abstract algebraTeaching: 33hr lectures and seminars, 167hr private study, including problem sheetsAssessment: 2hr examination

AimsTo investigate the problems of data compression and information transmission in both noiseless and noisy environments.

Learning outcomesOn completion of the course, students should be able to: state and derive a range of information-theoretic equalities and inequalities; explain data-compression techniques for ergodic as well as memoryless sources; explain the asymptotic equipartition property of ergodic systems; understand the proof of the noiseless coding theorem; define and use the concept of channel capacity of a noisy channel; explain and apply the noisy channel coding theorem; evaluate and understand a range of further applications of the theory.

ContentEntropy: Definition and mathematical properties of entropy, information and mutual information.Noiseless coding: Memoryless sources: proof of the Kraft inequality for uniquely decipherable codes, proof of the optimality of Huffman codes, typical sequences of a memoryless source, the fixed-length coding theorem.Ergodic sources: entropy rate, the asymptotic equipartition property, the noiseless coding theorem for ergodic sources.Lempel-Ziv coding.Noisy coding: Noisy channels, the noisy channel coding theorem, channel capacity.Further topics, such as hash codes, or the information-theoretic approach to cryptography and authentication.

Indicative TextsCodes and Cryptography D Welsh (Oxford UP). Library Ref. 001.5436 WELElements of Information Theory T M Cover and J A Thomas (Wiley). Library Ref. 001.539 COVInformation Theory, Inference and Learning Algorithms – D J C MacKay (Cambridge UP). Library Ref. 001.539 MAC

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MT5445 Quantum Information Theory (Term 2) Prerequisite: Undergraduate courses in linear algebra and probabilityTeaching: 33hr lectures, 167hr private study, including problem sheetsAssessment: 2hr written examination

Aims'Anybody who is not shocked by quantum theory has not understood it' (Niels Bohr). This course aims to provide a sufficient understanding of quantum theory in the spirit of the above quote. Many applications of the novel field of quantum information theory can be studied using undergraduate mathematics. The course relies almost exclusively on tools from linear algebra – prior knowledge of applied mathematics or quantum theory is neither required nor particularly useful.

Learning outcomesOn completion of the course the student should be able to: demonstrate a comprehensive understanding of the principles of quantum

superposition and quantum measurement; use the basic linear algebra tools of quantum information theory confidently; manipulate tensor-product states and use and explain the concept of entanglement; explain applications of entanglement such as quantum teleportation or quantum secret

key distribution; describe the Einstein-Podolsky-Rosen paradox and derive a Bell inequality; solve a range of problems involving one or two quantum bits; discuss Deutsch's algorithm and its implications for the power of a quantum

computer; understand and apply Grover’s search algorithm.

ContentLinear algebra: Complex vector space, inner product, Dirac notation, projection operators, unitary operators, Hermitian operators, Pauli matrices.One qubit: Pure states of a qubit, the Poincaré sphere, von Neumann measurements, quantum logic gates for a single qubit.Tensor products: 2 qubits, 3 qubits, quantum logic gates for 2 qubits, Deutsch's algorithm, the Schmidt decomposition.Mixed states: Partial trace, probability, entropy, von Neumann entropy.Entanglement: The Einstein-Podolsky-Rosen paradox, Bell inequalities, quantum teleportation, measures of entanglement, decoherence.Grover's search algorithm, and applications.Further applications, such as e.g. the quantum Fourier transform, Shor's factoring algorithm, the BB84 key distribution protocol, quantum channel capacity, the Holevo bound.

Indicative TextM A Nielsen and I L Chuang – Quantum Computation and Quantum Information (Cambridge 2000). Library Ref. 001.64 NIE

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MT5447 Advanced Financial Mathematics (Term 2)Prerequisite: An undergraduate course covering statistics, the ideas of risk and return in finance, stochastic calculus and the basics of derivative pricing.Teaching: 33hr lectures, 167hr private study, including problem sheetsAssessment: 2hr written examination

Aims To investigate the validity of various linear and non-linear time series occurring in

finance; To extend the use of stochastic calculus to interest rate movements and credit rating;

Learning outcomesOn completion of the course, students should: make use of some of the ARCH (autoregressive conditionally heteroscedastic) family of

models in time series; appreciate the ideas behind the use of the BDS test and the bispectral test for time series. understand the partial differential equation for interest rates and the assumptions that lead

to it; be able to model forward and spot rates; understand how a Poisson process can be included to model the possibility of default on a

bond or similar asset.

