Never has the collection of data been more widespread than it is now. The extraction of information from massive, often complex and messy, datasets brings many challenges to fields such as statistics, mathematics and computing.

Develop the skills and understanding to apply modern statistical and data-science tools to gain insight from contemporary data sets. By addressing challenges from a variety of applications, such as social science, public health, industry and environmental science, you will learn how to perform and present an exploratory data analysis and deploy statistical approaches to analyse data and draw conclusions. You will also develop judgement to critically evaluate the appropriateness of different methods for real-world challenges.

Building on your knowledge of vectors and matrices, this module explores the elegant framework of linear algebra, a powerful mathematical toolkit with remarkably diverse applications across statistical analysis, advanced algebra, graph theory, and machine learning.

You'll develop a comprehensive understanding of fundamental concepts, including vector spaces and subspaces, linear maps, linear independence, orthogonality, and the spectral decomposition theorem.

Through individual exploration, small-group collaboration, and computational exercises, you'll gain both theoretical insight and practical skills. The module emphasises how these abstract concepts translate into powerful problem-solving techniques across multiple disciplines, preparing you for advanced studies while developing your analytical reasoning abilities.

Machine learning is at the heart of modern AI systems, and it is a fundamentally mathematical subject. You will learn this mathematics by discovering how techniques are deployed in several AI systems, including the neural networks that have revolutionised the field.

You’ll start by building connections with previously encountered approaches through the unifying concept of a loss function of a parameter vector. For example, with a neural network model the vector input is the set of weights, and the loss function might be the prediction error on a dataset.

The goal is to find a vector input that produces a small loss; in the above example, this is known as training the neural net. You will learn and deploy some of the key mathematical ideas and numerical techniques, such as back propagation and stochastic gradient descent, that enable the automated iterative learning of a good vector input.

Statistics allows us to estimate trends and patterns in data and gives a principled way to quantify uncertainty in these estimates. The findings can lead to new insights and support decision-making in fields as diverse as cyber security, human behaviour, finance and economics, medicine, epidemiology, environmental sustainability and many more.

Dive into the behaviour of multivariate random variables and asymptotic probability theory, both of which are central to statistical inference. You will then be equipped to explore one of the most fundamental statistical models, the linear regression model, and learn how to apply general statistical inference techniques to multi-parameter statistical models. Statistical computing is embedded in the module, allowing you to investigate multivariate probability distributions, simulate random data, and implement statistical methods.

Continuing with your study into real numbers, you will explore their completeness (the idea that there are no ‘gaps’, unlike in the rationals). This completeness will be used to understand the limits of sequences, convergence of series, and power series.

This framework will allow for precision when exploring continuity, differentiability, and integrability of functions of a real variable, providing an improved foundation for calculus. That will enable you to understand when it is appropriate to use calculus; for instance, in proving theorems in other areas of mathematics, such as mathematical physics, probability and number theory.

The cornerstone of mathematical analysis is the construction of proofs involving arbitrarily small numbers, so-called epsilons and deltas. You will have opportunities to practise and improve your management of these quantities, in the process developing your skills in logic, communication and problem-solving.

Many real-world problems give rise to integrals and ordinary differential equation (ODEs) that cannot be solved analytically. To start, you will be introduced to techniques for tackling such problems, beginning with fundamental numerical methods, such as the trapezium rule and Euler’s method, before progressing to more advanced techniques and quantifying the accuracy, stability and limitations of these methods. Alongside numerical approaches, you will also develop heuristic methods to characterise a system's limiting behaviour.

Many familiar phenomena, from the pulses of light down a fibre optic cable to the shudder of turbulence on a plane, involve multiple variables, such as time and position. Their mathematical description requires differentiation with respect to each of these independent variables, leading to partial differential equations (PDEs). You will learn how to formulate PDEs for complex, real-world problems and practice core techniques for solving them.

Problem solving, gathering relevant information, writing, presenting and working as part of a group are essential in almost every professional career. You will develop and practice these skills as you undertake both individual and group-based, industry-linked and MARS-related projects.

