The module introduces you to the key abstract algebraic objects of groups, rings and fields and develops their fundamental theory. The theory will be illustrated and made concrete through numerous examples in settings that you will already have encountered.

How can we solve a problem that does not have a nice pen-and-paper solution? How do we ensure our computers use the available data efficiently to deliver accurate and reliable results? Understand the practical techniques for carrying out numerical computations on a range of mathematical problems. Build your knowledge of mathematical computing. Learn how to implement and execute algorithms in Matlab.

Number theory encompasses some of the most classical and important topics in mathematics, stemming from the study of integers, Diophantine equations, prime numbers and modular arithmetic. As well as introducing each of these, in this module it will be demonstrated how techniques from a range of mathematical disciplines such as algebra and geometry can be brought to bear.

Group theory is the study of symmetries, which are the actions that rotate polyhedrons such as the cube and they permeate science at large, playing an important role in physics (such as the standard model of particle physics ), chemistry (molecules, crystals, materials science…), cryptography or even music! In this module you will learn advanced constructions and techniques in modern group theory, with special emphasis on the study of finite groups.

Can you identify curves and regions in the complex plane defined by simple formulae? How do you evaluate residues at pole singularities? Study complex analysis, learning to apply the Residue Theorem to the calculation of real integrals.

How do you apply an algorithm or numerical method to a problem? What are the advantages? And the limitations? Understand the theory and application of nonlinear programming. Learn the principles of good modelling and know how to design algorithms and numerical methods. Critically assess issues regarding computational algorithms.

The subject of Ordinary Differential Equations (ODEs) is a very important and fascinating branch in mathematics. An abundance of phenomena in physics, biology, engineering, chemistry, finance and neuroscience to name a few, may be described and studied using such equations. The module will introduce you to advanced topics and theories in ODEs and dynamical systems.

Examine key definitions, proofs and proof techniques in graph theory. Gain experience of problems connected with chromatic number. Understand external graph theory, Ramsey theory and the theory of random graphs.

How do standard coding techniques in computer security work? And how does RSA cryptography work? Examine the principles of cryptography and the mathematical principles of discrete coding. Analsye the concepts of error detection and correction. Understand the algebra and number theory used in modern cryptography and coding schemes.

Commutative algebra is the cornerstone established by Hilbert to give a formal backing to intuitive arguments in geometry. This module will provide you with a solid foundation of commutative rings and module theory, as well as help developing foundational notions helpful in other areas such as number theory, algebraic geometry, and homological algebra. Examples will be key, many of them will be made ‘graphic’ thanks to Hilbert’s Nullstellensatz.

This module will enable you to expand your knowledge on multiple statistical methods. You will learn the concepts of decision theory and how to apply them, have the chance to explore “Monte Carlo” simulation, and develop an understanding of Bayesian inference, and the basic concepts of a generalised linear model.

Ever considered becoming an Actuary? This module covers the required material for the Institute and Faculty of Actuaries CT4 and CT6 syllabus. It explores the stochastic process and principles of actuarial modelling alongside time series models and analysis.

Why are arbitrage arguments important in modern finance? How can a binomial model evaluate derivatives? What are the main models for interest rates? Understand the mathematical techniques underlying the modelling of derivative pricing. Acquire skills in the development of pricing and risk management. Explore stochastic methods and credit risk.

What do you understand about Bayes’ theorem and Bayesian statistical modelling? Or about Markov chain Monte Carlo simulation? Focus on Bayesian and computational statistics. Understand the statistical modelling and methods available. Learn to develop a Monte Carlo simulation algorithm for simple probability distributions.

This module will cover partial differential equations (PDEs), which can describe a majority of physical processes and phenomena. You will learn the properties of first and second order PDEs, the concepts behind them and the methods for solving such equations.