��«Ӱҵ

Core modules:

Introduction to University Mathematics

This core module is designed to consolidate A-level material and explore topics in mathematics that you'll use throughout your degree. You'll also be introduced to core skills, such as mathematical literacy, communication and problem-solving.  

Throughout this module you'll develop a strong foundation in core mathematics. You'll consider techniques for solving equations, special functions, calculus, vectors, complex numbers, and finite and infinite series.

20 credits

Geometry, Matrices and Multivariate Calculus

This core module is designed to further develop your knowledge of the core mathematics you'll use across your degree.

You'll learn about two-dimensional coordinate geometry, discussing the theory of matrices geometrically and algebraically. You'll also define and evaluate derivatives and integrals for functions that depend on more than one variable, with an emphasis on functions of two variables.

Throughout this module you'll continue to develop your employability skills, exploring the career options open to mathematics graduates. You'll also work with your coursemates to undertake a group project on sustainability.

20 credits

Foundations of Pure Mathematics

The module aims to give an overview of basic constructions in pure mathematics; starting from the integers, we develop some theory of the integers, introducing theorems, proofs, and abstraction.  This leads to the idea of axioms and general algebraic structures, with groups treated as a principal example.  The process of constructing the real numbers from the rationals is also considered, as a preparation for “analysis”, the branch of mathematics where the properties of sequences of real numbers and functions of real numbers are considered.

20 credits

Probability and Data Science

Probability theory is branch of mathematics concerned with the study of chance phenomena. Data science involves the handling and analysis of data using a variety of tools: statistical inference, machine learning, and graphical methods. The first part of the module introduces probability theory, providing a foundation for further probability and statistics modules, and for the statistical inference methods taught here. Examples are presented from diverse areas, and case studies involving a variety of real data sets are discussed. Data science tools are implemented using the statistical computing language R.

20 credits

Optional modules:

A student will take 20 credits (two modules) from this group.

Mathematical Investigation Skills

This module introduces topics which will be useful throughout students’ time as undergraduates and in employment. These skills fall into two categories: computer literacy and presentation skills.  One aim of this module is to develop programming skills within Python to perform mathematical investigations.  Students will also meet the typesetting package LaTeX, the web design language HTML, and Excel for spreadsheets.  These will be used for making investigations, and preparing reports and presentations into mathematical topics.

20 credits

Mathematical modelling

Mathematics is the language of science.  By framing a scientific question in mathematical language, it is possible to gain deep insight into the empirical world.  This module aims to give students an appreciation of this astonishing phenomenon.  It will introduce them to the concept of mathematical modelling via examples from throughout science, which may include biology, physics, environmental sciences, and more.  Along the way, a range of mathematical techniques will be learned that tend to appear in empirical applications.  These may include (but not necessarily be limited to) difference and differential equations, calculus, and linear algebra.

20 credits

Students can also select 20 credits of Languages for All modules.

Core modules:

Mathematics and Statistics Project

This module forms the final part of the SoMaS project provision at Level 4 and involves the completion, under the guidance of a research active supervisor, of a substantial project on an advanced topic in Mathematics or Statistics.  Training is provided in the use of appropriate computer packages for the presentation of mathematics and statistics and guidance on the coherent and accurate presentation of technical information.

45 credits

Optional modules:

A student will take 75 credits from this group.

Machine Learning

Machine learning lies at the interface between computer science and statistics. The aims of machine learning are to develop a set of tools for modelling and understanding complex data sets. It is an area developed recently in parallel between statistics and computer science. With the explosion of 'Big Data', statistical machine learning has become important in many fields, such as marketing, finance and business, as well as in science. The module focuses on the problem of training models to learn from training data to classify new examples of data.

15 credits

Financial Mathematics

The discovery of the Capital Asset Pricing Model by William Sharpe in the 1960's and the Black-Scholes option pricing formula a decade later mark the beginning of a very fruitful interaction between mathematics and finance. The latter obtained new powerful analytical tools while the former saw its knowledge applied in new and surprising ways. (A key result used in the derivation of the Black-Scholes formula, Ito's Lemma, was first applied to guide missiles to their targets; hence the title 'rocket science' applied to financial mathematics). This course describes the mathematical ideas behind these developments together with their application in modern finance, and includes a project.

