###### MSci|Undergraduate

# Mathematics

**Entry year**

Academic Year 2023/24

**Entry requirements**

AAA/A*AB

**Duration**

4 years (Full Time)

**UCAS code**

G103

Mathematics is the universal language of science. It pervades technology, society, medicine, life, the universe and everything!

Mathematics is also one of the most powerful tools for analysis and problem solving known to mankind. As a result, mathematics graduates have the fifth highest employment rate of any degree subject in the UK, and the highest of all the 'pure' sciences. Thus, mathematics provides a unique combination of factors: a pure science with infinite intrigue awaiting a curious mind, and a hugely valuable and employable skill set enabling a broad range of possible future careers.

### Mathematics Degree highlights

100% satisfaction score (National Student Survey 2017 and 2018).

#### Global Opportunities

- We participate in the IAESTE and Turing student exchange programmes, which enable students to obtain work experience in companies and universities throughout the world.

#### Industry Links

- All students in the school have the option to include a year in industry as part of their studies. This is a fantastic opportunity to see mathematics at work in the real world, and to enhance your career prospects at the same time. Possible placements will include companies in the finance and technology sectors, and indeed we maintain strong links with local companies who hire mathematics graduates. Several local financial services companies (including Carus FT, Effex Capital and AquaQ Analytics) were founded by mathematics graduates from Queen’s.

#### World Class Facilities

- The school has its own dedicated teaching centre which opened in September 2016. This building houses lecture and group-study rooms, a hugely popular student social area and state-of-the-art computer and laboratory facilities. The centre is an exciting hub for our students and is situated directly adjacent to the Lanyon Building on the main university campus. This makes us the only school with a dedicated teaching space right at the heart of the university.

#### Internationally Renowned Experts

- The School of Mathematics and Physics is a large school with staff from 13 countries, including UK, US, Ireland, Italy, Spain, Bulgaria, Russia, Argentina, Cuba, Germany, China, Greece, Kenya, Niger, The Netherlands and Romania. Many of our staff are leading international experts in their fields of mathematical research. Thus the ethos of the School is one of excellence in research informing excellence in teaching.

#### Student Experience

- This 4-year integrated masters programme allows our students to advance their knowledge and skills to a much higher level of proficiency. The additional year also enables these capabilities to be applied in an extensive research project during which the students’ confidence and maturity grows markedly. This ultimately transforms the career prospects of our graduates
- Many students find the transition from school to university somewhat daunting. In order to help with this transition, mathematics students have pioneered a Peer Mentoring scheme that is generally regarded as one of the most effective in the University. As well as providing a forum for first year students to obtain support, it also provides mentors with transferable skills which will increase graduate employment opportunities.

Completing my Masters in Mathematics has made me more employable with a degree in my teaching subject, as well as giving me knowledge behind basic mathematical concepts taught in a school environment.

Sarah Wilson (MSci Mathematics 2019)

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Course content

#### Course Structure

Course Content | The course unit details given below are subject to change, and are the latest example of the curriculum available on this course of study. |
---|---|

Stage 1 | • Introduction to Algebra and Analysis • Mathematical Reasoning • Mathematical Methods 1 • Algorithmic Thinking • Introduction to Probability & Statistics • Introduction to Statistical & Operational Research Methods |

Stage 2 | Students have a choice from the modules listed, those with an * are compulsory • Employability for Mathematics • Linear Algebra * • Analysis * • Metric Spaces * • Group Theory • Mathematical Methods 2 * • Classical Mechanics • Statistical Inference • Methods of Operational Research |

Stage 3 | Students can take an approved selection of modules from the list below, those with an * are compulsory • Investigations * or • Mathematical Investigations * • Rings and Modules • Dynamical Systems • Discrete Mathematics • Numerical Analysis • Modelling and Simulation • Financial Mathematics • Classical Fields • Quantum Theory • Set Theory * • Metric and Normed Spaces * • Linear and Dynamic Programming • Linear Models • Statistical Data Mining with Machine Learning • Stochastic Processes and Risk • Functional Analysis • Topological Data Analysis • Statistical Mechanics • Geometry of Optimisation • Fourier Analysis and Partial Differential Equations • Quantum Fields |

