Mathematics with a Year in Industry MMath

The idea of proof is fundamental to all mathematics. We’ll look at mathematical reasoning using techniques from logic to deal with sets, functions, sequences and series.

This module links directly with your study in Calculus and Linear Mathematics. It provides you with the foundations for the broader area of Mathematical Analysis. This includes the rigorous study of the infinite and the infinitesimal.

You will also learn the basics of computer programming. This will give you the chance to use computational algorithms to explore many of the mathematical results you’ll encounter in your core modules.

Your study will include:

• propositional and predicate logic; set theory, countability
• proof: direct, indirect and induction
• sequences and infinite series (convergence and divergence)
• limits and continuity of functions
• programming in Python

How can the flight-path of a spacecraft to another planet be planned? How many fish can we catch without depleting the oceans? How long would it take a lake to recover after its pollution is stopped?

The real world is often too complicated to get exact information. Instead, mathematical models can help by providing estimates. In this module, you’ll learn how to construct and analyse differential equations which model real-life applications.

Your study will include:

• modelling with differential equations
• kinematics and dynamics of moving bodies
• Newton’s laws, balance of forces
• oscillating systems, springs, simple harmonic motion
• work, energy and motion

You'll be able to expand on these techniques later in your degree through topics such as:

• black holes, quantum theory
• fluid and solid mechanics
• mathematical medicine and biology
• mathematical finance

How do we define calculus? How is it used in the modern world?

The concept can be explained as the mathematics of continuous change. It allows us to analyse motion and change in time and space.

You will cover techniques for differentiating, integrating and solving differential equations. You’ll learn about the theorems which prove why calculus works. We will explore the theory and how it can be applied in the real world.

Your study will include:

• functions: limits, continuity and differentiability, rules of differentiation
• techniques for integration, fundamental theorem of calculus
• solution of linear and nonlinear differential equations
• multivariate calculus, Lagrange multipliers, stationary points
• multiple integrals, changes of variables, Jacobians

This module gives you the mathematical tools required for later modules which involve modelling with differential equations. These include:

• mathematical physics
• mathematical medicine and biology
• scientific computation

Pure mathematics at university is typically very different to the pure mathematics you've learnt at school or college. You'll use the language of sets, functions and relations to study some very abstract mathematical ideas.

In this module, we'll develop the skills of reading and writing the language of pure mathematics. You will learn techniques to build mathematical proofs in an abstract setting.

Your study will include:

• the language of set theory
• relations and functions
• rational and irrational numbers
• modular arithmetic
• prime factorisation

These topics will provide you with the basics you need for subsequent modules in algebra, number theory and group theory.

Vectors, matrices and complex numbers are familiar topics from A level Mathematics and Further Mathematics. Their common feature is linearity. A linear mathematical operation is one which is compatible with addition and scaling.

As well as these topics you’ll study the concept of a vector space, which is fundamental to later study in abstract algebra. We will also investigate practical aspects, such as methods for solving linear systems of equations.

The module will give you the tools to analyse large systems of equations that arise in mathematical, statistical and computational models. For example, in areas such as:

• fluid and solid mechanics
• mathematical medicine and biology
• mathematical finance

Your study will include:

• complex numbers, vector algebra and geometry
• matrix algebra, inverses, determinants
• vector spaces, subspaces, bases
• linear systems of simultaneous equations, Gaussian elimination
• eigenvalues and eigenvectors, matrix diagonalisation
• linear transformations, inner product spaces

Groups, rings and fields are abstract structures which underpin many areas of mathematics. For example, addition of integers fits the structure of a group. However, by analysing the general concept of a group, our proofs are relevant to many other areas of mathematics.

You will build on your understanding of Foundations of Pure Mathematics. Together we will develop a deeper knowledge of abstract algebraic structures, particularly groups. This provides the foundation for subsequent modules in abstract algebra and number theory.

Your study will include:

• symmetries
• groups, cyclic groups, Lagrange’s theorem
• rings and fields
• integer arithmetic, Euclid’s algorithm
• polynomial arithmetic, factorisation

What is the importance of probability in the modern world?

It allows us to assess risk when calculating insurance premiums. It can help when making investment decisions. It can be used to estimate the impact that government policy will have on climate change or the spread of disease.