ContentFinancial time series: Linear time series: ARMA and ARIMA models, stationarity, autoregressions. Testing of linearity, using spectral analysis. ARCH and GARCH models.Structure of financial series: The random walk model, trend and volatility, moments. Comparison with chaotic systems, dimensionality and memory effects in financial series. The nearest neighbour algorithm and the BDS test.Interest rate analysis: Revision of ideas in stochastic calculus. Modelling of interest rates, the bond pricing equation. Bond derivatives. The Heath-Jarrow-Morton model.Credit risk: Modelling of default probabilities. The equation for a risky bond.

Indicative TextsThe Econometric Modelling of Financial Time Series – T C Mills (Cambridge UP 1999) Library ref. 330.0151 MILPaul Wilmott Introduces Quantitative Finance – P Wilmott (Wiley 2001) Library ref. 332.632 WILMarket Models – C Alexander (Wiley 2001) Library ref. 332.6 ALE

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MT5454 Combinatorics (Term 1)Prerequisite : An undergraduate course in Discrete MathematicsTeaching: 33hr lectures and seminars, 167hr private study, including problem sheetsAssessment: 2hr written examination

Aims: To introduce the standard techniques of combinatorics, including methods of counting: generating functions, induction, subdivision; Principle of Inclusion and Exclusion; partitions, Ramsey and Polya Theory.

Learning Outcomes:On completion of the course, students should be able to: find small partition numbers; perform simple calculations with generating functions;. understand Ramsey numbers and calculate upper bounds for these (where practical); calculate sets by inclusion and exclusion and understand the applications to number

theory; calculate cycle indexes for the standard groups and the numbers of distinct configurations

of symmetrical objects.

ContentEnumeration: Binomial identities. The Principle of Inclusion-Exclusion with applications to number theory. Rook polynomials. Posets and lattices. The Möbius function of a lattice.Generating functions: Linear recursion. Power series and ordinary generating functions. Partitions and partition identities.Ramsey Theory: Monochromatic subsets, Ramsey numbers and Ramsey's Theorem.Polya Theory: Automorphisms of graphs. The Orbits-Stabiliser Theorem, and the Orbit Counting Lemma. Cycle index of a permutation group. Polya's Theorem.

Indicative TextsDiscrete Mathematics N.L. Biggs (Oxford UP, 1989); Library reference 510 BIG.Combinatorics: Topics, Techniques, Algorithms – P.J. Cameron (Cambridge UP, 1994); Library reference 512.23 CAM.

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MT5461 Theory of Error-Correcting Codes (Term 2)Prerequisite: Undergraduate courses covering linear algebra and probability.Teaching: 33hr lectures and seminars, 167hr private study, including problem sheetsAssessment: 2hr written examination

AimsTo provide an introduction to the theory of error correcting codes employing the methods of elementary enumeration, linear algebra and finite fields.

Learning OutcomesOn completion of the course, students should be able to: calculate the probability of error of the necessity of retransmission under various

assumptions for a binary symmetric channel with given cross-over probability; prove and apply various bounds on the number of possible code words in a code of

given length and difference; reduce a linear code to standard form, finding a parity check matrix, building standard

array and syndrome decoding tables, including for partial decoding; use MOLSs to construct large linear codes of certain parameters; know/prove/apply the theorem that a cyclic code of length n over a field consists of the

code words corresponding to all multiples of any factor of xn 1; understand the structure of BCH code; know/prove/apply the Peterson-Zierler decoding algorithm.

ContentDiscrete communication channels; Shannon’s coding theorem. Theory of linear block codes with special examples. Matrix description,. Standard arrays and Hamming codes, perfect codes. Packing points in Vn(q) - Hamming, Singleton, Plotkin, and Gilbert-Varshamov bounds. Structure of finite fields. Cyclic codes, polynomial description. BCH codes, RS codes. Decoding techniques.

Indicative TextsA First Course in Coding Theory Ray Hill (OUP). Library Ref. 001.539 HILCoding Theory – a First Course S Ling and C Xing (Cambridge UP 2004) Library Ref. 001.539 Lin

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MT5462 Advanced Cipher Systems (Term 1)Prerequisite: Undergraduate courses covering linear algebra and probability.Teaching: 33hr lectures and seminars, 167hr private study, including problem sheetsAssessment: 2hr written examination

AimsTo introduce both symmetric key cipher systems and public key cryptography covering methods of obtaining the two objectives of privacy and authentication.

