Mathematics

Core modules:

Dissertation

The dissertation is piece of extensive work (10-20,000 words) which provides students' with the opportunity to synthesise theoretical knowledge on a subject that is of interest to them. Students will gain experience of the phases of a relatively large piece of work: planning to a deadline; researching background information; problem specification; the carrying through of relevant analyses; and reporting, both at length through the dissertation and in summary through an oral presentation.

60 credits

Optional modules - six or seven from:

Topics in Advanced Fluid Mechanics

This module is lectured with the undergraduate module MAS411 of the same name. This module aims to describe advanced mathematical handling of fluid equations in an easily accessible fashion. A number of topics are treated in connection with the mathematical modelling of formation of the (near-)singular structures with concentrated vorticity in inviscid flows. After discussing prototype problems in one and two dimensions, we describe the three-dimensional flows in terms of vortex dynamics. Minimally required mathematical tools are explained during the course in a self-contained manner. Candidates are directed to read key original papers on some topics to deepen their understanding.

20 credits

Special Topics (Autumn)

There may be cases where students have already covered some of the material in the standard slate of courses, or where students need further background to cover prerequisite material. In addition, MAGIC courses, given to PGT students, may come under this heading.

10 credits

Mathematical methods and modelling of natural systems

This module will develop mathematical methods, including integral transforms, asymptotics and perturbation, applied to problems including differential equations and evaluation of integrals.Mathematical modelling enables insight in to a wide range of scientific problems. This module will provide a practical introduction to techniques for modelling natural systems. Students will learn how to construct, analyse and interpret mathematical models, using a combination of advanced mathematical and computational methods. Study systems will be drawn from throughout the environmental and life sciences.

20 credits

Special Topics (Spring)

There may be cases where students have already covered some of the material in the standard slate of courses, or where students need further background to cover prerequisite material. In addition, MAGIC courses, given to PGT students, may come under this heading.

10 credits

Stochastic Processes and Finance

A stochastic process is a mathematical model for phenomena unfolding dynamically and unpredictably over time. This module studies two classes of stochastic process particularly relevant to financial phenomena: martingales and diffusions. The module develops the properties of these processes and then explores their use in Finance. A key problem considered is that of the pricing of a financial derivative such as an option giving the right to buy or sell a stock at a particular price at a future time. What is such an option worth now? Martingales and stochastic integration are shown to give powerful solutions to such questions.

20 credits

Algebra I

This module is lectured with the undergraduate modules MAS438 and MAS442 and considers Fields and Galois Theory. The first half of the module covers basic field theory (field extensions, constructability, etc.) and the second gives the applications to the theory of equations: Galois groups of extensions, and of polynomials, culminating in one of the crowning glories of Galois's work classifying polynomials which are soluble by radicals.

20 credits

Algebra II

This module is lectured with the undergraduate modules on commutative algebra (MAS437 and MAS434), and considers rings and commutative algebra. The first half of the module covers basic ring and module theory (e.g., modules and ideals) and the second develops the theory of commutative algebra, an essential prerequisite for modern algebraic geometry, as well as being important in its own right.

20 credits

Analysis I

This module is lectured with the undergraduate module MAS435 Functional Analysis. Functional analysis is the study of infinite-dimensional vector spaces equipped with extra structure. Such spaces arise naturally as spaces of functions. As well as being a beautiful subject in its own right, functional analysis has numerous applications in other areas of both pure and applied mathematics, including Fourier analysis, study of the solutions of certain differential equations, stochastic processes, and in quantum physics. In this course we focus mainly on the study of Hilbert spaces- complete vector spaces equipped with an inner product- and linear maps between Hilbert spaces. Applications of the theory considered include Fourier series, differential equations, index theory, and the basics of wavelet analysis.

20 credits

Analysis II

This module is lectured with the undergraduate modules MAS331 Metric Spaces and MAS350 Probability and Measure. This unit explores ideas of convergence of iterative processes in the more general framework of metric spaces. A metric space is a set with a ¿distance function which is governed by just three simple rules, from which the entire analysis follows. It also will give students an additional opportunity to develop skills in modern analysis as well as providing a rigorous foundation for probability theory.

