## Papers

## 400 level (postgraduate) papers and modules

Not all papers/modules shown here will be available in any one year. Additional papers/modules may be offered. Check with the Director of Studies for a confirmed list of modules that are running, their prerequisites and semester.COMO480 Project 36 points Full year

A 36-point project in an agreed topic, supervised by one or more staff.

MATH401 Special topic (two 10-point modules) 20 points First Semester

A combination of two 10-point modules.

MATH402 Special topic (two 10-point modules) 20 points Second Semester

A combination of two 10-point modules.

MATH403 Special topic (two 10-point modules) 20 points First Semester

A combination of two 10-point modules.

MATH404 Special topic (two 10-point modules) 20 points First Semester

A combination of two 10-point modules.

MATH406 Special topic (two 10-point modules) 20 points Second Semester

A combination of two 10-point modules.

MATH485 Honours Project 36 points Full year

A 36-point project in an agreed topic, supervised by one or more staff.

MATH490 Honours Project 40 points Full year

A 40-point project in an agreed topic, supervised by one or more staff.

MATH495 MSc Preparation 18 points Full year

MSc Preparation.

## Modules

Introduction

General relativity, Albert Einstein’s theory of gravitation, is one of the most elegant theories of mathematical physics. It gives a geometric description of gravitation in terms of the curvature of space and time, using mathematical methods like tensor algebra, differential geometry and the theory of ordinary and partial differential equations.

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

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.

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.

This module is an introduction to General Relativity, the theory of gravity by Albert Einstein. Building on the module Differential Geometry (MATH4DG), which takes place in the first half of semester 1, we develop Einstein’s idea that “space” and “time” form a continuum “spacetime” described by a 4-dimensional Lorentzian manifold.

Introduction

This is a short course on group theory.

Introduction

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

Introduction

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.

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

This paper is an introduction to point-set topology, which underlies differential topology and algebraic topology, and is used all over mathematics (for example, operator algebra, functional analysis, topological group theory...). The main ideas are continuity of functions, and compactness and connectedness of sets. For example, you know that continuous functions take “nearby'” points to “nearby” points. In topology, there is a way to formulate what “nearby” is without using a distance function. The purpose of this module is to give students the background in topology that they need to pursue higher level mathematics.