Mathematics Department King's College, University of London The MSc in Mathematics

The Department offers a taught MSc degree in Mathematics, based on course work and a project. This course is centred on either Pure Mathematics or Mathematical Physics. The wide range of lectures available allows considerable flexibility. The degree course begins in September and requires one year of full-time study, or it may be taken part-time over two years. Candidates must satisfy the examiners in respect of both their written examinations and their project, and a Distinction will be awarded if the performance in both parts is of a sufficiently high standard.

Table of Contents:

• Taught Coursework
• The courses listed by core area
• The project
• Part-time study
• Entrance requirements
• Term dates
• How to apply and the application form
• Brief descriptions of the courses

Taught Coursework for the MSc

The taught course-work is organised on a unit system, and students normally offer eight half-unit courses for examination (courses usually carry 0.5 course units). Passes in a minimum of three units are required for the award of the degree. Merit or Distinction requires passes at the appropriate level in four units. Details of the courses available are listed below. Each student is assigned an advisor who will help him or her choose a coherent programme of courses. Typically this will be based on choosing predominantly mathematical physics or pure mathematics courses although a more even spread of courses is in principle allowed, as well as the possibility to take courses from the other MSc programmes.

As well as these postgraduate courses, students may offer up to two units from selected undergraduate courses if required as necessary background material. Students may also take courses at other London Colleges subject to the approval of their advisor.

Courses

The courses which are expected to be available in 2004/2005 are listed below. These are the current selection from the full list of approved courses which may be offered in different years. Some courses appear in both of the two core blocks, and this overlap, together with the many options available outside the core, allows considerable flexibility of choice.

Pure Mathematics

A (non-exhaustive) selection of undergraduate courses:

Linear systems and control theory, Geometric Structures, Galois Theory, Topology, Logic, Non-linear dynamics.

Mathematical Physics

The Project

Each MSc candidate must complete a project (normally about 30 pages) in some area of mathematics at the postgraduate level. This involves investigating a topic (not covered in the courses) using several sources (including relevant research papers) and writing a coherent exposition of it. The report must be the student's own work in the sense that it should be an original account of the material, but it need not contain any new mathematical results.

Project supervisors can be expected to give general guidance and advice, including a reading list, but they will not give a skeleton version of the report (although they will probably mention one or two major results that they think should be included). The report is normally completed after the written examinations in May for submission in September. An oral examination on the project topic may be required at the discretion of the examiners (although this has rarely been exercised).

Part-Time Study for the MSc Degree

A part-time student may opt to sit the complete written examination two years after entry, or may sit part I after one year and part II after the second year. In either case, the report must be submitted in September of the second academic year. However, students are advised to submit a project title in the first year and then work on the project during the summer vacation between the two years. Part-time candidates are not permitted to proceed to the second year if they have opted to be examined in two parts and have failed the first part.

Entrance Requirements

Applications from overseas students are welcomed.

Dates of Terms

The written examinations for the taught courses are (usually) held in May/June and the project report must be submitted in September.

Brief descriptions of the course contents

These are the courses which may be given in the MSc programme. The courses which are expected to be taught in 2004/2005 are listed above.

Advanced general relativity (CMMS38)
Einstein's field equations, physically significant solutions, black holes and gravitational waves. Lagrangian and Hamiltonian formulations of general relativity. The global structure of spacetime

Advanced neural nets (CMNN15)
Radial basis functions and function approximation. Self-organising maps and learning vector quantisation. Bayesian analysis of learning in layered networks; regularisation and generalisation. Gaussian processes. Support Vector Machines.

Algebraic curves (CMMS16)
Curves in the affine and projective planes. Bezout's theorem and the intersection of curves. Points of inflection. Branched coverings. The degree-genus formula. Riemann surfaces. The Weierstrass P-functions. Holomorphic differentials. The group law on a cubic and the addition formula for elliptic integrals

Algebraic number theory (CMMS03)
This course treats Ideal theory of Dedekind domains and the fact that the ring of integers in an algebraic number field is a Dedekind domain; lattices in rational vector spaces; the discriminant of an algebraic number field; ramification of primes; Galois number fields; Frobenius elements; Cyclotomic, quadratic and cubic fields, proof of quadratic reciprocity via prime decomposition in quadratic subfields of cyclotomic fields; discrete subgroups of R^m; Minkowski's theorem; finiteness of the ideal class group; characters of finite Abelian groups; modular characters; Dirichlet series; L-series; primes in arithmetic progression.

