Humans can often perform a task extremely well (e.g., telling cats from dogs) but are unable to understand and describe the decision process followed. Without this explicit knowledge, we cannot write computer programs that can be used by machines to perform the same task. “Machine learning” is the study and application of methods to learn such algorithms automatically from sets of examples, just like babies can learn to tell cats from dogs simply by being shown examples of dogs and cats by their parents. Machine learning has proven particularly suited to cases such as optical character recognition, dictation software, language translators, fraud detection in financial transactions, and many others.

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.

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.

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 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.