## 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

MATH4AC Algebraic Combinatorics First Semester 10 points

There is a rich interplay between abstract algebra and combinatorics. Algebraic combinatorics applies techniques from abstract algebra to combinatorial problems, or conversely applies combinatorial reasoning towards problems in algebra.

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.

MATH4HS Hilbert Spaces First Semester 10 points

This paper is an introduction to Hilbert spaces and linear operators on Hilbert spaces. It extends the techniques of linear algebra and real analysis to study problems of an intrinsically infinite-dimensional nature.

MATH4MA Modern Algebra First Semester 10 points

This paper introduces groups and rings. These are algebraic structures consisting of a set with one or more binary operations on that set satisfying certain conditions. These structures are ubiquitous throughout modern mathematics and this paper examines their properties and some applications.

MATH4MI Measure and Integration First Semester 10 points

In school and in introductory courses in mathematics, integral usually means Riemann integral of a real-valued function on the real line. This fundamental concept reveals its beauty in the Fundamental Theorem of Calculus which relates integration and derivation. However, the Riemann integral has some shortfalls that make it inadequate for many purposes in modern analysis. One of them is a gap in the fundamental theorem of calculus: the class of Riemann integrable functions does not coincide with the class of all functions that have an antiderivative. Another drawback is that the interchange of point-wise limits of function sequences and their integrals is only possible under rather restrictive conditions.

In this paper we introduce a modern theory of integration via measure theory that overcomes these shortfalls. It goes back to the French mathematician Henri Léon Lebesgue (1875–1941) and is the gate to many exciting branches of mathematics, like, for instance, modern probability theory, functional analysis, or the theory of partial differential equations.

MATH4NM Numerical Methods First Semester 10 points

This paper develops the theory and techniques required to apply computational methods in modelling, applied mathematics and data analysis. Topics include matrix computation, data fitting, and the numerical solution of differential equations.

MATH4NT Analytic Number Theory First 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.

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.

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

MATH4AA Asymptotic Analysis Second Semester 10 points

Asymptotic analysis is a branch of mathematical analysis concerned with the limiting behavior of functions, in particular the behavior of functions when a variable or parameter is "large" or "small". The tools of asymptotic analysis are useful in describing the unknown solution to a number of algebraic, differential, or integral equations, particularly for those cases where solving such an equation exactly is not possible. One can, in such contexts, view the approach as yielding results which lie somewhere between exact solutions (which are nice, yet uncommon for many equations) and numerical simulations (which can be quite useful for understanding the behavior of a solution for a specific collection of parameters, but often less useful for deducing general behavior or structure). In a number of cases, an asymptotic approximation for a given function may even provide more insight than does the exact solution.

MATH4CA Complex Analysis Second Semester 10 points

This paper develops the differential and integral calculus of functions of a complex variable, and its applications.

MATH4DE Partial Differential Equations Second Semester 10 points

This paper gives an introduction to the theory of partial differential equations by discussing the main examples (Laplace's equation, Poisson's equation, transport equation, wave equation) and their applications.

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 understood as differential geometry.

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.

MATH4GC Geometry of Curves and Surfaces Second Semester 10 points

This paper is an introduction to differential geometry; its focus is the structure of two-dimensional surfaces.

MATH4MP Mathematical Physics Second Semester 10 points

This paper presents the foundation theory for two major topics in Physics. The Classical Mechanics section introduces the formal framework of Classical Mechanics and illustrates its application to two-body problems, rotating systems, collisions, and chaos. The Special Relativity and Cosmology section covers the special theory of relativity with applications to relativistic mechanics, electrodynamics in covariant form, and cosmology. This paper is the same as the PHSI336 paper offered by the Physics Department. It is taught jointly by staff from both Departments.

MATH4PD Numerical Solution of PDEs Second 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.