We will look at the theory and practice of discrete and continuous probability. Your study will include:

• sample spaces, events and counting problems
• conditional probability, independence, Bayes’ theorem
• random variables, expectation, variance
• discrete and continuous probability distributions
• multivariate random variables
• sums of random variables, central limit theorem

These topics will help you prepare for later modules in:

• probability methods
• stochastic models
• uncertainty quantification
• mathematical finance

Statistics is concerned with methods for collecting, organising, summarising, presenting and analysing data. It enables us to draw valid conclusions and make reasonable decisions based on the analysis. It can be used to answer a diverse range of questions such as:

• Do the results of a clinical trial indicate that a new drug works?
• Is the HS2 rail project likely to be cost-effective?
• Should a company lend money to a customer with a given credit history?

In this module you’ll study statistical inference and learn how to analyse, interpret and report data. You’ll learn about the widely used statistical computer language R.

Your study will include:

• exploratory data analysis
• point estimators, confidence intervals
• hypothesis testing
• correlation, statistical inference
• linear regression, chi-squared tests

These first-year topics give you the foundations for later related modules in:

• statistical models and methods
• data analysis and modelling
• statistical machine learning

This course will develop in more detail the fundamental concepts in algebra such as groups and rings and will provide an introduction to elementary number theory.

We will consider how general algebraic concepts can be applied in concrete situations in number theory and after a review of primes, integer factorization and module arithmetic, the focus will be on classical problems. This includes Fermat’s Little Theorem and its application to primality testing, methods of factorization, primitive roots, discrete logarithms, some classical Diophantine equations (linear and polynomial), Fibonacci numbers, and continued fractions.

In this module you will learn about the theory and applications of functions of a complex variable using a method and applications approach. You will develop an understanding of the theory of complex functions and evaluate certain real integrals using your new skills.

This course is an introduction to Fourier series and integral transforms and to methods of solving some standard ordinary and partial differential equations which occur in applied mathematics and mathematical physics.

The course describes the solution of ordinary differential equations using series and introduces Fourier series and Fourier and Laplace transforms, with applications to differential equations and signal analysis. Standard examples of partial differential equations are introduced and solution using separation of variables is discussed.

This module introduces basic techniques in numerical methods and numerical analysis which can be used to generate approximate solutions to problems that may not be amenable to analysis. Specific topics include:

• Implementing algorithms in Matlab
• Discussion of errors (including rounding errors)
• Iterative methods for nonlinear equations (simple iteration, bisection, Newton, convergence)
• Gaussian elimination, matrix factorisation, and pivoting
• Iterative methods for linear systems, matrix norms, convergence, Jacobi, Gauss-Siedel
• Interpolation (Lagrange polynomials, orthogonal polynomials, splines)
• Numerical differentiation & integration (Difference formulae, Richardson extrapolation, simple and composite quadrature rules)
• Introduction to numerical ODEs (Euler and Runge-Kutta methods, consistency, stability) 

In this module you will build on the foundation of knowledge gained from your core year one modules in Analytical and Computational Foundations and Calculus. You will learn to follow a rigorous approach needed to produce concrete proof of your workings.

This course aims to provide students with tools which enable them to develop and analyse linear and nonlinear mathematical models based on ordinary and partial differential equations. Furthermore, it aims to introduce students to the fundamental mathematical concepts required to model the flow of liquids and gases and to apply the resulting theory to model physical situations. 

This module will give you an introduction to the theory of probability and random variables, with particular attention paid to continuous random variables. Fundamental concepts relating to probability will be discussed in detail, including well-known limit theorems and the multivariate normal distribution. You will then progress onto complex topics such as transition matrices, one-dimensional random walks and absorption probabilities.

The first part of this module provides an introduction to statistical concepts and methods and the second part introduces a wide range of techniques used in a variety of quantitative subjects. The key concepts of inference including estimation and hypothesis testing will be described as well as practical data analysis and assessment of model adequacy.

This course aims to give students a sound grounding in the application of both differential and integral calculus to vectors, and to apply vector calculus methods and separation of variables to the solution of partial differential equations. The module is an important pre-requisite for a wide range of other courses in Applied Mathematics.

This module involves the application of probability and statistics to a variety of practical, open-ended problems, typical of those that statisticians encounter in industry and commerce. Specific projects are tackled through workshops and student-led group activities.

The real-life nature of the problems requires students to develop skills in model development and refinement, report writing and teamwork. Students will have an opportunity to apply a variety of statistical methods and knowledge learned in previous modules.

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This module consists of a self-directed investigation of a project selected from a list of projects or, subject to prior approval of the School, from elsewhere.

