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.

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.

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.

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.

Can you calculate confidence intervals for parameters and prediction intervals for future observations? Represent a linear model in matrix form? Or adapt a model to fit growth curves? Learn to apply linear models to analyse data. Discuss underlying assumptions and standard approaches. Understand methods to design and analyse experiments.

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.

Are you interested in understanding how AlphaGo was able to beat a top Go player? In this module, you will learn about the models behind successful stories of Reinforcement Learning, where a machine (agent) makes sequential decisions to reach an optimal goal. The lectures will be complemented with Lab sessions where we will take advantage of the Open AI Gym environments, allowing us to train our agents to perform tasks such as playing videogames (Atari) and more.