Algebraic quantum theory (CMMS37)
Approach to quantum theory and statistical mechanics using algebraic methods. Algebraic structure of quantum theory, C*-algebras. Applications including symmetry breakdown, phase transitions, metastable states.

Algebraic topology (CMMS14)
An introduction to algebraic topology stressing geometric aspects; elementary homotopy theory and covering spaces, singular homology and cohomology: exact sequences, excision and Mayer-Veitoris. Also topics such as higher homotopy groups, the Hurewicz theorem, manifolds and duality; products and the Lefschetz fixed point theorem.

Applied Probability and stochastics (CMFM01)
Sample spaces, probabilities, random variables, conditional expectation. Moments, generating functions, characteristic functions. Gaussian random variables, central limit theorem, strong law of large numbers. Markov chains, Chapman Kolmogorov equation, classification of states, stationary distributions, ergodicity. Random processes, martingales, Brownian motion, Poisson process. Entropy and information.

Basic analysis (CMMS05)
A self-contained excursion through some fundamentals of functional analysis. Topics to be discussed: Banach spaces, open mapping and closed graph theorems, Hahn-Banach theorem, dual spaces, Banach-Alaoglu theorem.

C*-algebras (CMMS06)
C*-algebras, Gelfand theory, states, representations and further topics as time permits.

Commutative algebra (CMMS04)
Modules over a Noetherian ring, Hilbert Basis Theorem, primary decomposition, localisation, Nakayama's Lemma, Krull's Intersection Theorem, Principal Ideal Theorem, Cohen-Seidenberg Theorems.

Computational spectral theory (CMMS10)
Spectral theorem for self-adjoint matrices. Translation invariant operators and Fourier series. Periodic operators and spectral gaps. Spectral computations for tridiagonal matrices. Singular values. Fredholm operators. Stability of essential spectrum. Rayleigh-Ritz method. Temple-Lehmann method. Complex resources.

Fourier analysis (CMMS11)
Fourier Analysis on the circle; Fourier series and Fourier transform on the line and on Rⁿ. L₂-Fourier transform. Parseval's formula, Schwartz classes S and S'. Applications in analysis and theory of PDE.

Ideals of compact operators (CMMS07)
Ideals of B(H). The minimax theorem. The Hilbert-Schmidt class. Traces. The pre-dual of B(H). Classes C^p and their properties. Duality. Triangular forms.

Information theory in neural networks (CMNN14)
Information theory: entropy (joint and relative), Shannon's theorems relating entropy to optimal coding. Application to neural networks: Boltzmann machine learning, unsupervised learning in feed-forward networks by maximum information preservation, quantifying efficiency of associative memories.

Risk Neutral Valuation (CMFM02)
Forward prices, discounting, arbitrage-free pricing; binomial trees, derivatives pricing in discrete time by use of binomial lattices, geometric Brownian motion, volatility and drift, martingales and conditional expectation, Itô calculus, hedging portfolios, replication, Black-Scholes model, put-call parity. Risk premium, risk-neutral measure.

Lie groups and algebras (CMMS01)
Matrix (Lie) groups. Examples (GL_n, U_n, O_n, Sp_n, Heisenberg groups, Galilean, Lorentz, Poincaré groups). Exponential and logarithm of a matrix, 1-parameter subgroups, Lie algebras. Basic representation theory, examples (su(2), su(3), su(n)). Relation with spherical harmonics. Semisimple and solvable groups, Engel's and Lie's theorems.

Low-dimensional quantum field theory (CMMS33)
Quantum field theory in low dimensions, in particular, two dimensions. Conformal field theory and operator product expansion. Algebraic structures. Integrable systems.