Project modules are carried out in the Autumn and Spring semesters.

The project will be supervised by a member of staff and will be based on a substantial mathematical problem, an application of mathematics or investigation of an area of mathematics not previously studied by the student. The course includes training in the use of IT resources, the word-processing of mathematics and report writing.

This module involves the application of mathematics to a variety of practical, open-ended problems, typical of those that mathematicians encounter in industry and commerce. Specific projects are tackled through workshops and student-led group activities. The real-life nature of the problems requires students to develop skills in model development and refinement, report writing and teamwork.

In this module you will apply the general theory you learnt in Introduction to Mathematical Physics to more general problems. New topics will be introduced such as the quantum theory of the hydrogen atom and aspects of angular momentum such as spin.

In this module you will build on your theoretical knowledge of statistical inference by a practical implementation of the generalised linear model. You will move on to enhance your understanding of statistical methodology including the analysis of discrete and survival data. You will also be trained in the use of a high-level statistical computer program.

The course introduces and explores methods, concepts and paradigm models for classical and quantum mechanical dynamics exploring how classical concepts enter quantum mechanics, and how they can be used to find approximate semi-classical solutions.

In classical dynamics we discuss full integrability and basic notions of chaos in the framework of Hamiltonian systems, together with advanced methods like canonical transformations, generating functions and Hamiltonian-Jacobi theory. In quantum mechanics we recall Schrödinger's equation and introduce the semi-classical approximation. We derive the Bohr-Sommerfeld quantization conditions based on a WKB-approch to the eigenstates. We will discuss some quantum signatures of classical chaos and relate them to predictions of random-matrix theory. We will also introduce Gaussian states and coherent states and discuss their semi-classical dynamics and how it is related to the corresponding classical dynamics. An elementary introduction to complete descriptions of quantum mechanics in terms of functions on the classical phase space will be given.

This course provides an introduction to coding theory in particular to error-correcting codes and their uses and applications. It also provides an introduction to to cryptography, including classical mono- and polyalphabetic ciphers as well as modern public key cryptography and digital signatures, their uses and applications.

This course provides an opportunity for third-year students taking G100 and G103 to gain first-hand experience of being involved with providing mathematical education.

Students will work at local schools alongside practising mathematics teachers in a classroom environment and will improve their skills at communicating mathematics. Typically, each student will work with a class (or classes) for half a day a week for about sixteen weeks. Students will be given a range of responsibilities from classroom assistant to leading a self-originated mathematical activity or project. The assessment is carried out by a variety of means: on-going reflective log, contribution to reflective seminar, oral presentation and a final written report.

This course introduces various analytical methods for the solution of ordinary and partial differential equations, focussing on asymptotic techniques and dynamical systems theory. Students taking this course will build on their understanding of differential equations covered in Modelling with Differential Equations.

The course will start with several topics from the perspective of what can be explicitly calculated with an emphasis on applications to geometry and number theory.

Topics include:

• basic notions of projective geometry
• plane algebraic curves including elliptic curves
• addition of points on elliptic curves 
• results on the group of rational points on an elliptic curve
• properties of elliptic curves and their applications.

Number theory concerns the solution of polynomial equations in whole numbers, or fractions. For example, the cubic equation x³ + y³ = z³ with x, y, z non-zero has infinitely many real solutions yet not a single solution in whole numbers.

We shall establish the basic properties of the Riemann zeta-function to find out how evenly these primes are distributed in nature. This course will present several methods to solve Diophantine equations including analytical methods using zeta-functions and Dirichlet series, theta functions and their applications to arithmetic problems, and an introduction to more general modular forms.

The course provides an introduction to electromagnetism and the electrodynamics of charged particles. The aims of this course are:

• to develop an appropriate mathematical model of electromagnetic phenomena that is informed by observations
• to understand electromagnetic configurations of practical importance and to relate predictions made to everyday phenomena
• to illustrate the use of solutions of certain canonical partial differential equations for determining electrostatic fields and electromagnetic waves in vacuum and in matter
• to illustrate the interplay between experimental input and the development of a mathematical model, and the use of various mathematical techniques for solving relevant problems.

A graph (in the sense used in Graph Theory) consists of vertices and edges, each edge joining two vertices. Graph Theory has become increasingly important recently through its connections with computer science and its ability to model many practical situations. 

