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Graph theory is an aesthetically appealing branch of pure mathematics with strong links to other areas of mathematics (combinatorics, algebra, topology, probability, optimisation and numerics) and well-developed applications to a wide range of other disciplines (including operations research, data science, chemistry, systems biology, statistical mechanics and quantum field theory). This module provides an introduction to graph theory. There is an emphasis on theorems and proofs.

Asset prices in financial markets go up and down in accordance with how markets digest the flow of information. To understand the causal relation between information flow and price movements, it is necessary to model market information and use this to infer the price dynamics. In this way, market dynamics can be replicated artificially on a computer. This module explains the powerful process of artificially generating realistic market models.   The module begins with elements of probability theory. We will then learn the idea of conditional expectation and the Bayes formula, which gives the optimal inference under uncertainty. The meaning of the Bayes formula will be explained, leading to the understanding of what is meant by “intelligence”.   The module then covers the basics of stochastic process (specifically, the Brownian motion) and calculus (specifically, the Ito calculus), sufficient to follow the contents of the module. Then simple models for flows of information in financial markets will be introduced, and by use of the Bayes formula the associated price dynamics will be derived.   The module concludes with a brief application to the asset valuation problems in financial markets (such as options or other derivatives). 

Mathematics underpinning real-world uncertain events has become indispensable in many applications, including in particular financial markets. This module will begin with the introduction to probability theory and stochastic processes, with an emphasis on the Ito calculus for treating functions of Brownian motion. Such functions are commonly used in financial markets to model asset price dynamics, required for the valuation of financial contracts.   The module then discusses structures of financial markets, with an emphasis on the equity market. Several of the standard and exotic contingent claims will be introduced, and the need for mathematical models for the valuation and risk management of these products will be explained.   The pricing of a standard call option will then be worked out in a single-period binomial model, for which option price will be worked out in two ways: first using the portfolio replication and no arbitrage argument, and second using the risk-neutral expectation argument.   The model is then extended into multi-period binomial tree model, leading to the Cox-Ross-Rubinstein option pricing formula.   Finally, a continuous-time geometric Brownian motion model, originally introduced by Samuelson, will be considered, and used to deduce the famous Black-Scholes option pricing formula. This can be applied for the purpose of both pricing, as well as risk-management purposes, which will be demonstrated by working out the hedging strategy. The meaning of the pricing formula, and how it can be used in practical investment banking context, will be explained.

This module extends the abstract algebra introduced in MAT2048 Groups & Rings. A deeper insight into groups and rings is developed, and further algebraic structures are introduced. This module complements the module material in MATM035 Representation Theory and MATM011 Lie Algebras

Regular patterns arise naturally in many physical and biological systems, from hexagonal convection cells on the surface of the sun to stripes on a zebra's back. This module provides a mathematical framework for understanding the formation and evolution of these patterns using ordinary and partial differential equations and group theory. The module builds on group theory from MAT1031 Algebra and MAT2048 Groups & Rings. The module also builds on ordinary differential equations from MAT2007 Ordinary Differential Equation and partial differential equations from MAT2011 Linear PDEs.

This module introduces fundamental concepts in Quantum Mechanics and its applications to real-world quantum problems. The module covers the mathematics of Hilbert spaces and Dirac notation, the postulates of Quantum Mechanics, the uncertainty principle, the Schroedinger equation with one-dimensional applications to a particle in a potential well and the quantum harmonic oscillator, spin, and an introduction to the basics of quantum computation. This module utilises material from MAT1034 Linear Algebra and MAT2007 Ordinary Differential Equations. The module also builds on material from MAT1036 Classical Dynamics and MAT3008 Lagrangian & Hamiltonian Dynamics, although these modules are not pre-requisite.

This module introduces students to a range of mathematical methods and illustrates their application to real-world problems. Students will be introduced to residue calculus, integral transforms, variational calculus, and asymptotic approximations. Students will revise and extend their knowledge of methods of solving ordinary and partial differential equations.

