20 credits

This module introduces mathematical modelling and computational techniques used in financial derivatives pricing and risk assessment. You will explore:  

• fundamental tools for discrete and continuous-time modelling, including ODEs, PDEs, and stability analysis
• finite difference methods for solving PDEs, including Crank-Nicolson schemes
• probability theory and stochastic processes, including birth-death processes and Monte Carlo methods
• introduction to financial derivatives, including European and exotic options
• stochastic calculus, Brownian motion, and the Black-Scholes model
• computational methods for pricing options, including Monte Carlo simulation and binomial trees

20 credits

This module explores advanced topics in functional analysis, focusing on operator theory and its applications to integral and differential equations. You will study:

• continuous linear operators on normed vector spaces, including operator norms and eigenvalue-type equations
• key theorems in functional analysis, such as the Open Mapping Theorem and Uniform Boundedness Principle
• dual spaces, the Hahn–Banach Theorem, and weak convergence in Banach spaces
• linear operators on Hilbert spaces, including adjoint, self-adjoint, and normal operators
• spectral theory and compact operators, including the Fredholm Alternative
• applications to integral equations, differential operators, and Green’s functions

20 credits

This module explores the fundamental principles of viscous fluid flow and wave dynamics, covering both theoretical foundations and practical applications. You'll study:

• basics of viscous flow, including kinematics, constitutive relations, and the Navier–Stokes equations
• applications of viscous flow, including exact solutions, boundary layers, and low- and high-Reynolds-number flows
• linear wave theory, including travelling and standing waves, reflection, refraction, and dispersion
• nonlinear hyperbolic waves, including the method of characteristics and shock formation

20 credits

This module introduces the mathematical foundations of approximation, interpolation, and the finite element method (FEM) for solving boundary value problems. You will explore:

• approximation in normed vector spaces, orthogonal bases, and best approximation
• interpolation techniques, including Lagrange polynomials and finite element interpolation
• error analysis for approximation, interpolation, and finite elements
• quadrature formulas for numerical integration
• the Galerkin finite element method, weak formulation, and convergence analysis
• extension to two-dimensional problems, including triangulations and basis functions

10 credits

This module provides a rigorous foundation in the mathematical theory of optimisation, covering key theoretical results and techniques for constrained and unconstrained problems. You'll explore:

• fundamentals of variational calculus and optimisation principles
• necessary and sufficient conditions for extrema, including Euler-Lagrange equations
• constrained optimisation using Lagrange multipliers
• analytical methods for solving optimisation problems

20 credits

This module will enable you to develop a more fundamental understanding of the mathematics of machine learning and of the ideas underpinning some classical algorithms in the field. 

Following this module you'll be able to: 

• critically interpret new algorithms in machine learning
• understand convergence and properties of the computed solution
• work on real world problems using machine learning techniques

20 credits

This module introduces advanced asymptotic and complex analysis techniques for solving differential equations, with applications in physics and engineering. You'll study:

• asymptotic methods for ordinary differential equations (ODEs), including regular and singular expansions
• multiple scale analysis for nonlinear oscillators and parametric resonance
• matched asymptotic expansions, boundary layers, and WKBJ eigenvalue problems
• contour integral solutions of differential equations, including Watson’s Lemma and the method of steepest descents

10 credits

This module will provide the basic tools in statistical data analysis and the underlying theory. The module will enable you to:

• understand fundamentals of probability and distributions
• learn appropriate visualisations and summaries for different data types
• undertake appropriate statistical tests for different types of data, including producing confidence intervals

20 credits

This module introduces mathematical modelling techniques for biological and ecological systems, using differential equations and pattern formation theory. You'll explore:

• ordinary differential equation (ODE) models in biology, including bacterial growth, tumour-immune interactions, and marine ecosystems
• reaction kinetics, including enzyme dynamics, the Law of Mass Action, and biological oscillators
• difference equation models for population dynamics, fisheries management, and age-structured populations
• biological movement and pattern formation, including Turing mechanisms, chemotaxis, and animal pigmentation models
• travelling wave solutions of reaction-diffusion equations, with applications in wound healing, cancer growth, and epidemiology

20 credits

This module demonstrates the central role network theory plays in mathematical modelling.