Topics covered in the course include:

• paths and cycles
• the resolution of Euler’s Königsberg Bridge Problem
• Hamiltonian cycles
• trees and forests
• labelled trees,
• the Prüfer correspondence
• planar graphs
• Demoucron et al. algorithm
• Kruskal's algorithm
• the Travelling Salesman's problem
• the statement of the four-colour map theorem
• colourings of vertices
• chromatic polynomial
• colourings of edges.

This course builds on the basic ideas of group theory. It covers a number of key results such as the simplicity of the alternating groups, the Sylow theorems (of fundamental importance in abstract group theory), and the classification of finitely generated abelian groups (required in algebraic number theory, combinatorial group theory and elsewhere). Other topics to be covered are group actions, used to prove the Sylow theorems, and series for groups, including the notion of solvable groups that will be used in Galois theory.

This module gives an introduction into some basic ideas of functional analysis with an emphasis on Hilbert spaces and operators on them.

Many concepts from linear algebra in finite dimensional vector spaces (e.g. writing a vector in terms of a basis, eigenvalues of a linear map, diagonalisation etc.) have generalisations in the setting of infinite dimensional spaces making this theory a powerful tool with many applications in pure and applied mathematics

In this module the concepts of discrete time Markov chains are explored and used to provide an introduction to probabilistic and stochastic modelling for investment strategies, and for the pricing of financial derivatives in risky markets. You will gain well-rounded knowledge of contemporary issues which are of importance in research and applications.

Metric space generalises the concept of distance familiar from Euclidean space. It provides a notion of continuity for functions between quite general spaces.

The module covers metric spaces, topological spaces, compactness, separation properties like Hausdorffness and normality, Urysohn’s lemma, quotient and product topologies, and connectedness. Finally, Borel sets and measurable spaces are introduced.

This module is concerned with the analysis of multivariate data, in which the response is a vector of random variables rather than a single random variable. A theme running through the module is that of dimension reduction. Key topics to be covered include: principal components analysis, whose purpose is to identify the main modes of variation in a multivariate dataset; modelling and inference for multivariate data, including multivariate regression data, based on the multivariate normal distribution; classification of observation vectors into sub-populations using a training sample; canonical correlation analysis, whose purpose is to identify dependencies between two or more sets of random variables. Further topics to be covered include factor analysis, methods of clustering and multidimensional scaling.

In this module a variety of techniques and areas of mathematical optimisation will be covered including Lagrangian methods for optimisation, simplex algorithm linear programming and dynamic programming. You’ll develop techniques for application which can be used outside the mathematical arena. 

In this module you’ll have an introduction to Einstein’s theory of general and special relativity. The relativistic laws of mechanics will be described within a unified framework of space and time. You’ll learn how to compare other theories against this work and you’ll be able to explain new phenomena which occur in relativity.

Commutative rings and modules over them are the fundamental objects of what is often referred to as commutative algebra. Already encountered key examples of commutative rings are polynomials in one variable over a field and number rings such as the usual integers or the Gaussian integers.

There are many close parallels between these two types of rings, for example the similarities between the prime factorization of integers and the factorization of polynomials into irreducibles. In this module, these ideas are extended and generalized to cover polynomials in several variables and power series, and algebraic numbers.

You will learn how to use numerical techniques for determining the approximate solution of ordinary and partial differential equations where a solution cannot be found through analytical methods alone. You will also cover topics in numerical linear algebra, discovering how to solve very large systems of equations and find their eigenvalues and eigenvectors using a computer.

This course is concerned with the two main theories of statistical inference, namely classical (frequentist) inference and Bayesian inference. 

Topics such as sufficiency, estimating equations, likelihood ratio tests and best-unbiased estimators are explored in detail. There is special emphasis on the exponential family of distributions, which includes many standard distributions such as the normal, Poisson, binomial and gamma.

In Bayesian inference, there are three basic ingredients: a prior distribution, a likelihood and a posterior distribution, which are linked by Bayes' theorem. Inference is based on the posterior distribution, and topics including conjugacy, vague prior knowledge, marginal and predictive inference, decision theory, normal inverse gamma inference, and categorical data are pursued.

Common concepts, such as likelihood and sufficiency, are used to link and contrast the two approaches to inference. You will gain experience of the theory and concepts underlying much contemporary research in statistical inference and methodology.

In this module you will develop your knowledge of discrete-time Markov chains by applying them to a range of stochastic models. You will be introduced to Poisson and birth-and-death processes and then you will move onto more extensive studies of epidemic models and queuing models with introductions to component and system reliability.

