Mathematics and Statistics MMath

Core modules:

Mathematics Core

Mathematics Core covers topics which continue school mathematics and which are used throughout the degree programmes: calculus and linear algebra, developing the framework for higher-dimensional generalisation.  This material is central to many topics in subsequent courses.  At the same time, small-group tutorials with the Personal Tutor aim to develop core skills, such as mathematical literacy and communication, some employability skills and problem-solving skills.

40 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

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

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

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

Core modules:

Mathematics Core II

Building on Level 1 Mathematics Core, Mathematics Core II will focus on foundational skills and knowledge for both higher mathematics and your future life as a highly skilled, analytically-astute worker.  Mathematical content will focus on topics that are vital for all areas of the mathematical sciences (pure, applied, statistics), such as vector calculus and linear algebra.  This will help develop your analytic and problem solving skills.  Alongside this, you will continue to develop employability skills, building on Level 1 Core.  Finally, there will be opportunities to learn and reflect on social, ethical, and historical aspects of mathematics, which will enrich your understanding of the importance of mathematics in the modern world.

30 credits

Differential equations

Differential equations are perhaps the most important tool in applied mathematics.  They are foundational for modelling all kinds of physical and natural phenomena, including fluids and plasmas, populations of animals or cells, cosmological objects (via relativity), subatomic particles (via quantum mechanics), epidemics, even political and social opinions have been modelled using differential equations.  This module will build on the tools learned at Level 1 for analysing differential equations, extending them in a variety of ways.  This may include topics such as bifurcation analysis, partial differential equations (which are particularly valuable for modelling things that vary in both space and time), and the effects of boundaries on the dynamics of differential equations.  It will provide the foundation for essentially all applied maths modules taught at Levels 3 and 4.

20 credits

Stochastic Modelling

Many things about life are unpredictable.  Consequently, it often makes sense to incorporate some randomness in mathematical models of natural and physical processes. Such models are called 'stochastic models' and are the study object of this module.  We will learn about a number of general models for processes where the state of a system is fluctuating randomly over time. Examples might include the length of a queue, the size of a reproducing population, or the quantity of water in a reservoir. We will cover various techniques for analysis of such models, setting the student up for further study of stochastic processes and probability at levels 3 and 4.

10 credits

Scientific Computing

The ability to programme is a central skill for any highly-numerate person in the 21st century.  This module builds on skills learned at Level 1 in 'Mathematical Investigation Skills' by developing skills in computer programming and independent investigation. You will learn how to solve various mathematical problems in programming languages commonly-used by mathematicians, for example Python. You will learn basic computational and visualisation methods for exploring numerical solutions to equations (including differential equations), and then apply this knowledge to explore the behaviour of example physical systems that these equations might model.

10 credits

Statistical Inference and Modelling

Statistical inference and modelling are at the heart of data science, a field of rapidly-growing importance in the modern word.  This module develops methods for analysing data, and provides a foundation for further study of probability and statistics at higher Levels.  You will learn about a range of standard probability distributions beyond those met at Level 1, including multivariate distributions. You will learn about sampling theory and summary statistics, and their relation to data analysis. You will discover how to parametrise various types of statistical model, learn techniques for determining whether one model is 'better' than another for understanding a dataset, and learn how to ascertain how good a statistical model is at explaining trends in data. The software package R will be used throughout.

20 credits

Optional modules:

Analysis and Algebra

This module will build on the theory built in Level 1 'Foundations of Pure Mathematics', focusing on the twin pillars of analysis and algebra.  These are not only fundamental for pure mathematics at higher levels, but provide rigorous theory behind core concepts that are used throughout the mathematical sciences.  Whilst to some extent you have been doing analysis and algebra since you were at school, here you will be going much deeper.  You will examine why familiar tools, like differentiation and integration, actually work.  Familiar objects, such as vectors, differential operators, and matrices, will be unpacked; powerful, formal properties of these objects proved.  Ultimately, this rigorous foundation will enable you to extend these tools and concepts to tackle a far greater set of problems than before.

20 credits

Vector Calculus and Dynamics

Vector calculus is a fundamental tool for modelling the dynamics of all kinds of objects, both solid and fluid.  In this module, you will build on the tools of vector calculus from Mathematics Core II, combining them with tools of differential equations from the L1 Mathematical Modelling module, and applying them to understand the dynamics of physical systems.  Possible examples might include liquid, gases, plasmas, and/or planetary motion.  The tools developed here will build valuable knowledge for the study of fluid dynamics and other applied mathematics modules at higher levels.

10 credits

Group Theory

A group is one of the most foundational objects in mathematics.  It just consists of a set, together with a way of combining two objects in that set to create another object in an internally-consistent fashion.  Familiar examples abound: integers with addition, real numbers with multiplication, symmetries of the square, and so on.  In this module, you will learn about formal properties of groups in general, including famous results like the orbit-stabiliser theorem.  You will also learn about important foundational examples, such as number, matrices, and symmetries.  You will learn how the general framework of groups allows you to prove theorems that pertain to all these examples in one go.  This provides a great example of the power and beauty of abstraction, a feature of pure mathematics that underlies the entire module.