Stage 4 | The Level 4 modules open to MSci students offer students the opportunity to study a selection of topics in greater depth. The centrepiece of the fourth-year is the double-weighted investigations module, in which a student has the opportunity to study an aspect of mathematics close to the frontier of knowledge. The main choice of modules includes: (Students must choose 4 modules plus the Project) • Project • Topology • Applied Algebra and Cryptography • Information Theory • Mathematical Methods of Quantum Information Processing • Practical Methods for Partial Differential Equations • Advanced Quantum Theory • Functional Analysis • Topological Data Analysis • Statistical Mechanics • Integration Theory • Survival Analysis • Geometry of Optimisation • Fourier Analysis and Partial Differential Equations • Quantum Fields |

#### People teaching you

Dr Gleb GribakinAssociate Director of Education for Mathematics

**School of Maths and Physics**

Dr Gribakin is a Reader in Mathematics, with an active research program in the area of antimatter.

#### Contact Teaching Times

Medium Group Teaching | 4 (hours maximum) 4 hours of practical classes, workshops or seminars each week. |
---|---|

Personal Study | 21 (hours maximum) 21 hours studying and revising in your own time each week, including some guided study using handouts, online activities, etc. |

Small Group Teaching/Personal Tutorial | 1 (hours maximum) 1 hour of tutorials (or later, project supervision) each week. |

Large Group Teaching | 10 (hours maximum) 10 hours of lectures. |

#### Learning and Teaching

At Queen’s, we aim to deliver a high quality learning environment that embeds intellectual curiosity, innovation and best practice in learning, teaching and student support to enable student to achieve their full academic potential.

On the MSci in Mathematics we do this by providing a range of learning experiences which enable our students to engage with subject experts, develop attributes and perspectives that will equip them for life and work in a global society and make use of innovative technologies and a world class library that enhances their development as independent, lifelong learners. Examples of the opportunities provided for learning on this course are:

**Computer based modules**

These provide students with the opportunity to develop technical skills and apply theoretical principles to real-life or practical contexts.**E-learning technologies**

Information associated with lectures and assignments is often communicated via a Virtual Learning Environment (VLE) called Canvas. A range of e-learning experiences are also embedded in the degree programme through the use of, for example, interactive support materials and web-based learning activities.**Lectures**

These introduce basic information about new topics as a starting point for further self-directed private study/reading. Lectures also provide opportunities to ask questions, gain some feedback and advice on assessments (normally delivered in large groups to all year group peers).**Personal Tutor**

Undergraduates are allocated a Personal Tutor during Level 1 and Level 2 who meets with them on several occasions during the year to support their academic development.**Self-directed study**

This is an essential part of life as a Queen’s student when important private reading, engagement with e-learning resources, reflection on feedback to date and assignment research and preparation work is carried out.**Supervised projects**

In final year, students will be expected to carry out a significant piece of research on a topic or practical methodology that they have chosen. Students will receive support from a supervisor who will guide them in terms of how to carry out research and who will provide feedback on at least 2 occasions during the write up stage.**Tutorials**

Significant amounts of teaching are carried out in small groups (typically 10-20 students). These provide an opportunity for students to engage with academic staff who have specialist knowledge of the topic, to ask questions of them and to assess their own progress and understanding with the support of peers.

#### Assessment

The way in which students are assessed will vary according to the learning objectives of each module. Details of how each module is assessed are shown in the Student Handbook which is available online via the school website.

- Most modules are assessed through a combination of coursework and end of year examinations.
- Some modules (eg, final year Honours Project module) are assessed solely through project work or written assignments.
- Computer Based Assessment

Some modules use online quizzes/tests as part of the module assessment. This tests basic knowledge, understanding and problem solving.