This module will consist of self-directed but supervised study of an appropriate area of mathematics for the whole year. A list of possible topics will be supplied by the School and students choose a topic of interest to them, The study should result in a sustained piece of work assessed by an interim report, an oral presentation and a dissertation.

You will develop your knowledge and skills relevant to the mathematical modelling of investment and finance. Also, research experience will be broadened by undertaking some independent reading, computer simulations, group work and summarising the material in a project report.

The development of techniques for the study of nonlinear differential equations is a major worldwide research activity to which members of the School have made important contributions. This course will cover a number of state-of-the-art methods, namely:

• use of green function methods in the solution of linear partial differential equations
• characteristic methods, classification and regularization of nonlinear partial differentiation equations
• bifurcation theory

These will be illustrated by applications in the biological and physical sciences.

This module presents the fundamental features of algebraic number theory, the theory in which numbers are viewed from an algebraic point of view.

Numbers are often treated as elements of rings, fields and modules, and properties of numbers are reformulated in terms of the relevant algebraic structures. This approach leads to understanding of certain arithmetical properties of numbers (in particular, integers) from a new point of view.

In this module you’ll systematically study black holes and their properties, including astrophysical processes, horizons and singularities. You’ll have an introduction to black hole radiation to give you an insight into problems of research interest. You’ll gain knowledge to help you begin research into general relativity.

This module is largely concerned with infinite groups, especially free groups, although their role in describing and understanding finite groups is emphasized.

Following the basic definitions of free groups and group presentations, the fundamental Nielson-Schreier Theorem is covered in some detail. Methods for manipulating group presentations, and using them to read off properties of a given group, will also be covered.

This module introduces computational methods for solving problems in applied mathematics. You will develop knowledge and understanding to design, justify and implement relevant computational techniques and methodologies.

The increase in speed and memory capacity of modern computers has dramatically changed their use and applicability for complex statistical analysis. This module explores how computers allow the easy implementation of the standard, but computationally intensive, statistical methods and also explores their use in the solution of non-standard analytically intractable problems by innovative numerical methods.

The material builds on the theory of the course to cover several topics that form the basis of some current research areas in computational statistics. Particular topics to be covered include a selection from simulation methods, Markov chain Monte Carlo methods, the bootstrap and nonparametric statistics, statistical image analysis, and wavelets. You will gain experience of using a statistical package and interpreting its output.

In this module you’ll be equipped with the tools and knowledge to extend your understanding of general relativity. You’ll explore more abstract and powerful concepts using examples of curved space-times such as Lie groups and manifolds among others. 

The first part of the module introduces no-arbitrage pricing principle and financial instruments such as forward and futures contracts, bonds and swaps, and options. The second part of the module considers the pricing and hedging of options and discrete-time discrete-space stochastic processes. The final part of the module focuses on the Black-Scholes formula for pricing European options and also introduces the Wiener process. Ito integrals and stochastic differential equations.

This module gives you a mathematical introduction to quantum information theory. The aim is to provide you with a background in quantum information science which will facilitate further independent learning and allow you to understand the scope and nature of current research topics.

In this year-long module you’ll be introduced to the study of the quantum dynamics of relativistic particles. You’ll learn about the quantum description of electrons, photons and other elementary particles leading to an understanding of the standard model of particle physics. 

Machine Learning is a topic at the interface between statistics and computer science that concerns models that can adapt to and make predictions based on data.

This module builds on principles of statistical inference and linear regression to introduce a variety of methods of regression and classification, trade-off, and on methods to measure and compensate for overfitting. The learning approach is hands on, with students using R to tackle challenging real world machine learning problems.

This module will provide a general introduction to the analysis of data that arise sequentially in time. You will discuss several commonly-occurring models, including methods for model identification for real-time series data. You will develop techniques for estimating the parameters of a model, assessing its fit and forecasting future values. 

This module illustrates the applications of advanced techniques of mathematical modelling using ordinary and partial differential equations. A variety of medical and biological topics are treated bringing you close to active fields of mathematical research.

You will learn to apply ideas from probability and statistics and various mathematical tools in traditional areas of applied and computational mathematics involving ordinary differential equation (ODE) and partial differential equation (PDE) models.  The module will provide an introduction to a variety of techniques which are useful in UQ and will focus on a more in-depth study of selected application areas