Topics include:

• the connection between linear algebra and graph theory
• the use of theory as a tool for revealing structure in networks
• application of algorithms on a network using programming

10 credits

This module introduces key mathematical techniques for optimisation, with applications in analytics and decision-making. You'll explore:

• the distinction between local and global optimisation problems
• methods for unconstrained optimisation, including gradient-based and Newton-type methods
• techniques for constrained optimisation, including penalty methods and relaxation techniques
• theoretical foundations of constrained optimisation, including Kuhn-Tucker conditions and Lagrangian methods
• introduction to variational principles and functional differentiation, with applications to optimisation problems

20 credits

This module provides advanced numerical techniques for solving differential equations, including stochastic, partial, and ordinary differential equations, with applications in finance, biology, and electromagnetism. You'll explore:

• simulating stochastic differential equations (SDEs), including Monte Carlo methods and stability analysis
• finite difference methods for PDEs, focusing on parabolic and hyperbolic equations
• computational methods for electromagnetic wave propagation, including Maxwell’s equations and finite difference schemes
• optimisation techniques, including steepest descent and Newton’s method
• subspace methods in numerical linear algebra, including Krylov methods and eigenvalue computations
• advanced numerical analysis of ordinary differential equations (ODEs), including Runge-Kutta methods and stability considerations

20 credits

This module will cover the fundamental statistical methods necessary for the application of classical statistical methods to data collected for healthcare research. There will be an emphasis on the use of real data and the interpretation of statistical analyses in the context of the research hypothesis under investigation.

Topics covered include:

• survival analysis
• analysing categorical data using hypothesis tests
• experimental Design and sampling
• clinical measurement

10 credits

This module covers all aspects of statistical consultancy skills necessary for being a successful statistician working in any research environment. You'll work on real-life problems in small groups and have the opportunity to interact with healthcare researchers to formulate hypotheses.

This module will cover:

• how to engage with professionals working in business, industry and the public sector
• how to apply their statistical knowledge in different situations
• how to effectively communicate statistical results to non-statisticians

10 credits

Surveys are an important way of collecting data. This module will introduce you to the methods that are commonly used in health care to design questionnaires and analyse data resulting from these questionnaires.

You will consider:

• how to design appropriate survey questions
• a variety of sampling methods
• analysing data for different sampling methods

10 credits

This module will cover the theory of assessing risks under uncertainty. It will focus on the practical assessment of risk using simulation methods such as Monte Carlo simulation. You'll develop skills in communicating risk to risk managers as well as formulating practical risk questions that can influence policy decisions.

You can expect to learn about:

• uncertainty and variability
• bootstrapping
• Monte Carlo Simulation
• selecting appropriate probability distributions based on given scenarios

10 credits

This module will introduce you to Bayesian statistics and the modern Bayesian methods that are used in a variety of applications. Like with other modules, the focus is on real-life data and using statistical software packages for analysis.
You will gain experience in working with the following:

• visualising spatial data
• geospatial data, including methods for prediction
• bayesian modelling using software to implement Markov Chain Monte Carlo
• areal unit modelling

10 credits

This module provides you with the basic theories of machine learning and how to construct a machine model for a real dataset using R. You will also understand the ethical issues regarding data processing and management.

On completion of this module, you will be able to:

• clean data using RStudio and the tidyverse
• understand missing data and the role it plays
• understand ethical issues regarding data processing and management
• carry out single-value imputation
• carry out multiple imputed chained equations in R
• understand and implement artificial neural networks
• understand and implement support vector machines
• understand and implement tree-based classification and regression techniques
• understand and implement ensemble methods

10 credits

This module will develop your skills in data presentation and statistical communication. You will learn to develop data dashboards, which are increasingly used to allow key stakeholders (and the public) to gain key insights into data via interactive visualisation.

Topics covered will include:

• Creating a data dashboard in RStudio
• User interface design with respect to accessibility
• Creating interactive data visualisations which reflect a specific aim
• Reactive programming  in RStudio
• Static programming in R

10 credits

This module introduces deep Learning prediction algorithms covering background theory and practical application in Python.

Topics covered include:

• activation functions
• gradient descent, backpropagation and optimisation
• convolutional Neural Networks
• recurrent Neural Networks
• generative Adversarial Networks

60 credits

You'll undertake a 60-credit research project, applying mathematical modelling to a practical problem. Projects may involve numerical analysis, fluid dynamics, optimisation, or biological systems. Working with academic or industry partners is possible.