Mathematics (MMath)

The notion of limit and convergence are two key ideas on which rest most of modern Analysis. This module aims to present these notions building on the experience gained by students in first year Calculus module. The context of our study is: limits and co...

Differential equations occupy a central role in mathematics because they allow us to describe a wide variety of real-world systems. The module will aim to stress the importance of both theory and applications of differential equations. The module begin...

Algorithms are systematic methods for solving mathematical problems, such as sorting numbers in ascending order, finding the cheapest way to ship goods on the road network or finding the shortest path in a graph. They can be regarded as practical applicat...

Over the last four hundred years progress in understanding the physical world (theoretical physics) has gone hand in hand with progress in the mathematical sciences, so much so that the terms applied mathematics and theoretical physics have come to be alm...

This module provides a solid mathematical introduction to the subject of Compound Interest Theory and its application to the analysis of a wide variety of complex financial problems, including those associated with mortgage and commercial loans, the valua...

Geometry has grown out of efforts to understand the world around us, and has been a central part of mathematics from the ancient times to the present. Topology has been designed to describe, quantify, and compare shapes of complex objects. Together, geome...

Group theory is one of the great simplifying and unifying ideas in modern mathematics. It was introduced in order to understand the solutions to polynomial equations, but only in the last one hundred years has its full significance, as a mathematical for...

A variety of OR techniques are covered in lectures and assessed by examination. Workshops develop skills with computer modelling software (discrete-event simulation and linear programming). Other skills that are developed within the module are group w...

Functions of one and several random variables are considered such as sums, differences, products and ratios. The central limit theorem is proved and the probability density functions are derived of those sampling distributions linked to the normal distrib...

Simple linear regression is developed for one explanatory variable using the principle of least squares. The extension to two explanatory variables raises the issue of whether both variables are needed for a well-fitting model, or whether one is sufficien...

The module will introduce the basic ideas in modelling, solving and simulating stochastic processes.

In the first part of this module we build on multivariate calculus studied in the first year and extend it to the calculus of scalar and vector functions of several variables. Line, surface and volume integrals are considered and a number of theorems inv...

This subject arises through a fusion of compound interest theory with probability theory, and provides the mathematical framework necessary for analysing such contracts, which are essentially long term financial transactions in which the various cash flow...

Synopsis: The module extends the mathematical framework developed in MATH3063 in order to enable modelling of long term financial transactions where the various cash flows are contingent on the death or survival of several lives, or where there are sever...

Partial Differential Equations (PDEs) occur frequently in many areas of mathematics. This module extends earlier work on PDEs by presenting a variety of more advanced solution techniques together with some of the underlying theory.

Topology is concerned with the way in which geometric objects can be continuously deformed to one another. It can be thought of as a variation of geometry where there is a notion of points being "close together" but without there being a precise measure o...

Algorithms are systematic methods for solving mathematical problems, such as sorting numbers in ascending order, finding the cheapest way to ship goods on the road network or finding the shortest path in a graph. They can be regarded as practical applicat...

Complex Analysis is the theory of functions in a complex variable. While the initial theory is very similar to Analysis (i.e, the theory of functions in one real variable as seen in the second year), the main theorems provide a surprisingly elegant, found...

Some of the biggest uncertainties in climate change predictions come from our lack of understanding of the impact of some of the smallest airborne dust particles. These dust particles are transported between continents from some of the hottest and most u...

This module will develop your understanding and knowledge of development across childhood, adolescence and adulthood. We will present historical and current research studies and explain how these link to related theoretical frameworks. The module will e...

Over the last four hundred years progress in understanding the physical world (theoretical physics) has gone hand in hand with progress in the mathematical sciences, so much so that the terms applied mathematics and theoretical physics have come to be alm...

This module provides a solid mathematical introduction to the subject of Compound Interest Theory and its application to the analysis of a wide variety of complex financial problems, including those associated with mortgage and commercial loans, the valua...

Number Theory is the study of integers and their generalisations such as the rationals, algebraic integers or finite fields. The problem more or less defining Number Theory is to find integer solutions to equations, such as the famous Fermat equation x^n ...

This module is designed for students in their third year and aims to introduce the basic concepts and techniques of Galois theory, building on earlier work at level 2. As such, it will provide an introduction to core concepts in rings, fields, polynomials...

Graph theory was born in 1736 with Euler’s solution of the Königsberg bridge problem, which asked whether it was possible to plan a walk over the seven bridges of the town without re-tracing one’s steps. Euler realised that the problem could be rephrased ...

This module is an introduction to functional analysis on Hilbert spaces. The subject of functional analysis builds on the linear algebra studied in the first year and the analysis studied in the second year. The module introduces the concept of Hilbert...

Many classes of problems are difficult to solve in their original domain. An integral transform maps the problem from its original domain into a new domain in which solution is easier. The solution is then mapped back to the original domain with the inver...

Biology is undergoing a quantitative revolution, generating vast quantities of data that are analysed using bioinformatics techniques and modelled using mathematics to give insight into the underlying biological processes. This module aims to give a flavo...

Following an initial discussion of the assessment and measurement of investment risk, mean-variance portfolio theory is introduced and used to determine the risk and return for a portfolio of risky assets, the composition of the optimal such portfolio, an...

- Linear programs: their basic properties; the simplex algorithm. - Duality: the relationship between a linear program and its dual, duality theorems, complementarity, and the alternative; sensitivity analysis. - The interior point method for convex op...

