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.

This module introduces the mathematics of the bizarre world of quantum physics, where physical systems are described by waves, energies and probabilities. You will develop skills in solving quantum mechanical problems associated with atomic and molecular systems.

" 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.

In this module you will learn techniques underpinning algorithms for studying integer-valued systems, and apply these algorithms to solve integer and mixed integer problems with cutting-plane algorithms.

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.

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. 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 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.