10 credits

Mathematics and Statistics in Action

In this project module, you will investigate one or more case studies of using mathematics and statistics for solving empirical (i.e. 'real world') problems. These case studies will illustrate the process of mathematical and statistical modelling, whereby real-world questions are translated to mathematical and/or statistical questions. Students will see how techniques learned earlier in their degree can be used to explore these problems. There will be a mix of individual and group projects to choose from, and some projects may  involve the use of R or Python, but 'MAS116 Mathematical Investigations Skills' is not a prerequisite. Students will be expected to work independently (either individually or in a small group). However, the topic and scope of each piece of project work will be clearly defined by the lecturer in charge of the topic.

10 credits

Core modules:

Skills Development in Mathematics and Statistics

This module consolidates skills development across a number of areas of the SoMaS curriculum, allowing students choice from a range of application areas in mathematics and statistics.  Students will complete a portfolio, comprising a range of outputs from project work, group work, and outputs from digital learning.  Possible areas of mathematics include statistical investigations, history of mathematics, mathematical modelling, and sustainability, while the outputs might take the form of written projects, for example.  The module will involve considerable independent study, but staff will be available to guide you in your work.

20 credits

Medical Statistics

This course comprises sections on Clinical Trials and Survival Data Analysis. The special ethical and regulatory constraints involved in experimentation on human subjects mean that Clinical Trials have developed their own distinct methodology. Students will, however, recognise many fundamentals from mainstream statistical theory. The course aims to discuss the ethical issues involved and to introduce the specialist methods required. Prediction of survival times or comparisons of survival patterns between different treatments are examples of paramount importance in medical statistics. The aim of this course is to provide a flavour of the statistical methodology developed specifically for such problems, especially with regard to the handling of censored data (eg patients still alive at the close of the study). Most of the statistical analyses can be implemented in standard statistical packages.

10 credits

Sampling Theory and Design of Experiments

The results of sample surveys through opinion polls are commonplace in newspapers and on television. The objective of the Sampling Theory section of the module is to introduce several different methods for obtaining samples from finite populations. Experiments which aim to discover improved conditions are commonplace in industry, agriculture, etc. The purpose of experimental design is to maximise the information on what is of interest with the minimum use of resources. The aim of the Design section is to introduce some of the more important design concepts.

10 credits

Time Series

Time series are observations made in time, for which the time aspect is potentially important for understanding and use. The course aims to give an introduction to modern methods of time series analysis and forecasting as applied in economics, engineering and the natural, medical and social sciences. The emphasis will be on practical techniques for data analysis, though appropriate stochastic models for time series will be introduced as necessary to give a firm basis for practical modelling. Appropriate computer packages will be used to implement the methods.

10 credits

Optional modules include:

Stochastic Processes and Finance

A stochastic process is a mathematical model for phenomena unfolding dynamically and unpredictably over time. This module studies two classes of stochastic process particularly relevant to financial phenomena: martingales and diffusions. The module develops the properties of these processes and then explores their use in Finance. A key problem considered is that of the pricing of a financial derivative such as an option giving the right to buy or sell a stock at a particular price at a future time. What is such an option worth now? Martingales and stochastic integration are shown to give powerful solutions to such questions.

20 credits

Undergraduate Ambassadors Scheme in Mathematics

This module provides an opportunity for Level Three students to gain first hand experience of mathematics education through a mentoring scheme with mathematics teachers in local schools. Typically, each student will work with one class for half a day every week for 11 weeks. The classes will vary from key stage 2 to sixth form. Students will be given a range of responsibilities from classroom assistant to the organisation and teaching of self-originated special projects. Only a limited number of places are available and students will be selected on the basis of their commitment and suitability for working in schools.

20 credits

Codes and Cryptography

The word 'code' is used in two different ways. The ISBN code of a book is designed in such a way that simple errors in recording it will not produce the ISBN of a different book. This is an example of an 'error-correcting code' (more accurately, an error-detecting code). On the other hand, we speak of codes which encrypt information - a topic of vital importance to the transmission of sensitive financial information across the internet. These two ideas, here labelled 'Codes' and 'Cryptography', each depend on elegant pure mathematical ideas: codes on linear algebra and cryptography on number theory. This course explores these topics, including the real-life applications and the mathematics behind them.

10 credits

Combinatorics

Combinatorics is the mathematics of selections and combinations. For example, given a collection of sets, when is it possible to choose a different element from each of them? That simple question leads to Hall's Theorem, a far-reaching result with applications to counting and pairing problems throughout mathematics.

10 credits

Complex Analysis

It is natural to use complex numbers in algebra, since these are the numbers we need to provide the roots of all polynomials. In fact, it is equally natural to use complex numbers in analysis, and this course introduces the study of complex-valued functions of a complex variable. Complex analysis is a central area of mathematics. It is both widely applicable and very beautiful, with a strong geometrical flavour. This course will consider some of the key theorems in the subject, weaving together complex derivatives and complex line integrals. There will be a strong emphasis on applications. Anyone taking the course will be expected to know the statements of the theorems and be able to use them correctly to solve problems.