You will also integrate prose, mathematics, computer code and its output into a clear, well-structured and appropriately linked Markdown document - a key skill in reproducibly documenting your research.

Differential equations are fundamental to mathematical modelling, with countless applications in engineering, biology and the environment.

Explore both ordinary and partial differential equations (ODEs and PDEs), with an introduction to advanced solution techniques and real-world applications. You will understand the deeper theory of ODEs and how solutions lead to special functions through series expansion, including Bessel functions. You will be introduced to Fourier series as a foundational tool in modern science.

Shifting your focus to PDEs, you will classify second-order equations and key equations in diverse geometries, noting the importance of boundary conditions. Applications will include acoustics, pollution dispersal, groundwater flow and forms of medical imaging.

By the end of the module, you will have mastered analytical methods for solving differential equations and gained the intuition to interpret their solutions in scientific and engineering contexts.

Models of dynamical systems are fundamental to our understanding of the physical and natural world.

Explore a new class of model, the Markov jump process, for the time evolution of dynamical systems such as the evolution of species populations in the wild and the spread of infectious diseases. You will learn how to simulate from these processes and will study methods for understanding their properties and behaviours. Unlike deterministic differential equation models, Markov jump processes are random, allowing for different behaviour every time they are simulated. You will discover how it is often possible to associate a jump process with a related differential equation approximation and that this can provide important insights into the behaviour of the jump process and the original real-world system.

Statistical techniques are often applied to environmental data, such as air temperatures, rainfall or wildfire locations. You will learn about some of the common features of such datasets and how these features are used to design statistical models. You will first be introduced to the Gaussian process model for continuous spatial processes. You will learn about the properties of the Gaussian process and implement this model for spatial data analysis, before investigating methods for point-reference data, such as earthquake or wildfire locations.

You will also dip into natural hazard risk management, which seeks to mitigate the effects of events, such as flooding or storms, in a manner that is proportionate to the risk. You will learn basic concepts from extreme value theory, including the appropriate distributions for extremes, and how to use these as statistical models for estimating the probability of events more extreme than those in the dataset.

The study of graphs (mathematical objects used to model networks and pairwise relations between objects) is a cornerstone of discrete mathematics. Graphs can represent important real-world situations, and the study of algorithms for graph-theoretical problems has strong practical significance.

You will learn about structural and topological properties of graphs, including graph minors, planarity and colouring. We will introduce several theoretical tools, including matrices relating to graphs and the Tutte polynomial. We will also study fundamental algorithms for network exploration, routing and flows, with applications to the theory of connectivity and trees, considering implementation, proofs of correctness and efficiency of algorithms.

You will gain experience in following and constructing mathematical proofs, correctly and coherently using mathematical notation, and choosing and carrying out appropriate algorithms to solve problems. The module will enable you to develop an appreciation for a range of discrete mathematical techniques.

Linear systems of differential and integral equations provide a mathematical model for a wide range of real-world devices, including communication systems, 5G networks, electrical circuits, heating systems and economic processes. Mathematical analysis of these models gives insight into the behaviour of these devices, with applications in automatic control, signal processing, wireless communications and numerous other areas.

Linear systems are considered in continuous time that reduce to a standard (A,B,C,D) state space representation. Via the Laplace transform, these are reduced further to rational transfer functions. Linear algebra enables us to classify and solve (A,B,C,D) models, while we describe their properties via diagrams in standard computer software. You will consider feedback control for linear systems, describing the rational controllers that stabilise an (A,B,C,D) system. Alongside the development of analytic methods to study linear systems, you will also gain experience in modelling real-world devices by such systems.

AI is suddenly everywhere and the methods for training and using AI tools are fundamentally mathematical, and fascinating in their own right. By understanding what goes on ‘under the hood’ you will open up a plethora of exciting opportunities for both further study and employment.

We will introduce you to the deep learning architectures used in modern AI. You will investigate how different architectures work with different data types and tasks, and what the computationally specified architectures actually mean in a modelling sense.

