Mathematics (with optional Placement/Project year) MSc

Computers are increasingly all pervasive throughout mathematics, science and technology.  This ranges from every-day tasks, such as surfing the web, or communicating securely with your bank, to more specialist scientific questions such as using mathematical computer simulations to model real-world phenomena.  As a result, possessing proficient computer and programming skills is indispensable not only in academic and scientific research but also for future-focused business and industry.

This module is designed with dual objectives. Firstly to provide an in-depth introduction to algorithms and the process of translating these into computer programs, using state-of-the-art software tools. This will provide you with a solid foundational understanding to tackle any future computational and programming challenges.  Secondly, you will develop important research and writing skills.  You will learn to use LaTeX, an industry-standard typesetting system for produce professional scientific documents.

• Introduction to algorithms and how to translate them into computer programs.
• Introduction to computer programming software, for example Python.
• Learn about basic programming concepts including algorithms, loops, conditional statements and functions.  Also learn about more advanced scientific functions.
• Introduction to numerical algorithms from a variety of mathematical areas.
• Develop research and writing skills through LaTeX and online mathematics resources.

Note: Applications within this course will be chosen to suit the interests and background of the students.  Typical application areas may be to the numerical and symbolic solution of equations of various type, and to the graphical representation of mathematical models.

1. Introduce the student to new opportunities for researching and tackling mathematical problems which are provided by modern software tools.

Ordinary differential equations (ODEs) play a crucial role in modelling many problems in science and engineering.  Despite their significance, finding analytic solutions for these differential equations is often challenging. In this module, we will study the methods for numerically solving ODEs, analysing their behaviour, and gaining practical experience in their application. Our focus will be on first-order ODEs, examining a variety of algorithms such as forward and backward Euler, the family of Runge-Kutta methods, and multistep methods. We will discuss the zero stability, absolute stability, and convergence of the proposed numerical methods. To implement these methods in practice, we will utilise ODE solvers in MATLAB and Python to address different types of differential equations. Additionally, we will consider the finite difference method for solving the boundary value problems and the heat equation.

• Concepts of convergence, consistency and zero stability of the numerical methods.
• Forward Euler method, backward Euler method, Runge-Kutta method
• Multistep methods
• absolute stability
• Finite difference method for solving boundary value problem
• Finite different methods for solving heat equation
• Discussion of examples drawn from: difference equations; non-linear equations; ordinary differential equations; partial differential equations.

We aim to:

1. Introduce students to some modern numerical methods and to provide an opportunity for them to undertake some convergence and stability analysis, assisted by appropriate software.

Functional analysis is a field with widespread applications throughout applied mathematics and science.  It provides the fundamental underpinnings which allows us to analyse and find approximate solutions for many challenging problems in ordinary and partial differential equations, such as the heat equation, wave equation and various quantum phenomena.

In this module students will discover that the formal notions and techniques developed in analysis can be applied more generally to infinite-dimensional spaces endowed with notions of distance that generalise the properties of Euclidean distance. Throughout the module, students will gain familiarity with the definitions of these more general spaces, including Metric Spaces, Normed Spaces and Inner Product Spaces. We will explore examples where the points in these spaces are functions, sequences or even operators between spaces, rather than vectors of real or complex numbers.

1. Introduce students to the theory and application of functional analysis.

Stochastic differential equations (SDEs) model evolution of systems affected by randomness. They offer a beautiful and powerful mathematical language in an analogous way to what ordinary differential equations (ODEs) do for deterministic systems. SDEs have found many applications in diverse disciplines such as biology, physics, chemistry and the management of risk. Replacing the classical Newton-Leibnitz calculus with (Ito) stochastic calculus, we are able to build a new and complete theory of existence and uniqueness of solutions to SDEs. Ito's formula proves to be a powerful tool to solve SDEs. This leads to many new and often surprising insights about quantities that evolve under randomness.This module provides the student with the necessary language and methods for investigating applications of and solutions to stochastic differential equations.