#### Feedback

As students progress through their course at Queen’s they will receive general and specific feedback about their work from a variety of sources including lecturers, module co-ordinators, placement supervisors, personal tutors, advisers of study and your peers. University students are expected to engage with reflective practice and to use this approach to improve the quality of their work. Feedback may be provided in a variety of forms including:

- Feedback provided via formal written comments and marks relating to work that you, as an individual or as part of a group, have submitted.
- Face to face comment. This may include occasions when you make use of the lecturers’ advertised “office hours” to help you to address a specific query.
- Placement employer comments or references.
- Online or emailed comment.
- General comments or question and answer opportunities at the end of a lecture, seminar or tutorial.
- Pre-submission advice regarding the standards you should aim for and common pitfalls to avoid. In some instances, this may be provided in the form of model answers or exemplars which you can review in your own time.
- Feedback and outcomes from practical classes.
- Comment and guidance provided by staff from specialist support services such as, Careers, Employability and Skills or the Learning Development Service.
- Once you have reviewed your feedback, you will be encouraged to identify and implement further improvements to the quality of your work.

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Overview

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Modules

#### Modules

The information below is intended as an example only, featuring module details for the current year of study (2023/24). Modules are reviewed on an annual basis and may be subject to future changes – revised details will be published through Programme Specifications ahead of each academic year.

- Year 1
#### Core Modules

Mathematical Methods 1 (30 credits)## Mathematical Methods 1

### Overview

Review of A-level calculus: elementary functions and their graphs, domains and ranges, trigonometric functions, derivatives and differentials, integration. Maclaurin expansion. Complex numbers and Euler’s formula.

Differential equations (DE); first-order DE: variable separable, linear; second-order linear DE with constant coefficients: homogeneous and inhomogeneous.

Vectors in 3D, definitions and notation, operations on vectors, scalar and vector products, triple products, 2x2 and 3x3 determinants, applications to geometry, equations of a plane and straight line. Rotations and linear transformations in 2D, 2x2 and 3x3 matrices, eigenvectors and eigenvalues.

Newtonian mechanics: kinematics, plane polar coordinates, projectile motion, Newton’s laws, momentum, types of forces, simple pendulum, oscillations (harmonic, forced, damped), planetary motion (universal law of gravity, angular momentum, conic sections, Kepler’s problem).

Curves in 3D (length, curvature, torsion). Functions of several variables, derivatives in 2D and 3D, Taylor expansion, total differential, gradient (nabla operator), stationary points for a function of two variables. Vector functions; div, grad and curl operators and vector operator identities. Line integrals, double integrals, Green's theorem. Surfaces (parametric form, 2nd-degree surfaces). Curvilinear coordinates, spherical and cylindrical coordinates, orthogonal curvilinear coordinates, Lame coefficients. Volume and surface integrals, Gauss's theorem, Stokes's theorem. Operators div, grad, curl and Laplacian in orthogonal curvilinear coordinates.### Learning Outcomes

On completion of the module, the students are expected to be able to:

• Sketch graphs of standard and other simple functions;

• Use of the unit circle to define trigonometric functions and derive their properties;

• Integrate and differentiate standard and other simple functions;

• Expand simple functions in Maclaurin series and use them;

• Perform basic operations with complex numbers, derive and use Euler's formula;

• Solve first-order linear and variable separable differential equations;

• Solve second-order linear differential equations with constant coefficients (both homogeneous and inhomogeneous), identify complementary functions and particular integrals, and find solutions satisfying given initial conditions;

• Perform operations on vectors in 3D, including vector products, and apply vectors to solve a range of geometrical problems; derive and use equations of straight lines and planes in 3D;

• Calculate 2x2 and 3x3 determinants;

• Use matrices to describe linear transformations in 2D, including rotations, and find eigenvalues and eigenvectors for 2x2 matrices.

• Define basis quantities in mechanics, such as velocity, acceleration and momentum, and state Newton’s laws;

• Use calculus for solving a range of problems in kinematics and dynamics, including projectile motion, oscillations and planetary motion;

• Define and recognise the equations of conics, in Cartesian and polar coordinates;

• Investigate curves in 3D, find their length, curvature and tension;

• Find partial derivatives for a function of several variables;

• Expand functions of one and two variables in the Taylor series and investigate their stationary points;

• Find the partial differential operators div, grad and curl for scalar and vector fields;

• Calculate line integrals along curves;

• Calculate double and triple integrals, including surface and volume integrals;

• Transform between Cartesian, spherical and cylindrical coordinate systems;

• Investigate simple surfaces in 3D and evaluate surface for the shapes such as the cube, sphere, hemisphere or cylinder;

• State and apply Green's theorem, Gauss's divergence theorem, and Stokes's theorem### Skills

• Proficiency in calculus and its application to a range of problems.