Learning OutcomesOn completion of the course the student should be able to: understand the concepts of secure communications and cipher systems; understand and use statistical information and the concept of entropy in the cryptanalysis

of cipher systems; understand the structure of stream ciphers and block ciphers; know how to construct as well as have an appreciation of desirable properties of key

stream generators, understand and manipulate the concept of perfect secrecy; understand the modes of operation of block ciphers and their properties; understand the concept of public key cryptography, including details of the RSA and

ElGamal cryptosystems both in the description of the schemes and in their cryptanalysis; understand the concepts of authentication, identification and signature, be familiar with

techniques that provide these, including one way functions, hash functions and interactive protocols, including the Fiat-Shamir scheme;

understand the problems of key management, be aware of key distribution techniques.

ContentExamples of ciphers. Mathematical and statistical aspects of cipher systems. Substitution ciphers; Shannon’s theory; stream and block ciphers; public key systems; authentication/identification; digital signatures.

Indicative TextCodes and Cryptography – D Welsh (Oxford UP 1988). Library Ref. 001.5436 WELCipher Systems – H J Beker and F C Piper (Van Nostrand 1982)

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MT5464 Operational Research Methodology (Term 1)Prerequisite: Undergraduate courses covering linear algebra and probability.Teaching: 33hr lectures, discussion classes and seminars, 167hr private study.Assessment: The candidate writes two short reports (about 2500 words) on case studies chosen from a list provided by the lecturer (25% each), a longer report (about 5000 words) on a topic chosen by the student with the agreement of the lecturer (40%), and an oral examination (10%).

Aims to show how the mathematical techniques of OR (such as mathematical programming,

combinatorial optimization, statistics and others), have been used in real situations; to develop a critical faculty, so that students can distinguish between good and bad

approaches to a problem; to look at some of the factors important in the relationship between analyst and client; to give practice in obtaining information from journals to improve communication skills.

Learning OutcomesOn completion of the course the student should be able to: be able to make constructive criticism based on the published description of an OR

project; decide whether a problem has been tackled by a suitable method, and whether or not a

satisfactory solution has been found; see what the client would understand by ‘a good solution’; collect information from journals, books, people and other sources and decide what is

appropriate; write a coherent report on what they have found; talk about their report.

ContentThis course discusses the problems faced by the OR analyst; problem formulation, choice of methodology, communication with the client, social aspects and also how techniques work in practice, and is based on case studies. These are normally taken from papers published in the twelve months before the course begins.

Indicative TextNone: the course involves reading recent papers, mostly in the Journal of the Operational Research Society.

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MT5465 Network Algorithms (Term 2)Prerequisite: An undergraduate course in Discrete MathematicsTeaching: 33hr lectures and seminars, 167hr private study, including problem sheetsAssessment: 2hr written examinationAims To introduce the formal idea of an algorithm, when it is a good

algorithm and techniques for constructing algorithms and checking that they work.

To explore connectivity and colourings of graphs, from an algorithmic perspective.

To show how algebraic methods such as path algebras and cycle spaces may be used to solve network problems.

Learning OutcomesOn completion of the course the students should be able to: use particular algorithms which optimize various properties for

graphs and networks and prove that they work; understand ideas of complexity exemplified in particular by the

Travelling Salesman Problem; apply Fleury’s and Tucker’s algorithms to find Eulerian trails; find chromatic polynomials and illustrate Vizing’s theorem on edge

colourings; use path algebra methods to find maximal flows, critical paths and

similar problems.Content

Trees: Algorithms for minimum spanning trees.Sorting and searching: Sorting methods including bubble sort and heap sort. Depth first search and breadth first search. Shortest paths.The Travelling Salesman Problem: Branch and bound method, upper and lower bounds, approximate methods. Flows in networks: The max-flow min-cut theorem. An algorithm for finding maximum flows. Matching problems: Hall's theorem. Maximum and complete matchings. Alternating paths and applications. Menger's theorems on edge and vertex connectivity.Eulerian trails: Algorithms for finding them: Fleury's algorithm; Tucker's algorithm.Hamiltonian paths: Ore’s and Dirac’s theorems on Hamiltonian cycles.Colouring graphs: Vertex and edge colourings; chromatic polynomials. Brook's, Vizing's and König's theorems. Colouring maps, the four-colour theorem. Path algebras: Definitions, strong and weak closure, matrices over path algebras, absorptive matrices, applications to various network problems, including critical path analysisCycle spaces: Definitions, feasible flows, displacement networks. Maximum flow minimum cost flows, cost-reducing cycles.