15 credits

Further Topics in Mathematical Biology

This module focuses on the mathematical modelling of biological phenomena.  The emphasis will be on deterministic models based on systems of differential equations.  Examples will be drawn from a range of biological topics, which may include the spread of epidemics, predator-prey dynamics, cell biology, medicine, or any other biological phenomenon that requires a mathematical approach to understand.  Central to the module will be the dynamic consequences of feedback interactions within biological systems.  In cases where explicit solutions are not readily obtainable, techniques that give a qualitative picture of the model dynamics (including numerical simulation) will be used.  If you did not take Scientific Computing at Level 2, you may still be able to enrol on this module, but you will need to obtain permission from the module leader first.

15 credits

Advanced Quantum Mechanics

Quantum mechanics at an intermediate to advanced level, including the mathematical vector space formalism, approximate methods, angular momentum, and some contemporary topics such as entanglement, density matrices and open quantum systems. We will study topics in quantum mechanics at an intermediate to advanced level, bridging the gap between the physics core and graduate level material.   The syllabus includes a formal mathematical description in the language of vector spaces; the description of the quantum state in Schrodinger and Heisenberg pictures, and using density operators to represent mixed states; approximate methods: perturbation theory,  variational method and time-dependent perturbation theory;  the theory of angular momentum and spin; the treatment of identical particles; entanglement; open quantum systems and decoherence. The problem solving will provide a lot of practice at using vector and matrix methods and operator algebra techniques. The teaching will take the form of traditional lectures plus weekly problem classes where you will be provided with support and feedback on your attempts.

15 credits

Generalised Linear Models

This module introduces the theory and application of generalised linear models. These models can be used to investigate the relationship between some quantity of interest, the 'dependent variable', and one or more 'explanatory' variables; how the dependent variable changes as the explanatory variables change. The term 'generalised' refers to the fact that these models can be used for a wide range of different types of dependent variable ,continuous, discrete, categorical, ordinal etc. The application of these models is demonstrated using the programming language R.

15 credits

Sampling Theory and Design of Experiments

Whereas most statistics modules are concerned with the analysis of data, this module is focussed on the collection of data. In particular, this module considers how to collect data efficiently: how to ensure the quantities of interest can be estimated sufficiently accurately, using the smallest possible sample size. Three settings are considered: sample surveys (for example when conducting an opinion poll), physical experiments, as may be used in industry, and experiments involving predictions from computer models, where there is uncertainty in the computer model prediction.

15 credits

Time Series

This module considers the analysis of data in which the same quantity is observed repeatedly over time (e.g., recordings of the daily maximum temperature in a particular city, measured over months or years). Analysis of such data typically requires specialised methods, which account for the fact that successive observations are likely to be related. Various statistical models for analysing such data will be presented, as well as how to implement them using the programming language R.

15 credits

Advanced Particle Physics

The module provides students with a comprehensive understanding of modern particle physics. Focusing on the standard model, it provides a theoretical underpinning of this model and discusses its predictions. Recent developments including the discovery of the Higgs Boson and neutrino oscillation studies are covered. A description of the experiments used to probe the standard model is provided. Finally the module looks at possible physics beyond the standard model.

15 credits

Probability with Measure Theory

Probability is a relatively new part of mathematics, first studied rigorously in the early part of 20th century. This module introduces the modern basis for probability theory, coming from the idea of 'measuring' an object by attaching a non-negative number to it. This might refer to its length or volume, but also to the probability of an event happening. We therefore find a close connection between integration and probability theory, drawing upon real analysis. This rigorous theory allows us to study random objects with complex or surprising properties, which can expand our innate intuition for how probability behaves. The precise material covered in this module may vary according to the lecturer's interests.

15 credits

Topics in Mathematical Physics

This unit will introduce students to advanced concepts and techniques in modern mathematical physics, in preparation for research-level activities.

It is assumed that the student comes equipped with a working knowledge of analytical dynamics, and of non-relativistic quantum theory.

We will examine how key physical ideas are precisely formulated in the language of mathematics. For example, the idea that fundamental particles arise as excitations of relativistic quantum fields finds its mathematical realisation in Quantum Field Theory. In QFT, particles can be created from the vacuum, and destroyed, but certain other quantities such as charge, energy, and momentum are conserved (after averaging over quantum fluctuations).

We will examine links between conservation laws and invariants, and the underlying (discrete or continuous) symmetry groups of theories. We will also develop powerful calculation tools. For example, to find the rate of creation of new particles in a potential, one must evaluate the terms in a perturbative (Feynman-diagram) expansion.

For details of the current syllabus, please consult the module leader.