• Constructing and clearly presenting mathematical and logical arguments.

• Mathematical modelling and problem solving.

• Ability to manipulate precise and intricate ideas.

• Analytical thinking and logical reasoning.Coursework

15%

Examination

85%

Practical

0%

###### Stage/Level

1

###### Credits

30

###### Module Code

MTH1021

###### Teaching Period

Full Year

###### Duration

12 weeks

Algorithmic Thinking (10 credits)## Algorithmic Thinking

### Overview

Basic programming skills (e.g. in Python); introduction of software to present mathematical contents (e.g. LaTex) and to solve mathematical problems (e.g. Mathematica, R or packages like numpy and matplotlib); basic understanding of the complexity of algorithms (Big Oh notation).

### Learning Outcomes

By the end of this module students should be able to

1. Use python and/or Mathematica and/or R to

1. solve simple mathematical problems

2. visualise results with suitable plots

2. Construct, implement and follow simple algorithms and analyse their worst case complexity

3. Use Latex to present and disseminate mathematical results### Skills

Basic computer programming; basic analysis of algorithms; basic skills in the presentation of mathematical results.

Coursework

100%

Examination

0%

Practical

0%

###### Stage/Level

1

###### Credits

10

###### Module Code

MTH1025

###### Teaching Period

Spring

###### Duration

12 weeks

Mathematical Reasoning (10 credits)## Mathematical Reasoning

### Overview

The notion of mathematical statements and elementary logic. Mathematical symbols and notation. The language of sets. The concept of mathematical proof, and typical examples. Communicating mathematics to others.

### Learning Outcomes

By the end of the module, students are expected to be able to: state key mathematical statements and definitions and be familiar with standard mathematical notation and its meaning; describe the role played by mathematical proof and reproduce the proofs of key mathematical statements using the methods of induction, proof by contradiction and direct proof; identify patterns within proofs that can be used in other contexts and use them successfully in the construction of new proofs; use natural language to communicate key mathematical concepts to fellow students in a rigorous way.

### Skills

The key skills that will be developed through the module are: problem solving skills, presentation skills and logical thinking skills.

The final module mark is determined by various components: tutorial participation, homework assignments, oral presentations, and a report. As a guideline, one can expect four pieces of written homework, and two oral presentations (the first being a practice run carrying few marks). Details of the assessment scheme will be made available at the start of the semester.Coursework

65%

Examination

0%

Practical

35%

###### Stage/Level

1

###### Credits

10

###### Module Code

MTH1015

###### Teaching Period

Autumn

###### Duration

12 weeks

Introduction to Algebra and Analysis (30 credits)## Introduction to Algebra and Analysis

### Overview

Elementary logic and set theory, number systems (including integers, rationals, reals and complex numbers), bounds, supremums and infimums, basic combinatorics, functions.

Sequences of real numbers, the notion of convergence of a sequence, completeness, the Bolzano-Weierstrass theorem, limits of series of non-negative reals and convergence tests.

Analytical definition of continuity, limits of functions and derivatives in terms of a limit of a function. Properties of continuous and differentiable functions. L'Hopital's rule, Rolle's theorem, mean-value theorem.

Matrices and systems of simultaneous linear equations, vector spaces, linear dependence, basis, dimension.### Learning Outcomes

It is intended that students shall, on successful completion of the module, be able:

• to understand and to apply the basic of mathematical language;

• use the language of sets and maps and understand the basic properties of sets (finiteness) and maps (injectivity, surjectivity, bijectivity);

• demonstrate knowledge of fundamental arithmetical and algebraic properties of the integers (divisibility, prime numbers, gcd, lcm) and of the rationals;

• Solve combinatorial counting problems in a systematic manner.