20 credits

Algebraic Topology I

This will be a first course on algebraic topology, following on from the introduction to topology given in the Level 3 course Metric spaces. Topology studies the shape of space, with examples such as spheres, the Mobius Band, the Klein bottle, the torus and other surfaces. The first task is to formalise this notion of space, and then the challenge is to work out when a given space can be smoothly deformed into another, where stretching and bending is allowed, but cutting and glueing is not. Algebraic topology builds a powerful bridge between space and algebra, enabling us to use familiar algebraic techniques from group theory to study spaces and their deformations.

20 credits

Analytical Dynamics and Classical Field Theory

This module is lectured alongside the undergraduate module MAS412 of the same name. Mathematicians like Lagrange, Hamilton and Jacobi discovered that underlying Newton's mechanics are wonderful mathematical structures. In the first semester we discuss this work, its influence on the subsequent formulation of field theory, and Noether's theorem relating symmetries and conservation laws.In the second semester, Einstein's theory of gravity, General Relativity, will be introduced, preceded by mathematical tools such as covariant derivatives and curvature tensors. Einstein's field equations, and two famous solutions, will be derived. Two classic experimental tests of General Relativity will be discussed.

20 credits

Waves and Magnetohydrodynamics

This module is lectured with the undergraduate modules MAS315 Waves and MAS422 Magnetohydrodynamics. Studying wave phenomena has had a great impact on Applied Mathematics. This module looks at some important wave motions with a view to understanding them by developing from first principles the key mathematical tools. In the second part it gives an introduction to classical magnetohydrodynamics. Magnetohydrodynamics has been successfully applied to a number of astrophysical problems (e.g., to problems in Solar and Magnetospheric Physics), as well as to problems related to laboratory physics, especially to fusion devices.

20 credits

Directed Reading in Mathematics

This module will be used as a reading module for the MSc and may cover material deemed useful by a dissertation supervisor, or background material for a student. Precise content and subject matter will depend on the interests of the student, as well as the level and background of the individual student.

20 credits

Special Topics I

There may be cases where students have already covered some of the material in the standard slate of courses, or where students need further background to cover prerequisite material. In addition, MAGIC courses, given to PGT students, may come under this heading.

20 credits

Analytic Number Theory

The module will discuss the distribution of prime numbers (Bertrand's Postulate, prime counting function, the statement of the Prime Number Theorem and some of its consequences), basic properties of the Riemann zeta function, and Euler products of L-series. A big chunk of the module will be dedicated to Dirichlet's Theorem on primes in arithmetic progressions and it's proof.

10 credits

Machine Learning

Machine learning lies at the interface between computer science and statistics. The aims of machine learning are to develop a set of tools for modelling and understanding complex data sets. It is an area developed recently in parallel between statistics and computer science. With the explosion of 'Big Data', statistical machine learning has become important in many fields, such as marketing, finance and business, as well as in science. The module focuses on the problem of training models to learn from training data to classify new examples of data.

10 credits

Financial Mathematics

The discovery of the Capital Asset Pricing Model by William Sharpe in the 1960's and the Black-Scholes option pricing formula a decade later mark the beginning of a very fruitful interaction between mathematics and finance. The latter obtained new powerful analytical tools while the former saw its knowledge applied in new and surprising ways. (A key result used in the derivation of the Black-Scholes formula, Ito's Lemma, was first applied to guide missiles to their targets; hence the title 'rocket science' applied to financial mathematics). This course describes the mathematical ideas behind these developments together with their application in modern finance, and includes a computational project where students further explore some of the ideas of option pricing.

10 credits

Advanced Operations Research

Mathematical Programming is concerned with the algorithms that deal with constrained optimisation problems. We consider only constraints which are linear, and for which the objective function to be maximised or minimised is also linear. These problems are not continuously differentiable and so they do not fall into the category of problems considered in organisation; special algorithms have to be developed. The module considers not only the solution of such problems but also the important area of post-optimality analysis; i.e. given the solution can one answer questions about the effect of small changes in the parameters of the problem (such as values of the cost coefficients)? Additional topics will include the transportation and assignment problems.

10 credits

Applied Probability

The unit will link the idea of probability modelling to Statistics. It will explore a range of models that can be constructed for random phenomena that vary in time or space - the evolution of an animal population, for example, or the number of cancer cases in different regions of the country. It will illustrate how probability models are built and how they might be applied: how likelihood functions for a model may be derived and used to fit the model to data, and how the result may be used to assess model adequacy and suggest improvements.

10 credits