15 credits

Mathematical Modelling of Natural Systems

Mathematical modelling enables insight into a wide range of scientific problems. This module will provide a practical introduction to techniques for modelling natural systems. Students will learn how to construct, analyse and interpret mathematical models, using a combination of differential equations, scientific computing and mathematical reasoning. Students will learn the art of mathematical modelling: translating a scientific problem into a mathematical model, identifying and using appropriate mathematical tools to analyse the model, and finally relating the significance of the mathematical results back to the original problem. Study systems will be drawn from throughout the environmental and life sciences.

15 credits

Further Topics in Number Theory

Elementary number theory has been seen in a number of earlier modules.  To go further, however, additional input is needed from other areas of pure mathematics - analysis and algebra.  For example, the distribution of prime numbers is intricately related to the complex analytic properties of the Riemann zeta function   And one can ask similar questions to those we ask about prime numbers for the rational numbers over, for example, quadratic fields.  This module will treat examples of further topics in number theory, accessible with the aid of advanced mathematical background.

15 credits

Bayesian Statistics and Computational Methods

This module develops the Bayesian approach to statistical inference. The Bayesian method is fundamentally different to the approach taken in earlier statistics courses. It is a more general and more powerful approach, and it is widely used, but it relies on modern computers for much of its implementation. It is based on the idea that if we take a (random) statistical model, and condition this model on the event that it generated the data that we actually observed, then we will obtain a better model. This course covers the foundations of Bayesian statistics and the incorporation of prior beliefs, as well as computational tools for practical inference problems, specifically Markov Chain Monte Carlo and Gibbs sampling. Computational methods will be implemented using R and Python. Advanced computational techniques will be explored, in the second semester, using STAN.

30 credits

Advanced Topics in Algebra A

Algebra is a very broad topic, relating many disparate areas of mathematics, from mathematical physics to abstract computer science.  It was noticed that the same underlying structures arose in a number of different areas, and this led to the study of the abstract structures.  This module studies some of the algebraic structures involved: fields, groups and Galois Theory; rings and commutative algebra, and gives applications to, for example, number theory, roots of polynomials and algebraic geometry.

30 credits

Advanced Topics in Algebra B

Algebra is a very broad topic, relating many disparate areas of mathematics, from mathematical physics to abstract computer science.  It was noticed that the same underlying structures arose in a number of different areas, and this led to the study of the abstract structures.  This module studies some of the algebraic structures involved: fields, groups and Galois Theory; rings and commutative algebra, and gives applications to, for example, number theory, roots of polynomials and algebraic geometry.

30 credits

Algebraic Topology

This unit will cover algebraic topology, following on from metric spaces. Topology studies the shape of spaces, with examples such as spheres, the Möbius Band, the Klein bottle, the torus and other surfaces. The first task is to formalise the notion of space, and to work out when a given space can be continuously deformed into another, where stretching and bending is allowed, but cutting and glueing is not. Algebraic topology builds a powerful bridge between shapes and algebra, enabling the use of familiar algebraic techniques from linear algebra and group theory to study spaces and their deformations.

30 credits

Analytical Dynamics and Classical Field Theory

Newton formulated his famous laws of mechanics in the late 17th century. Later, mathematicians like Lagrange, Hamilton and Jacobi discovered that underlying Newton's work are wonderful mathematical structures. In the first semester we discuss this work, its influence on the subsequent formulation of field theory, and Noether's theorem relating symmetries and conservation laws. In the second semester, Einstein's theory of gravity, General Relativity, will be introduced, preceded by mathematical tools such as covariant derivatives and curvature tensors. Einstein's field equations, and two famous solutions, will be derived. Two classic experimental tests of General Relativity will be discussed.

30 credits

Stochastic Processes and Finance

Stochastic processes are models that reflect the wide variety of unpredictable ways in which reality behaves. In this course we study several examples of stochastic processes, and analyse the behaviour they exhibit. We apply this knowledge to mathematical finance, in particular to arbitrage free pricing and the Black-Scholes model.

30 credits

Medical Statistics

This module introduces an important application of statistics: medical research, specifically, the design and analysis of clinical trials. For any new drug to be approved by a regulator (such as the Medicines and Healthcare products Regulatory Agency in the UK) for use on patients, the effectiveness of the drug has to be demonstrated in a clinical trial. This module explains how clinical trials are designed and how statistical methods are used to analyse the results, with a particular focus on 'survival' or 'time-to-event' analysis.

15 credits