However, deep neural nets need more than just an appropriate architecture; they need to be both trained and deployed. You’ll study the interesting maths at each of these stages: the most recent approaches to loss function minimisation, and the techniques to sequentially learn and adapt to new data and observations, a critical component of modern AI methods.

A metric space consists of a set, whose elements are called points, and a notion of distance between points governed by three simple rules, abstracted from basic properties of Pythagorean distance in the Euclidean plane. In examples, ‘points’ may be functions where uniformity of convergence can be captured, or binary sequences with applications in computer science, or even subsets of a Euclidean space delivering fractal sets as limits.

Topology goes further, abstracting the notions of continuity and convergence, rendering a teacup and doughnut indistinguishable. A topological space equips each of its ‘points’ with its so-called ‘neighbourhoods’. The few simple principles governing these unlock a robust theory that now pervades the mathematical sciences and theoretical physics.

You will learn the fundamental concepts of completeness, total boundedness for metric spaces, compactness, and the Hausdorff property and metrisability for topological spaces.

From the complicated behaviours of ecosystems to the rapid spread of wildfires, nonlinear dynamics govern many fascinating phenomena around us. They underpin vital technological applications, such as the functioning of chemical reactors and the design of efficient transport networks. Nonlinear systems exhibit a rich variety of behaviours including sudden transitions, chaos and self-organised pattern-formation.

This module will combine theoretical insights with computational approaches to explore the nonlinear world in which we live, introducing key ideas in the geometric theory of dynamical systems and nonlinear partial differential equations. You will uncover how simple rules give rise to intricate structures, why some deterministic systems seem to have probabilistic behaviour, and how nonlinear models help describe real-world problems in diverse areas, such as traffic flow and population dynamics. Topics will range from stability, oscillation and chaos to nonlinear waves and shock formation, revealing the beauty and complexity of nonlinear systems in action.

Building on the statistical techniques explored so far, you deepen your understanding of both the theoretical underpinnings and practical application of frequentist statistical inference. You will then be introduced to an alternative paradigm: Bayesian statistics.

The frequentist perspective views all probabilities in terms of the proportions of outcomes over repeated experimentation and has been the foundation of hypothesis testing and experimental design over years of data-driven science and research. Meanwhile, the increasingly popular Bayesian approach arises directly from Bayes theorem, avoiding hypothetical repeated sampling. As a result, Bayesian statistics is often more intuitive and easier to communicate and naturally takes all forms of uncertainty into account.

With this in mind, you will compare and contrast these two perspectives and their associated tools. You will learn to select and justify an appropriate methodology for inference and model selection, and to reason about the uncertainty in your findings within each paradigm.

Stochastic processes are fundamental to probability theory and statistics and appear in many places in both theory and practice. For example, they are used in finance to model stock prices and interest rates, in biology to model population dynamics and the spread of disease, and in physics to describe the motion of particles.

During this module, you will focus on the most basic stochastic processes and how they can be analysed, starting with the simple random walk. Based on a model of how a gambler's fortune changes over time, it questions whether there are betting strategies that gamblers can use to guarantee a win. We will focus on Markov processes, which are natural generalisations of the simple random walk, and the most important class of stochastic processes. You will discover how to analyse Markov processes and how they are used to model queues and populations.

Statistics and machine learning share the goal of extracting patterns or trends from very large and complex datasets. These patterns are used to forecast or predict future behaviour or interpolate missing information. Learn about the similarities and differences between statistical inference and machine learning algorithms for supervised learning and how the two approaches can be used for classification and prediction.

You will explore the class of generalised linear models, which is one of the most frequently used classes of supervised learning model. You will learn how to implement these models, how to interpret their output and how to check whether the model is an accurate representation of your dataset. Lastly, you will have the opportunity to see how regularisation and dimension-reduction techniques can be used to apply these models to the case of the ‘large p, small n’ question. This phrase refers to datasets with many more variables than samples.