This module introduces the students to topological spaces and manifolds; the core structures in pure mathematics. The module begins with fundamental concepts of point-set topology, homeomorphisms and their invariants - the sigma-algebra of open sets, the Hausdorff property, connectedness and compactness. We then move onto algebraic topology, introduce topological surfaces as quotient spaces, and discuss their invariants - the Euler characteristic and orientability. Finally, we progress to smooth manifolds and diffeomorphisms, discuss their construction method, that include two fundamental results of analysis, and the concepts of tangent bundles and transversality.

Riemannian geometry is the study of geometric properties of spaces, called manifolds, typically in higher dimensions, which are described smoothly by sets of well-defined co-ordinates. The emphasis is on how mathematically to understand the notion of distance via a metric on such spaces, and how this is used to investigate key associated geometric concepts.Students are introduced to the key concepts of manifolds and metrics, with illustrative examples such as spheres, hyperbolic spaces and Lie groups. The module continues with geodesics, isometries, covariant derivatives, and concludes with curvature and its properties.Modules in Curves and Surfaces and in Manifolds and Topology contain related material.

This module introduces programming in Python for data science, with a focus on data pre-processing, data mining and analysis, machine learning and deep learning. Besides the practical hands-on experience with writing code, this course also covers the theoretical background on different data analysis techniques and machine learning approaches. The goal is to develop an understanding of how information can be extracted from data and how this information can be further used to make predictions, but importantly how this is done practically in terms of writing clear and transparent source code. Using real-world data sets and illustrative examples, this course will help to develop a theoretical understanding of data science as well as practical experience by developing useful software tools. Many of the techniques acquired through this module are likely to be of potential use in the dissertation project.

This module introduces fundamental concepts in analytical dynamics and illustrates their application to real-world problems. The module covers the calculus of variations, Lagrangian and Hamiltonian formulations of dynamics, Poisson brackets, canonical transformations and symplectic manifolds. The module also leads to a deeper understanding of the role of symmetries and conservation laws in dynamical systems. This module builds on material from MAT1036 Classical Dynamics and lays the foundations for MAT3039 Quantum Mechanics.

This module provides an introduction to the applications of ordinary, delay and partial differential equations to ecology and epidemiology.

This module will introduce the students to the principles and formalism of General Relativity and its applications to Black Holes and astrophysical phenomena.

Data science is the study of data to extract meaningful and actionable insights at all levels of society such as dynamical systems and social media networks. This module introduces the role of data in society and provides students with the underpinning mathematics that drives data methodology and algorithms. This module then covers wide-ranging topics with a focus on the ����Vlog brand of data, as research into data is part of the School's research agenda

In many practical settings it is important to be able to deduce optimal states and/or solutions to problems that can be posed mathematically. Early such problems include determining the brachistochrone, which is the shape of a curve in space that that minimises the time taken for a particle to move between two fixed points (first posed by Johann Bernoulli and solved shortly afterwards by Isaac Newton and other great scholars of the day.) In more modern applications, we might want to control a dynamical system (e.g. the motion of a spacecraft) in such a way as to minimize an associated cost (e.g. fuel usage), and so determine optimal `path' (e.g. a flight trajectory). In still other settings, we might be concerned with the optimal transport of mass from one region to another, which in mathematical terms is a type of constrained minimisation problem. In this module, students are introduced to ideas, results and techniques from the calculus of variations, optimal control theory and optimal mass transport. The underpinning theory will be developed in detail and illustrated through examples wherever possible.

This module introduces the ideas of viscous fluid flows which build on the ideas some students would have seen in MAT2050: Inviscid Fluid Dynamics. By the end of the module, students should be familiar with the Navier-Stokes equations, and should be able to solve these equations in various simplified situations as well as in a variety of geometries. A selection of advanced topics will also be covered, enabling students to appreciate some of the finer points of viscous fluid mechanics.