Introduce the students to the practical application of a relatively wide spectrum of numerical techniques and familiarise the students with numerical coding. Often in mathematics, it is possible to prove the existence of a solution to a given problem, ...

A variety of OR techniques are covered in lectures and assessed by examination. Workshops develop skills with computer modelling software (discrete-event simulation and linear programming). Other skills that are developed within the module are group w...

Module Contents: This module discusses continuous optimization problems where either the objective function or constraint functions or both are nonlinear. It explains optimality conditions, that is, which conditions an optimal solution must satisfy. It in...

The module will cover basic topics in perceptual psychology. We will discuss how the visual brain uses information from the two eyes in combination with learnt heuristics and other information sources to construct a perceptual representation of the world....

This is a module principally on Einstein's general theory of relativity, a relativistic theory of gravitation which explains gravitational effects as coming from the curvature of space-time. It provides a comprehensive introduction to material which is cu...

Functions of one and several random variables are considered such as sums, differences, products and ratios. The central limit theorem is proved and the probability density functions are derived of those sampling distributions linked to the normal distrib...

Statistical inference involves using data from a sample to draw conclusions about a wider population. Given a partly specified statistical model, in which at least one parameter is unknown, and some observations for which the model is valid, it is possibl...

Simple linear regression is developed for one explanatory variable using the principle of least squares. The extension to two explanatory variables raises the issue of whether both variables are needed for a well-fitting model, or whether one is sufficien...

The module Statistical Modelling II covers in detail the theory of linear regression models, where explanatory variables are used to explain the variation in a response variable, which is assumed to be normally distributed. However, in many practical situ...

The module will introduce the basic ideas in modelling, solving and simulating stochastic processes.

Networks are ubiquitous in the modern world: from the biological networks that regulate cell behaviour, to technological networks such as the Internet and social networks such as Facebook. Typically real-world networks are large, complex, and exhibit both...

This module introduces some of the fundamental ideas and issues of lifetime and time-to-event data analysis, as used in actuarial practice, biomedical research and demography.

This module provides a compulsory two semester project in the fourth year of the MMath programme. The project entails a weekly meeting with a supervisor during term-time who advises and assists the student in the study of a particular topic. The dissertat...

General relativity is the theory of spacetime and gravity developed by Einstein. This module aims to develop a geometric understanding of general relativity and explore advanced applications of GR. In particular, we will study the physics of black holes (...

Partial Differential Equations (PDEs) occur frequently in many areas of mathematics. This module extends earlier work on PDEs by presenting a variety of more advanced solution techniques together with some of the underlying theory.

When planning experiments, it is essential that the data collected are as relevant and informative as possible. The statistical principles for the design of experiments include the choice of optimal or good treatments sets and appropriate replication of t...

This is a compulsory lecture module for MMath students in their fourth year. The module will begin by looking at differential manifolds and the differential calculus of maps between manifolds. Manifolds are an abstraction of the idea of a smooth surfa...

This module will introduce and develop flexible statistical modelling methods that allow for general and complex forms of data to be modelled, extending ideas already encountered in earlier modules on linear and/or generalised linear modelling. The two ma...

Students will be able to apply logic and reason to prove basic results in geometric group theory. They will be able to use the geometry of the hyperbolic plane and other related simple geometric artefacts to demonstrate many of the standard features of ge...

This module is designed for MMath and MPhys students in their fourth year, and builds directly upon MATH3006 Relativity, Blackholes and Cosmology. Gravitational waves are tiny ripples in space-time, first predicted by Einstein himself in 1916. These ...

Harmonic analysis extends key ideas of Fourier analysis from Euclidean spaces to general topological groups. A fundamental goal is understanding algebras of functions on a group in terms of elementary functions. These correspond t the idea representing ...

Homotopy theory is the study of continuous deformations. A geometric object may be continuously deformed by pulling, stretching, pressing or compressing, but not by tearing or puncturing (which are discontinuous). Two objects can then be regarded as equiv...

This is a structured self-study module designed for MMath students in their fourth year. In this module, we take a guided tour of the hyperbolic plane, H, exploring its geometry and the group preserving its geometric structure. The hyperbolic plane arises...

Many classes of problems are difficult to solve in their original domain. An integral transform maps the problem from its original domain into a new domain in which solution is easier. The solution is then mapped back to the original domain with the inver...

This is a structured module, partly delivered by self-study and partly by lectures, designed for MMath students in their fourth year. A semigroup is a non-empty set on which is defined an associative binary operation. Unlike a group, a semigroup needn'...

The purpose of the module will be to introduce students to the fundamentals of machine learning, i.e. computational methods for statistical learning, prediction and decision-making using data. The basic principles of predictive modelling will be outlined,...

The emphasis of this module is on the methods required to develop mathematical models using differential equations to understand physical problems. The module involves both conventional lectures as well as discussion lectures. The discussion lectures comp...

This module provides an introduction to the theory of modules over a principal ideal domain and the representation theory of finite groups, two basic tools in advanced mathematics.

This module consists of lecturers and associated practical sessions. The first part will focus on basic statistical programming in R. The second part will provide an introduction to some modern computational statistical methods and their implementation in...

Networks are ubiquitous in the modern world: from the biological networks that regulate cell behaviour, to technological networks such as the Internet and social networks such as Facebook. Typically real-world networks are large, complex, and exhibit both...

This module introduces some of the fundamental ideas and issues of lifetime and time-to-event data analysis, as used in actuarial practice, biomedical research and demography Co-Requisite: MATH6122