10 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.

10 credits

Game Theory

The module will give students the opportunity to apply previously acquired mathematical skills to the study of Game Theory and to some of the applications in Economics.

10 credits

Graph Theory

A graph is a simple mathematical structure consisting of a collection of points, some pairs of which are joined by lines. Their basic nature means that they can be used to illustrate a wide range of situations. The aim of this course is to investigate the mathematics of these structures and to use them in a wide range of applications.

10 credits

Introduction to Relativity

Einstein's theory of relativity is one of the corner stones of our understanding of the universe. This course will introduce some of the ideas of relativity, and the physical consequences of the theory, many of which are highly counter-intuitive. For example, a rapidly moving body will appear to be contracted as seen by an observer at rest. The course will also introduce one of the most famous equations in the whole of mathematics: E=mc^2.

10 credits

Knots and Surfaces

The course studies knots, links and surfaces in an elementary way. The key mathematical idea is that of an algebraic invariant: the Jones polynomial for knots, and the Euler characteristic for surfaces. These invariants will be used to classify surfaces, and to give a practical way to place a surface in the classification. Various connections with other sciences will be described.

10 credits

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.

10 credits

Mathematical Methods

This course introduces methods which are useful in many areas of mathematics. The emphasis will mainly be on obtaining approximate solutions to problems which involve a small parameter and cannot easily be solved exactly. These problems will include the evaluation of integrals. Examples of possible applications are: oscillating motions with small nonlinear damping, the effect of other planets on the Earth's orbit around the Sun, boundary layers in fluid flows, electrical capacitance of long thin bodies, central limit theorem correction terms for finite sample size.

10 credits

Mathematical modelling of natural systems

Mathematical modelling enables insight in to 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.

10 credits

Metric Spaces

This unit explores ideas of convergence of iterative processes in the more general framework of metric spaces. A metric space is a set with a distance function which is governed by just three simple rules, from which the entire analysis follows. The course follows on from MAS207 'Continuity and Integration', and adapts some of the ideas from that course to the more general setting. The course ends with the Contraction Mapping Theorem, which guarantees the convergence of quite general processes; there are applications to many other areas of mathematics, such as to the solubility of differential equations.

10 credits

Operations Research

Mathematical Programming is the title given to a collection of optimisation algorithms that deal with constrained optimisation problems. Here the problems considered will all involve constraints which are linear, and for which the objective function to be maximised or minimised is also linear. These problems are not continuously differentiable; special algorithms have to be developed. The module considers not only the solution of such problems but also the important area of post-optimality analysis; i.e. given the solution can one answer questions about the effect of small changes in the parameters of the problem (such as values of the cost coefficients)?

10 credits

Quantum Theory

The development of quantum theory revolutionized both physics and mathematics during the 20th century. The theory has applications in many technological advances, including: lasers, super-conductors, modern medical imaging techniques, transistors and quantum computers. This course introduces the basics of the theory and brings together many aspects of mathematics: for example, probability, matrices and complex numbers. Only first year mechanics is assumed, and other mathematical concepts will be introduced as they are needed.

10 credits

Topics in Number Theory

In this module we study intergers, primes and equations. Topics covered include linear and quadratic congruences, Fermat Little Theorem and Euler's Theorem, the RSA cryptosystem, Quadratic Reciprocity, perfect numbers, continued fractions and others.

10 credits

Core modules:

Bayesian Statistics and Computational Methods

This module introduces the Bayesian approach to statistical inference. The Bayesian method is fundamentally different in philosophy from conventional frequentist/classical inference, and has been the subject of some controversy in the past, but is now widely used. The module also presents various computational methods for implementing both Bayesian and frequentist inference, in situations where obtaining results 'analytically' would be impossible. The methods will be implemented using the programming languages R and Stan, and some programming is taught alongside the theory lectures.

30 credits

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

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

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

Optional modules:

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

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

Functional Analysis

Functional analysis is the study of infinite-dimensional vector spaces equipped with extra structure. Such spaces arise naturally as spaces of functions. As well as being a beautiful subject in its own right, functional analysis has numerous applications in other areas of both pure and applied mathematics, including Fourier analysis, study of the solutions of certain differential equations, stochastic processes, and in quantum physics. In this unit we focus on Banach spaces and especially Hilbert spaces - complete vector spaces equipped with an inner product - and linear maps between Hilbert spaces. Applications of the theory we examine include Fourier series and the Fourier transform, and differential equations.

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

Probability and Random Graphs

Random graphs were studied by mathematicians as early as the 1950s. The field has become particularly important in recent decades as modern technology gives rise to a vast range of examples, such as social and communication networks, or the genealogical relationships between organisms. This course studies a range of models of random trees, graphs and networks, alongside probabilistic ideas that are needed to analyse their different properties. The precise material covered in this module may vary according to the lecturer's interests.

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

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 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

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