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Indicative TextsAlgorithmic Graph Theory A. Gibbons (Cambridge UP). Library Ref. 512.23 GIBDiscrete Mathematics N.L. Biggs (Oxford UP). Library Ref. 510 BIGGraphs and Networks – B. Carré (Oxford UP). Library Ref. 512.23 CAR

MT5466 Public Key Cryptography (Term 2)Prerequisite: MT5485Teaching: 33hr lectures and seminars, 167hr private study, including problem sheetsAssessment: 2hr written examination

Aims To introduce some of the mathematical ideas essential for an understanding of public

key cryptography, such as discrete logarithms, integer factorisation, lattices and elliptic curves;

To introduce several important public key cryptosystems, such as RSA, Rabin, ElGamal, Diffie-Hellman, Schnorr signatures;

To discuss modern notions of security and attack models for public key cryptosystems.

Learning outcomesOn completion of the course, students should:

be familiar with the RSA and Rabin cryptosystems, the hard problems on which their security relies and certain attacks on them;

have a basic knowledge of finite fields and elliptic curves over finite fields, and the discrete logarithm problem in these groups;

be familiar with cryptosystems based on discrete logarithms, and some algorithms for solving the discrete logarithm problem;

know the definition of a lattice and be familiar with the LLL algorithm and some applications of lattices in cryptography and cryptanalysis;

be able to define security notions and attack models relevant for modern theoretical cryptography, such as indistinguishability and adaptive chosen-ciphertext attack.;

be able to critically analyse cryptosystems; have experience with implementing cryptosystems and cryptanalytic methods using a

computer algebra package such as Mathematica.

ContentBackground: Integers modulo n; Chinese remainder theorem; finite fields; fast exponentiation; public key cryptography and security; complexity theory; primality testing and certificates.RSA/Rabin: Key generation; implementation; encryption and signatures with OAEP; the RSA problem and relationship with factoring; square roots modulo a prime; Hastad attack; Wiener attack; smooth numbers; survey of integer factorisation methods such as method and index calculus.Discrete logarithms: Diffie-Hellman; ElGamal encryption; Schnorr signatures; Diffie-Hellman problem and decision Diffie-Hellman; methods to solve discrete logarithms such as baby-step-giant-step, Pollard rho and lambda, index calculus.

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Lattices: Definition of a lattice; GGH cryptosystems; LLL algorithm; lattice attacks on RSA with small public or private exponents.Elliptic curves: Group law; Hasse bound; group structure; ECC protocols; elliptic curve factorisation and primality certificates; Maurer equivalence of DH and DL.

Indicative TextsCryptography: an introduction – Nigel Smart (McGraw Hill) Library Ref. On orderCryptography theory and practice – Doug Stinson (CRC press, 2nd ed.) Library Ref. 001.5436 STI

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MT5473 Advanced Control Theory (Term 1)Prerequisite: Undergraduate courses in linear algebra and complex variableTeaching: 40 hours of lectures and examples classes. 160 hours of private study, including work on problem sheets and examination preparation. Assessment: 2hr written examination

Aims To show how physical control systems can lead to systems of equations. To extend the methods of solution available from earlier courses. To study the stability of these systems, especially using complex analysis. To develop a method for analysing non-linear behaviour of systems.

Learning outcomesOn completion of the course the student should be able to to show how physical systems can be described in terms of block diagrams. to get the differential equations describing linear systems in the standard form. to show how these equations can be solved (i) using Laplace transforms or z-

transforms; (ii) using a matrix exponential in the time-independent case; (iii) using transfer functions; (iv) expressing the solution in terms of the Fundamental Matrix.

to define different types of stability including asymptotic and BIBO stability; to apply the methods to examples.

to examine stability using poles of the transfer function, the Routh criterion, the circle criterion, and Nyquist plots.

to establish criteria (in terms of rank) for the complete controllability and for the complete observability of linear systems, and to apply these to examples.

to use describing functions to deal with non-linear systems.

ContentState space: the underlying concepts of state, control and output variables, open and closed loop control. Discrete and continuous time systems. Solution of time-independent and time-dependent controlled equations, using matrix, Laplace transform and z-transform techniques as appropriate. Transfer functions. Realizations of a system.Controllability and observability: criteria for controllability and observability: duality. Minimal realizations.Stability: eigenvalue analysis. Routh analysis. Root locus method. Nyquist criterion and the circle criterion. Relocation of poles.Optimal control: dynamic programming approach; time-independent and dependent systems. Pontryagin's Theorem (statement). Constrained control, using Lagrange multiplier approach. Bang-bang control.Non-linear systems: periodic behaviour, use of Fourier series, describing functions, relays, saturation and hysteresis.

