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 fulltime study, or it may be taken parttime 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.
The taught coursework is organised on a unit system, and students normally offer eight halfunit 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.
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.
A (nonexhaustive) selection of undergraduate courses:
Linear systems and control theory, Geometric Structures, Galois Theory, Topology, Logic, Nonlinear dynamics.
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).
A parttime 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. Parttime 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.
Applications from overseas students are welcomed.
The written examinations for the taught courses are (usually) held in May/June and the project report must be submitted in September.
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. Selforganising
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 degreegenus formula. Riemann surfaces. The Weierstrass
Pfunctions. 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;
Lseries; 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 MayerVeitoris. 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 selfcontained excursion through some fundamentals of functional
analysis. Topics to be discussed: Banach spaces, open mapping and
closed graph theorems, HahnBanach theorem, dual spaces,
BanachAlaoglu 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, CohenSeidenberg Theorems.
Computational spectral theory (CMMS10)
Spectral theorem for selfadjoint 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. RayleighRitz
method. TempleLehmann method. Complex resources.
Fourier analysis (CMMS11)
Fourier Analysis on the circle; Fourier series and Fourier transform
on the line and on R^{n}.
L_{2}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 HilbertSchmidt class.
Traces. The predual 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 feedforward
networks by maximum information preservation, quantifying efficiency
of associative memories.
Risk Neutral Valuation (CMFM02)
Forward prices, discounting, arbitragefree 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, BlackScholes model, putcall parity.
Risk premium, riskneutral 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, 1parameter 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.
Lowdimensional 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.
Lowdimensional topology (CMMS15)
Knots, links and projections; braids, braid groups, Markov
equivalence; knot polynomials, Hecke algebras and skein relations;
Alexander polynomial and Seifert surfaces; 3manifolds, 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 sigmafields 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,
BanachTarski paradox, measuretheoretic 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, selfadjointness, 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
YangBaxter 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 RiemannRoch 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.
OrnsteinUhlenbeck process, geometric Brownian motion, Brownian bridge,
Bessel processes. Diffusions, FokkerPlanck equation, FeynmanKac
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.
Noninteracting systems. Phase transitions and critical exponents.
Transfer matrix methods. Meanfield theory. An introduction to
realspace 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.
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