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.

In this module, you will learn how to extend techniques from calculus to vector-valued systems, through classical concepts such as gradient, divergence and curl. You will learn central theorems about these operators, and examine various applications and examples.

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 actions that preserve structure (such as rotations of the cube). These permeate science at large, playing an important role in physics (particularly particle physics and astrophysics), chemistry (molecules and crystals), cryptography and 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.

This module extends analytical and algebraic techniques to functions of complex variables, and their applications. You will develop powerful tools for studying functions via their zeroes and poles, including the powerful Residue Theorem for calculating real integrals.

The subject of Ordinary Differential Equations (ODEs) is a very important and fascinating branch of mathematics. These equations describe many phenomena, for instance in physics, biology, engineering, chemistry, finance and neuroscience and elsewhere. This module will introduce you to advanced topics in ODEs and dynamical systems.

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. Analyse 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, and will develop foundational notions used in other areas such as number theory, algebraic geometry and homological algebra. Examples will be key, and many will be made visual thanks to Hilbert’s Nullstellensatz.

This module will cover partial differential equations (PDEs), which can describe a wide array 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.