## 400 level (postgraduate) papers and modules

Please contact the Director of Studies for any questions or queries.

### Papers and enrolment

Which paper(s) you need to enrol for depends on your program of study. Here are a few common cases. If your particular situation is not covered by these, please contact the Director of Studies.

**BSc(Hons), BA(Hons), or PGDipSci in Mathematics**: Enrol for the papers MATH401 and MATH403 (Semester 1), MATH402 and MATH406 (Semester 2) and MATH490 (Full Year). Then select eight modules from the list below, either five in Semester 1 and three in Semester 2, or, four in Semester 1 and four in Semester 2.**MSc in Mathematics (two years)**: For year 1, enrol for the papers MATH401 and MATH403 (Semester 1), MATH402 and MATH406 (Semester 2) and MATH495 (Full Year). Then select eight modules from below, either five in Semester 1 and three in Semester 2, or, four in Semester 1 and four in Semester 2. See here for information about year 2.**Some other program (e.g., Statistics or Physics) which allows you to do 20-points in 400-level Mathematics**: Enrol for the paper MATH401 in Semester 1 or MATH402 in Semester 2. Then select two modules from the list below. The two modules can be either in the same semester or in two consecutive semesters.

Please send your choice of modules to the Director of Studies (you do not need to formally enrol for these modules).

We may be able to offer some of our 300-level papers as 10-point 400-level modules. All the degrees in Mathematics above allow you to take 20 points in 400-level Statistics. We particularly recommend the 20-point paper STAT444 Stochastic Processes.

### First semester Modules

MATH4FA Functional Analysis First Semester 10 points

The main focus of this course is the analysis of linear mappings between normed linear spaces. It turns out that many problems in analysis can be studied from this abstract point of view, which recognizes important underlying principles without getting lost in technical details. The applications of the approach ranges from differential and integral equations through problems in optimal control theory and numerical analysis to probability theory to name a few. This introductory course covers some of the basic constructions and principles of functional analysis: completions of metric/normed spaces, the Hahn-Banach Theorem and its consequences, dual spaces, bounded linear operators and their adjoints, closed operators, the Open Mapping and Closed Graph Theorems, the Principle of Uniform Boundedness and some elements of the spectral theory for closed linear operators. The applications of the abstract concepts are demonstrated through various examples from different branches of analysis.

MATH4MI Measure and Integration First Semester 10 points

This course introduces a modern theory of integration of real-valued functions via measure theory.

MATH4OP Optimization First Semester 10 points

Optimization is a core tool of applied mathematics, computational modelling, statistics, operation research, finance, engineering, indeed almost any application of the mathematical sciences. This paper focuses on convex optimization, covering a few key algorithms, the theory behind them, and applications.

MATH4PD Numerical Solution of PDEs First Semester 10 points

Partial differential equations (PDEs) are used to model a wide range of phenomena in engineering, economics and the natural sciences. They generalise ordinary differential equations (ODEs) modelling functions of one variable to describe quantities depending on multiple variables, such as time and space, or several spatial dimensions. Examples are the heat equation, the Laplace equation and the wave equation.

Because closed-form analytical solutions can seldom be found, even for the simplest PDEs, approximate solutions are usually sought. This course gives an overview of approximation methods for solving PDEs on finite domains, the so-called boundary-value problems, ranging from the semi-analytical separation of variable technique to the numerical finite difference method. The focus of the paper will be on implementing methods to generate approximate solutions using computational software Matlab and analyse their properties.

MATH4TO Topology First Semester 10 points

The subject of topology is rather abstract which means that it can be used in many different contexts. As a consequence it is fundamental for many branches in Mathematics. This paper is an introduction to point set topology. The notion of “closeness” is formalised and developed to define continuity of maps, connectedness of sets, convergence of sequences etc. There will be many familiar and unfamiliar examples to illustrate the power of this abstraction.

### Second semester Modules

MATH4DG Differential Geometry Second Semester 10 points

Since the time of the ancient Greeks, mathematicians and philosophers have been interested in the geometry of curves and surfaces, for example the Euclidean plane and the surface of the earth. From the end of the 19th century onwards with the work of Riemann, however, a powerful mathematical theory of much more general classes of curved spaces arose; this is what is nowadays under- stood as differential geometry.

MATH4FC Fractional Calculus Second Semester 10 points

Over the last 20 years fractional calculus has experienced a renais- sance due to the observation that fractional models capture power-law properties observed in natural systems. We will develop these models using some probability theory and solve them using numerical and analytical tools.

MATH4FD Fourier Analysis and Distribution Theory Second Semester 10 points

Fourier analysis is the study of representing functions as sums or integrals of simple waves. It has applications across a broad range of mathematical and physical sciences such as the analysis of solutions to partial differential equations, inverse problems and data processing. The natural setting for this decomposition is on the space of generalised functions, known as distributions.

MATH4NT Analytic Number Theory Second Semester 10 points

Number theory, which is probably the oldest branch of mathematics, is concerned with properties of integers. Of particular interest are prime numbers. Analytic number theory studies these using methods from analysis. This module gives a basic introduction with a focus on arithmetic functions, asymptotic formulae and the distribution of prime numbers.

MATH4PT Probability Theory Second Semester 10 points

This course is an introduction to probability theory based on measure theory.