• Review of probability spaces, random variables & stochastic processes.
• Brownian Motion, Weiner Processes.
• The Ito stochastic integral & Ito’s formula.
• Other stochastic integrals.
• Linear Stochastic Differential Equations & methods of solution.
• Weak & strong solutions to SDEs.
• Existence & uniqueness of solutions.
• Applications – topics may be chosen from, for example, mathematical finance, stochastic control, boundary value problems.
• Introduction to numerical methods.
• Markov property.
• Stopping times, optimal stopping.

1. Introduce students to the role of stochastic calculus and its applications.
2. Develop skills in solving stochastic differential equations.
3. Develop an understanding of theory relating to stochastic calculus.

Mathematical ecology harnesses advanced models and analytical tools to understand and describe the dynamics of individual species, ranging from the propagation of COVID-19 to the spread of wildfires, as well as the relationships between different species and their environment in ecosystems, for example in predator-prey dynamics and the invasive behaviour of cancer cells.  Mathematical ecology can also help us to understand natural patterns, such as the arrangement of leaves on plants and the markings on animal coats, by employing models such as reaction-diffusion systems.  So we can finally answer: "How the Leopard got its spots?"

Topics include:

• Continuous models for a single species; analysis of models using linear stability theory, discrete models and cobwebbing; and discrete logistic growth.

• Two-dimensional models; introduction to simple phase plane analysis; realistic models for various cases (e.g. predator-prey interactions, predator-prey competition).

• Bifurcation: how the behaviour of dynamical systems such as ODEs and maps changes when parameters are varied.

• Mutualism: where two species benefit from their association with each other.

• Reaction-diffusion problems and biological waves; the Fisher equation; Turing instabilities and diffusion-driven instabilities in two-component systems; generation of patterning by domain geometry; minimal domains for stable pattern formation.

1. Introduce students to mathematical methods for solving ecological problems.
2. Develop the students' abilities for formulating ecological problems as mathematical problems.

Partial differential equations
(PDEs) serve as mathematical models for a wide range of physical, biological, and economic phenomena and are foundational tools across various branches of pure and applied mathematics. In 1822, Fourier provided uniform solutions for significant PDEs, such as the wave and heat equations, along with Laplace's equation. This course will concentrate on these three equations, considering auxiliary initial or boundary conditions.  Throughout the course, we will explore diverse techniques, including separation of variables, Fourier methods, Laplace transform methods, among others, to effectively solve various types of partial differential equations.

• Mathematical techniques relevant to the solution of PDEs; e.g. Fourier series, Laplace Transforms.
• Introduction to partial differential equations. First order partial differential equations (linear and quasi-linear). Well-posedness.
• Linear partial differential operators: characteristic curves and surfaces.
• Classification of second order partial differential equations. Canonical form and reduction to canonical form.
• Initial value and boundary value problems.
• Existence and uniqueness of solutions.
• Laplace's equation; The Heat equation; The Wave equation; The Diffusion equation.
• Methods for solving PDEs: e.g. separation of variables, difference methods, transform methods, Fourier's method, Green's functions.
• Applications of partial differential equations.
• Systems of first-order partial differential equations.
• An introduction to the numerical Solution of PDEs.

1. Develop an understanding of theory relating to partial differential equations.
2. Develop skills in solving partial differential equations.
3. Introduce students to modelling using partial differential equations.
4. Give students an appreciation of the importance and relevance of numerical methods in solving partial differential equations.

The research project gives the student an opportunity to apply theory learned on the programme and to develop skills of self-discipline, project management and written communication.

Students will negotiate with tutors the precise title and objectives of the project. Students will study the art of mathematical writing and communication. Tutors will provide appropriate levels of support and advice.

The 3 module dissertation is designed to enable students to complete an independent research project written up in the form of a thesis that conforms with normal requirements for presentation of mathematical results.