MSc — 2025 entry Mathematics

This module is an introduction to classical and modern methods in the Calculus of Variations. Variational methods have many applications in Mathematics, Physics and Engineering. Examples will be included in the module to illustrate the theory of the Calculus of Variations in practice. This module complements module material in MAT3004 Introduction to Function Spaces, and MAT3008 Lagrangian & Hamiltonian Dynamics, but these modules are not pre-requisite.

This module introduces students to topological spaces and manifolds. Topology is the study of properties of spaces which are invariant under continuous transformations, and forms one of the cornerstones of pure mathematics. Study of topology leads to a deeper understanding of the concepts of continuity and connectedness. After introducing these key ideas, with appropriate examples, smooth manifolds are introduced. Differential forms and cohomology are developed, which enables the classification of manifolds using topologically invariant numbers. MAT2047 Curves and Surfaces is a recommended prior module, but is not pre-requisite. Students will also find the material in MAT3044 Riemannian Geometry complementary.

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.

Understanding the motions in our atmosphere and oceans is key to understanding, and hence solving, the global climate crisis. By applying a range of mathematical techniques to the governing fluid flow equations we are able to consider simplified models of a hugely complex system which will allow us to gain a deeper understanding about the transports of energy and heat on our planet.

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

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.

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.

The module is an introduction to nonlinear partial differential equations (PDEs) with a focus on hyperbolic and dispersive PDEs. The module takes key classes of equations as the organising centre. Each class of PDEs is considered and the properties, analytical techniques, and analysis of each is taken in turn.

Group theory is a branch of mathematics developed to understand symmetries, which are powerful tools for understanding the properties of complex mathematical problems and physical systems. Group representations map abstract group elements to linear transformations on vector spaces. Representation theory allows us to better understand the properties of symmetry groups, and leads to powerful and compact solutions of otherwise difficult and intractable problems. This module builds on groups theory in MAT1031 Algebra and MAT2048 Groups & Rings.

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, the Levi Civita connection, 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.

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 Surrey brand of data, as research into data is part of the department research agenda. 

This 15-credit M-Level module introduces important topics and techniques in theoretical physics that have a wide range of applications in many areas physics and engineering and which the students will not have met before. Both the mathematical techniques and their applications are covered at a level appropriate for Masters level students coming to the end of their degree and who should be able to pull many different ideas in theoretical physics together.

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.

The presentations of the module will focus on data-driven methods for the analysis of dynamical systems and time-series data and on related machine learning problems such as dimensionality reduction, manifold learning, regression, and classification.  Python will be used to implement data-driven methods. The methods will then be applied to typical benchmark problems such as chaotic dynamical systems, metastable stochastic systems, and fluid dynamics problems, but also, for instance, to image classification problems to highlight similarities with classical supervised learning applications.