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

Develop an understanding of partial differential equations (PDE) appearing in physics and geometry, and classical techniques to study them analytically.

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

Broaden your knowledge on asymptotic methods and learn more about the techniques introduced that find wide applications in engineering and natural sciences.

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

Build your understanding of key elements of classical dynamics including the calculus of variations, Lagrangian Mechanics and Hamiltonian Mechanics.

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.

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 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.

Become familiar with concepts in geometric mechanics including Fermat’s principle and subsequent developments in the field.

Examine basic concepts and techniques of modern hydrodynamic stability theory and learn how to predict hydrodynamic instabilities in a variety of fluid flows.

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

Study the theory and applications of stochastic differential equations and touch on issues including stochastic optimal control, and the applications of SDEs to partial differential equations.

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.

Discover more about machine and deep learning as you gain an insight into the impact of optimisation algorithms and network architecture through mathematical tools.

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

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

Gain a thorough grounding in optimisation models, theory, algorithms and software

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 1 and further your understanding of key areas of contemporary physics.

Assess the theory and applications of random dynamical systems and ergodic theory before exploring more advanced developments in this field.

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

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

Examine the use of valuations, hedging and risk management of financial options by utilising stochastic differential equations.

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

Analyse the theory of partial differential equations and its related concepts, and build familiarity with conservation laws.

Focus on the dynamics of vorticity in an ideal fluid in two and three dimensions and explore various techniques in dynamical systems theory.