• Understand the fundamental properties of the real numbers (existence of irrational numbers, density of Q, decimal expansion, completeness of R).

• Understand the notions of a sequence of real numbers, including limits, convergence and divergence.

• Define convergence of infinite series.

• Investigate the convergence of infinite series using convergence tests.

• Define limits of functions and define continuous functions.

• Prove that a function is continuous or discontinuous.

• Prove and apply basic properties of continuous functions including the intermediate value theorem and the existence of a maximum and a minimum on a compact interval.

• Define a differentiable function and a derivative.

• Prove whether a function is differentiable.

• Calculate (using analysis techniques) derivatives of many types of functions.

• Understand, apply and prove Rolle's theorem and the Mean Value Theorem.

• Prove the rules of differentiation such as the product.

• Understand and apply the theory of systems of linear equations.

• Produce and understand the definitions of vector space, subspace, linear independence of vectors, bases of vector spaces, the dimension of a vector space.

• Apply facts about these notions in particular examples and problems.

• Understand the relation between systems of linear equations and matrices.### Skills

• Understanding of part of the main body of knowledge for mathematics: analysis and linear algebra.

• Logical reasoning.

• Understanding logical arguments: identifying the assumptions made and the conclusions drawn.

• Applying fundamental rules and abstract mathematical results, equation solving and calculations; problem solving.Coursework

0%

Examination

100%

Practical

0%

###### Stage/Level

1

###### Credits

30

###### Module Code

MTH1011

###### Teaching Period

Full Year

###### Duration

12 weeks

#### Optional Modules

## Introduction to Statistical and Operational Research Methods

### Overview

Introduction to statistical software for applying the following topics in Operational Research and Statistical Methods:

Linear Programming: Characteristics of linear programming models, general form. Graphical solution. Simplex method: standard form of linear programming problem, conversion procedures, basic feasible solutions. Simplex algorithm: use of artificial variables.

Decision Theory: Characteristics of a decision problem. Decision making under uncertainty: maximax, maximin, generalised maximin (Hurwicz), minimax regret criteria. Decision making under risk: Bayes criterion, value of perfect information. Decision tree; Bayesian decision analysis.

Random Sampling and Simulation: Random sample from a finite population, from a probability distribution. Use of random number tables. General method for drawing a random sample from a discrete distribution. Drawing a random sample from a continuous distribution: inverse transformation method, exponential distribution. Dynamic simulation techniques: application to queueing problems. Computer aspects: random number generators, sampling from normal distributions.

Initial Data Analysis: Scales of measurement. Discrete and continuous variables. Sample mean, variance, standard deviation, percentile for ungrouped data; boxplot. Frequency table for grouped discrete data: relative frequency, cumulative frequency, bar diagram; sample mean, variance, percentile. Frequency table for grouped continuous data: stem-and-leaf plot, histogram, cumulative percentage frequency plot; sample mean, variance, percentile. Linear transformation. Bivariate data; scatter diagram, sample correlation coefficient.### Learning Outcomes

Perform linear programming using computer software.

Utilise decision analysis methods, such as decision trees.

Simulate date and produce random samples.

Calculate descriptive statistics for a given data set identifying the key characteristics and any unusual features.

Summarise data using appropriate graphical and tabular techniques.### Skills

Apply a range of statistical and OR techniques to data using an appropriate method. Computational skills in statistical software to manage and analyse data. Ability to interpret results and add meaning to the analysis. Understanding sampling processes and the appropriate process to undertake.

Coursework

0%

Examination

0%

Practical

100%

###### Stage/Level

1

###### Credits

10

###### Module Code

SOR1021

###### Teaching Period

Spring

###### Duration

12 weeks

Introduction to Probability & Statistics (30 credits)## Introduction to Probability & Statistics

### Overview

This is a fundamental module which provides an introduction to probability theory and the key concepts found in statistics. The topics covered include the laws of probability, discrete and continuous random variables, standard discrete and continuous distributions, bivariate distributions, statistical models, sampling, estimation, hypothesis testing and statistical quality control.