Low-dimensional topology (CMMS15)
Knots, links and projections; braids, braid groups, Markov equivalence; knot polynomials, Hecke algebras and skein relations; Alexander polynomial and Seifert surfaces; 3-manifolds, examples; Dehn surgery; Lickorish's theorem; homology spheres; Casson's invariant; Kirby's theorem.

Manifolds (CMMS18)
Manifolds and functions; vectors and vector fields; tensors and tensor fields; Lie derivative; differential forms and exterior derivative; covariant derivatives, connections, torsion and curvature; metrics and Riemannian geometry; integration; symplectic geometry; further topics in differential geometry.

Mechanics, Relativity and Quantum Theory (CMMS30)
Introduction to the concepts of modern mathematical phsyics.

Measure theory (CMMS13)
This is an advanced course setting up the basic ingredients of measure theory. It includes the definition of sigma-fields and countably additive measures and examples of such measures; the construction of the Lebesgue integral with respect to a given measure; the construction of product measures. Further applications include: standard Brownian motion, aspects of probability theory, Fourier Analysis, completeness of L_p spaces, Hausdorff measures, Banach-Tarski paradox, measure-theoretic aspects of dynamical systems.

Operator theory (CMMS08)
Hilbert and Banach spaces. Riesz representation theorem. The adjoint. Orthogonal projections. Spectral theory of bounded linear operators. Spectral theorem for bounded linear operators. Spectral theorem for compact operators. Applications to differential and integral equations.

Partial differential equations (CMMS12)
Harmonic and subharmonic functions. Second order elliptic operators, quadratic forms, self-adjointness, spectrum. Estimation of eigenvalues, minimax, the uncertainty principle. The heat equation and Schrödinger equation.

Point particles and strings (CMMS34)
An introduction to modern string theory starting from a relativistic point particle. The bosonic string is defined and quantised. The superstring is introduced.

Supersymmetry (CMMS40)
Elementary supersymmetry is discussed and applied to the particle case.

Quantum field theory (CMMS32)
A basic course in quantum field theory which develops the subject using path integrals. Perturbation theory, Feynman diagrams and an introduction to renormalisation are covered.

Quantum groups (CMMS02)
Introduction to Hopf algebras; quantised universal enveloping algebras; quantised function algebras; classical and quantum Yang-Baxter equations; representation theory of quantum groups.

Riemann surfaces (CMMS17)
Background facts from topology and complex analysis. Solution of the basic existence problems. Construction of the Jacobian and Picard varieties, Abel's theorem. The Riemann-Roch theorem. Some special curves of low genus. The moduli problem.

Stochastic Analysis (CMFM04)
Filtrations, stopping times and martingales in continuous time. Itô's isometry and the Itô integral. Random time changes, martingale representation theorems. Local time, Tanaka's formula. Reflected Brownian motion. Change of measure. Stochastic differential equations. Ornstein-Uhlenbeck process, geometric Brownian motion, Brownian bridge, Bessel processes. Diffusions, Fokker-Planck equation, Feynman-Kac formula; Cauchy and Dirichlet problems.

Statistical dynamics (CMMS35)
Non equilibrium statistical dynamics is treated as a dynamical system. Application to chemical kinematics, heat conduction. Quantised and second quantised Boltzmann equation. Derivation of equilibrium statistical mechanics and thermodynamics.

Statistical mechanics (CMMS36)
The basis principles of thermodynamics and statistical mechanics. Non-interacting systems. Phase transitions and critical exponents. Transfer matrix methods. Mean-field theory. An introduction to real-space renormalisation method.

Statistical mechanics of neural nets (CMNN13)
Application of statistical mechanics methods to analyse and quantify operation and learning in neural networks. Exact solutions, obtained by replica theory, theory of stochastic processes, path integrals, including: statics and dynamics of attractor neural networks, Gardner theory, dynamics of learning.

Spectral theory of Markov chains (CMMS09)
The course will study the spectral properties of operators associated with Markov Chains and random walks on graphs. Topics to be discussed: Markov transition matrices, invariant states, ergodicity, continuous time Markov chains, the infinitesimal generator, etc.