Indicative TextsIntroduction to Mathematical Control Theory S. Barnett (Oxford). Library Ref. 519.3 BARIntroduction to Control Theory O.L.R. Jacobs (Macmillan). Library Ref. 519.3 JAC

MT5485 Applications of Field Theory (Term 1)

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Prerequisite: An undergraduate course covering the elementary theory of groups, rings and fields.Teaching: 33hr lectures and seminars, 167hr private study, including problem sheetsAssessment: 2hr written examination

Aims: To introduce some of the basic theory of extension fields, with special emphasis on finite fields and their applications.

Learning Outcomes:On completion of the course, students should be able to: understand simple field extensions of finite degree; classify finite fields and determine the number of irreducible polynomials over a finite

field; state the fundamental theorem of Galois theory for finite fields; compute in a finite field; understand some of the applications of fields.

Content (Other than the first section, all topics refer to finite fields only, unless otherwise specified.)Extension theory: Polynomial factorisation. Field extensions. Simple extensions. The degree of an extension. Applications to ruler and compass constructions.Classifying finite fields: The number of irreducible polynomials. Existence and uniqueness of finite fields. Concrete representations of a finite field.The structure of finite fields: Roots of irreducible polynomials and the Frobenius automorphism. Cyclotomic polynomials. The Galois correspondence for finite fields. An indication of Galois correspondence for general fields. The norm and trace of an element. Applications to m-sequences. Dual and self-dual bases. Normal bases and the normal basis theorem. Applications to multiplication in finite fields.Discrete logarithms: The discrete log problem and its applications. The Pohlig-Hellman and baby step, giant step algorithms.

Indicative TextsIntroduction to Finite Fields and their Applications – R. Lidl and H. Niederreiter (Cambridge UP, 1994); Library ref. 512.4 LID.Galois Theory – I. Stewart (Chapman and Hall, 1989); Library ref. 512.4 STE.

Academic Session: 2007-8

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Course Code: MT5492 Course Value: 200 hours Status:(ie:Core, or Optional) Optional

Course Title: Integration and Function SpacesAvailability: (state which teaching terms)

Term 2

Prerequisites: An undergraduate course in real analysis Recommended: None

Aims: To provide a unified introduction to integration and to a functional analytic account of the most important function spaces.

Learning Outcomes:

On completion of the course, students should be able to: understand and be able to apply the basic ideas of the Lebesgue integral; understand how the basic ideas of functional analysis apply to the classical function

spaces.

CourseContent:

Measure: Lebesgue outer measure on sets in the real line; it is sub-additive and gives an interval its length. Measurable sets: these form a sigma algebra on which outer measure is additive on disjoint countable unions. Hence Lebesgue measure. Regularity of the measure; approximation by finite sets of intervals. Existence of non-measurable setsMeasurable functions and the Lebesgue integral: Lebesgue integral for measurable simple functions, hence for non-negative measurable functions.Fatou’s lemma; Lebesgue Monotone Convergence Theorem. Non-negative measurable functions are monotone limits of simple functions.Integration of general measurable functions. Lebesgue’s Dominated Convergence Theorem, with examples. Comparison of Riemann and Lebesgue integrals. Differentiation of indefinite integrals, the Lebesgue Density Theorem. Integration of series of functions. Spaces of functions: Lp space and its norm. Jensen, Holder and Minkowski’s inequalities (

). Lp spaces are complete normed spaces . Continuous and step functions are dense in Lp. Normed space: linear transformations and their continuity and norms. Dual spaces. The dual of Lp is Lq where 1/p + 1/q = 1. The dual of L1. L2 as a Hilbert space. The least distance theorem and orthogonal complements. Orthonormal sets. and Legendre Polynomials. Fourier series coefficients; in L2 , Fourier series converge and obey Parseval’s identity.

Teaching & Learning Methods:

33 hours of lectures and examples classes.167 hours of private study, including work on problem sheets and examination preparation. This may include discussions with the course leader if the student wishes.

Key Bibliography:

G. de Barra – Measure Theory and Integration (Ellis Horwood 1981) Library reference 515.52 DEBW. Rudin – Real and Complex Analysis (McGraw Hill 1966) Library reference 515.23 RUDW. Rudin – Functional Analysis (McGraw Hill 1974) Library reference 515.61 RUD

Formative Assessment & Feedback:

Formative assignments in the form of 8 problem sheets. The students will receive feedback as written comments on their attempts.

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