### Learning Outcomes

- Demonstrate an understanding of the concepts of probability, conditional probability, multiplicative law, independence, Bayes theorem and their interpretations.

- Be able to apply set theory to the proof and use of the axioms of probability.

- Understand and use combinatorial methods: counting rules; sampling with and without replacement; ordered and non-ordered samples.

- Be able to define discrete and continuous random variables and the corresponding probability distributions, probability functions, cumulative distribution functions and probability density functions.

- Understand and use transformations in the discrete and continuous variable context.

- Be able to define expectation and calculate expected values for the mean and variance of specific discrete and continuous distributions.

- Be able to define, interpret and apply the properties of the expectation and variance operators for discrete and continuous cases.

- Demonstrate an understanding of key discrete and continuous distributions including the specific circumstances when distributions may be applied.

- Demonstrate an ability to use statistical tables and deal with linear combinations of independent normal random variables.

- Be familiar with the Central limit theorem for the approximate distribution of sample mean and be able to utilise this theorem in the approximation of binomial and Poisson distributions.

- Understand and be able to define bivariate distributions, their joint probability (density) functions, cumulative distribution functions, marginal distributions, conditional distributions of discrete and continuous random variables.

- Demonstrate an understanding of independence for bivariate data.

- Be able to define expectation and to calculate expected values: means, variances and covariances, correlation coefficients for bivariate distributions and for linear combinations of random variables.

- Be able to define statistical models, experimental, systematic and random errors; precision and accuracy.

- Be able to describe and utilise the following methods of sampling: accessibility, judgement, quota, sequential, random, systematic, stratified and cluster sampling methods.

- Understand the concept of estimation, the definition of a statistic, sampling distribution, sample estimator, sample estimate and the desirable properties for an estimator.

- Be able to define and calculate an estimate of the population mean and variance from a single sample and from several samples.

- Demonstrate an understanding of and be able to implement the method of moments, maximum likelihood estimation and the method of least squares, in particular, the likelihood function, asymptotic variance, normal equations, and linear regression.

- Understand and be able to define the null and alternative hypotheses; one and two-sided tests; test statistic; critical region, P-value, significance level; type I and type II errors; power function and confidence intervals.

- Know when to apply the correct method for significance testing based on given circumstances.

- Be able to interpret results of a significance test and confidence intervals.

- Demonstrate an understanding and be able to describe non-parametric methods and their advantages and disadvantages.

- Understand, be able to carry out and interpret significance tests, in particular key parametric tests based on the Normal distribution, t-distribution, F-distribution and Chi squared distribution, and key non-parametric tests.

- Know when to apply and how to calculate nonparametric statistics and how to choose the appropriate technique to use for a practical example.### Skills

- Understanding when and how to apply probability theory and reasoning with uncertainty.

- Knowing how to apply estimation approaches and the appropriate technique to use.

- Being able to apply probability theory to a practical example.

- Understanding the principles of hypothesis testing.

- Knowing when to apply the correct method for significance testing.

- Calculating test statistics and being able to use these to draw a conclusion about a null hypothesis.

- Knowing when to apply and how to calculate nonparametric statistics and choosing the appropriate technique to use for a practical example.Coursework

0%

Examination

100%

Practical

0%

###### Stage/Level

1

###### Credits

30

###### Module Code

SOR1020

###### Teaching Period

Full Year

###### Duration

24 weeks

- Year 2
#### Core Modules

Metric Spaces (20 credits)## Metric Spaces

### Overview

- definition and examples of metric spaces (including function spaces)

- open sets, closed sets, closure points, sequential convergence, density, separability

- continuous mappings between metric spaces

- completeness### Learning Outcomes

It is intended that students shall, on successful completion of the module, be able to: understand the concept of a metric space; understand convergence of sequences in metric spaces; understand continuous mappings between metric spaces; understand the concepts and simple properties of special subsets of metric spaces (such as open, closed and compact); understand the concept of Hilbert spaces, along with the basic geometry of Hilbert spaces, orthogonal decomposition and orthonormal basis.

### Skills

Analytic argument skills, problem solving, analysis and construction of proofs.

