Transition towards the way you will be expected to think about and approach mathematics during your degree, with emphasis on the importance of precise definitions and rigorous proofs.

Undertake a rigorous treatment of the concept of a limit, as applied to sequences, series and functions.

Generalise what you already know about systems of linear equations and matrices and view them in more abstract, and more geometric, frameworks of vector spaces and linear transformations.

Explore a selection of mathematical tools that will enable you to solve more complex problems in applied mathematics than you will have tackled previously.

Focus on probability concepts, within an axiomatic framework. There will be a strong emphasis on principles of modelling and data analysis and you will learn to use the formal language of probability.

Gain knowledge of programming in Python through illustrative examples, practice questions and assessment tasks, guided by computational principles and their underlying mathematical concepts.

Identify how mathematical ideas can be used to underpin a range of scientific problems and familiarise yourself with a mathematical framework that embraces multiple disciplines from engineering to economics and statistics.

Develop elementary research skills in mathematics while exploring your personal interests in a specific area of mathematics.

Examine how to find matrices for a linear transformation which reflect its important features, culminating in the rational and Jordan canonical forms, and prove the fundamental Cayley-Hamilton Theorem.

Explore higher-dimensional derivatives, leading to the inverse and implicit function theorems, alongside metric and topological spaces as generalisations of n dimensional spaces, and limiting behaviour of sequences in such spaces.

Gain an understanding of advanced topics in calculus and ordinary differential equations, including an introduction to multi-dimensional vector calculus and differential operators.

Further your mathematical research and communication skills while developing transferable teamwork and presentation skills.

Deepen your knowledge in a brand new subject area, chosen from a huge range of for-credit modules.

Study examples of groups and homomorphisms, along with applications of group theory and rings, a class of algebraic object equipped with both addition and multiplication.

Explore Lebesgue's theory of measure and integration, a powerful extension of the Riemann integral and an essential tool in all aspects of analysis and its applications, including probability, stochastic processes and PDEs.

Investigate probability concepts that are useful in applications, particularly to statistics, focusing on the joint behaviour of several random variables.

Develop the traditional concepts of statistical inference, including maximum likelihood, hypothesis testing and interval estimation and apply them to the linear model that arises in many practical situations.

Foray into an ever-evolving interdisciplinary field in which concepts and methods from mathematics play a central role in the analysis of real-world systems from computing, science, and engineering.

Learn about partial differential equations and the concept of modelling in applied mathematics, with a focus on practical application.

Build a core of programming skills beyond those encountered in your first year through the object-oriented programming model, which is very widely used and taught in Python.

Analyse boundary value problems for elliptic and parabolic PDEs using the variational approach.

Further your appreciation of algebra and explore the basics of homological algebra, complexes and exact sequences.

Explore a variety of combinatorial techniques that have wide applications to other areas of mathematics.

Discover algebraic number theory, with emphasis on quadratic fields. In such fields, the familiar unique factorisation enjoyed by the integers may fail, but the extent of the failure is measured by the class group.

Engage with algebraic topology and advance your knowledge of homology theory, fundamental groups and the Galois correspondence for covering spaces.

Further your knowledge of mathematical techniques including the Wiener-Hopf method, Plemelj formulae, Hilbert problem, Orthogonal polynomials, the GrammSchmidt process, and the Rodrigues formula.

Gain an understanding of the basics of stochastic processes and how they can be applied. Lay the groundwork for further deep work, especially in such areas as genetics, industrial applications, and medicine.

Study a range of asymptotic series and expansions, including the asymptotic expansion of integrals, the Laplace method and Watson's lemma.

Advance your understanding of bifurcation theory, the study of how the behaviour of dynamical systems (ODEs, maps) changes when parameters are varied.

Assess the different methods used to solve linear systems and other linear algebra computations, with a focus on both mathematical analysis and practical applications.

Investigate a variety of computational approaches for solving partial differential equations, including finite difference methods, finite volume and spectral methods.

Develop the theory and practice of developing credit risk models in retail finance, exploring credit scoring and application scoring.

Examine the foundations of discrete-time dynamical systems and discover why they are so important to areas such as finance, physics, biology and social sciences.

Explore modelling learning through studying the dynamics of games and analyse learning models used by technology companies.

Develop a deep understanding of the finite element method by spanning both its analysis and implementation.

Be presented with a derivation of the Navier-Stokes and learn techniques to simplify and solve the equation with the purpose of describing the motion of fluids at different conditions.

Seek out simplifications in the Navier-Stokes formulation and learn how to solve Prandtl’s boundary-layer equations.

Broaden your knowledge of basic function spaces and the basic methodologies needed to undertake the analysis of partial differential equations.

Assess ideas of continuity and linear algebra, with a particular emphasis on vector spaces and linear maps.

Learn why the formula for the solution to a quadratic equation and similar formulae for cubic and quartic equations are well-known but no formula is possible for quintics.

Analyse the study and functions of complex numbers and look at solving two-dimensional problems in physics such as hydrodynamics and Ferromagnetism.

Explore the representations of groups and the character of a group representation.

Familiarise yourself with key concepts on this introduction to some of the more advanced topics in the theory of groups.

Develop intermediate proficiency in the use of the C++ programming language, as well as the skills to write efficient parallel programs and use of software engineering tools and best practices.

Advance your knowledge of fluid-dynamics relating to geophysics through a mix of theory and applications.

Discover why Markov processes are widely used to model random evolutions and how the theory connects with many other subjects in mathematics.

Learn how to describe the application of mathematical models to biological phenomena and give consideration to problems typical of particular organisms and environments.

Engage with the mathematical modelling of derivatives securities, which focuses on the pricing and hedging of derivatives using a probabilistic representation of market uncertainty.

Analyse mathematical structures by looking into the foundational issues of mathematics such as formal logic and set theory.

Discover microeconomics and macroeconomics and consider the motivations and optimal behaviours of firms and consumers in the marketplace.

Develop a deeper understanding with a supervised independent project in a particular area or topic. The project may be theoretical or computational.

Learn how to visualise and explore data in R, and understand the fundamental concepts and challenges of learning from data.

Focus on the properties of natural numbers, and in particular of prime numbers, which can be proved by elementary methods.

Explore a range of techniques for solving ordinary differential equations, including the Runge-Kutta, extrapolation and linear multistep methods.

Explore the fundamental properties of probability, including probability measures. independent random variables and elements of Brownian motion.

Untangle the theory of quantum mechanics and discover why it provides the basis for many areas of contemporary physics.

Build on topics covered in Quantum Mechanics I and further your understanding of key areas of contemporary physics.

Engage with the efficient algorithms deployed to solve mathematical problems using computation and become comfortable with the Python programming language.

Discover how advances in modern theoretical physics are being made and become familiar with the classical theory of fields.

Develop the traditional concepts of statistical inference, including maximum likelihood, hypothesis testing and interval estimation and apply them to the linear model that arises in many practical situations.

Familiarise with the criteria and the theoretical results necessary to develop and evaluate optimum statistical procedures in hypothesis testing, point and interval estimation.

Build an up-to-date understanding of computational simulation methods, covering areas from basic random variate generation to the Monte Carlo methodology.

Examine survival models that are fundamental to actuarial work, as well as being a key concept in medical statistics, with particular emphasis on actuarial applications.

Advance your understanding of Tensor calculus and explore concepts including Riemannian geometry, Einstein's field equations and the Newtonian approximation.

Analyse model complex systems using the basic foundation of mathematical methodology.

Learn the basic statistical concepts and reasoning relevant to atmospheric science and gain experience in the proper use of statistical methods for the analysis of weather and climate data.