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

2

###### Credits

20

###### Module Code

MTH2013

###### Teaching Period

Spring

###### Duration

12 weeks

Mathematical Methods 2 (20 credits)## Mathematical Methods 2

### Overview

Functions of a complex variable: limit in the complex plane, continuity, complex differentiability, analytic functions, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, Taylor and Laurent series, residues, Cauchy residue theorem, evaluation of integrals using the residue theorem.

Series solutions to differential equations: Frobenius method.

Fourier series and Fourier transform. Basis set expansion.

Introduction to partial differential equations. Separation of variables. Wave equation, diffusion equation and Laplace’s equation.### Learning Outcomes

On completion of the module, the students are expected to be able to:

• determine whether or not a given complex function is analytic;

• recognise and apply key theorems in complex integration;

• use contour integration to evaluate real integrals;

• apply Fourier series and transforms to model examples;

• solve the wave equation, diffusion equation and Laplace’s equation with model boundary conditions, and interpret the solutions in physical terms.### Skills

• Proficiency in complex calculus and its application to a range of problems.

• Constructing and presenting mathematical and logical arguments.

• Mathematical modelling and problem solving.

• Ability to manipulate precise and intricate ideas.

• Analytical thinking and logical reasoning.Coursework

40%

Examination

60%

Practical

0%

###### Stage/Level

2

###### Credits

20

###### Module Code

MTH2021

###### Teaching Period

Spring

###### Duration

12 weeks

Linear Algebra (20 credits)## Linear Algebra

### Overview

- Recap and extend to fields such as C, the notions of abstract vector spaces and subspaces, linear independence, basis, dimension.

- Linear transformations, image, kernel and dimension formula.

- Matrix representation of linear maps, eigenvalues and eigenvectors of matrices.

- Matrix inversion, definition and computation of determinants, relation to area/volume.

- Change of basis, diagonalization, similarity transformations.

- Inner product spaces, orthogonality, Cauchy-Schwarz inequality.

- Special matrices (symmetric, hermitian, orthogonal, unitary, normal) and their properties.

- Basic computer application of linear algebra techniques.

Additional topics and applications, such as: Schur decomposition, orthogonal direct sums and geometry of orthogonal complements, Gram-Schmidt orthogonalization, adjoint maps, Jordan normal form.### Learning Outcomes

It is intended that students shall, on successful completion of the module: have a good understanding and ability to use the basics of linear algebra; be able to perform computations pertaining to problems in these areas; have reached a good level of skill in manipulating basic and complex questions within this framework, and be able to reproduce, evaluate and extend logical arguments; be able to select suitable tools to solve a problem, and to communicate the mathematical reasoning accurately and confidently.

### Skills

Analytic argument skills, computation, manipulation, problem solving, understanding of logical arguments.

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

2

###### Credits

20

###### Module Code

MTH2011

###### Teaching Period

Autumn

###### Duration

12 weeks

Analysis (20 credits)## Analysis

### Overview

Cauchy sequences, especially their characterisation of convergence. Infinite series: further convergence tests (limit comparison, integral test), absolute convergence and conditional convergence, the effects of bracketing and rearrangement, the Cauchy product, key facts about power series (longer proofs omitted). Uniform continuity: the two-sequence lemma, preservation of Cauchyness (and the partial converse on bounded domains), equivalence with continuity on closed bounded domains, a gluing lemma, the bounded derivative test. Mean value theorems including that of Cauchy, proof of l'Hôpital's rule, Taylor's theorem with remainder. Riemann integration: definition and study of the main properties, including the fundamental theorem of calculus.

### Learning Outcomes

It is intended that students shall, on successful completion of the module, be able to: understand and apply the Cauchy property together with standard Level 1 techniques and examples in relation to limiting behaviour for a variety of sequences; understand the relationships between sequences and series, especially those involving the Cauchy property, and of standard properties concerning absolute and conditional convergence, including power series and Taylor series; demonstrate understanding of the concept of uniform continuity of a real function on an interval, its determination by a range of techniques, and its consequences; understand through the idea of differentiability how to develop and apply the basic mean value theorems; describe the process of Riemann integration and the reasoning underlying its basic theorems including the fundamental theorem of calculus, and relate the concept to monotonicity and continuity.

### Skills

Knowledge of core concepts and techniques within the material of the module. A good degree of manipulative skill, especially in the use of mathematical language and notation. Problem solving in clearly defined questions, including the exercise of judgment in selecting tools and techniques. Analytic and logical approach to problems. Clarity and precision in developing logical arguments. Clarity and precision in communicating both arguments and conclusions. Use of resources, including time management and IT where appropriate.

Coursework

10%

Examination

90%

Practical

0%

###### Stage/Level

2

###### Credits

20

###### Module Code

MTH2012

###### Teaching Period

Autumn

###### Duration

12 weeks

#### Optional Modules

Statistical Inference (20 credits)## Statistical Inference

### Overview

Statistical investigations. Initial data analysis. Sample diagnostics. Point estimation: maximum likelihood, least squares. Multiple linear regression. Significance tests: Neyman-Pearson approach, likelihood ratio tests. Confidence intervals. Introduction to Experimental design. Bayesian methods. Oral presentation of aspects of statistics.

### Learning Outcomes

On completion of the module, it is intended that students will be able to: understand how to build statistical models, know the issues involved with using real data and use sample diagnostic methods to test data for independence, normality and goodness of fit; understand the difference between estimates and estimators in terms of finding an unknown parameter and understand the assessment of an estimator's unbiasedness, relative efficiency, mean square error, sufficiency and whether a distribution belongs to the regular exponential class; understand and use the method of moments, of maximum likelihood and of least squares to provide unbiased estimates of distribution parameters; understand and use experimental design, the different forms of analysis of variance techniques and use and interpret methods of association; rates, relative risk and the odds ratio; through expanding their knowledge of significance and hypothesis tests, formulate confidence intervals and regions and use pivotal quantities; understand Bayesian inference; the prior and posterior distributions, and degree of belief; be able to carry out a statistical investigation of real data in group work using SAS and present the results in a written and oral presentation.

### Skills

Statistical modelling and problem solving. Application of statistical methods in data analysis. Presentation skills.

Coursework

20%

Examination

70%

Practical

10%

###### Stage/Level

2

###### Credits

20

###### Module Code

SOR2002

###### Teaching Period

Spring

###### Duration

12 weeks

Group Theory (20 credits)## Group Theory

### Overview

- definition and examples of groups and their properties

- countability of a group and index

- Lagrange’s theorem

- normal subgroups and quotient groups

- group homomorphisms and isomorphism theorems

- structure of finite abelian groups

- Cayley’s theorem

- Sylow’s theorem

- composition series and solvable groups### Learning Outcomes

It is intended that students shall, on successful completion of the module, be able to: understand the ideas of binary operation, associativity, commutativity, identity and inverse; reproduce the axioms for a group and basic results derived from these; understand the groups arising from various operations including modular addition or multiplication of integers, matrix multiplication, function composition and symmetries of geometric objects; understand the concept of isomorphic groups and establish isomorphism, or otherwise, of specific groups; understand the concepts of conjugacy and commutators; understand the subgroup criteria and determine whether they are satisfied in specific cases; understand the concepts of cosets and index; prove Lagrange's theorem and related results; understand the concepts and basic properties of normal subgroups, internal products, direct and semi-direct products, and factor groups; establish and apply the fundamental results about homomorphisms - including the first, second and third isomorphism theorems - and test specific functions for the homomorphism property; perform various computations on permutations, including decomposition into disjoint cycles and evaluation of order; apply Sylow's theorem.

### Skills

Numeracy and analytic argument skills, problem solving, analysis and construction of proofs.

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

2

###### Credits

20

###### Module Code

MTH2014

###### Teaching Period

Spring

###### Duration

12 weeks

Employability for Mathematics (0 credits)## Employability for Mathematics

### Overview

Introduction to placement for mathematics and physics students, CV building, international options, interview skills, assessment centres, placement approval, health and safety and wellbeing. Workshops on CV building and interview skills. The module is delivered in-house with the support of the QUB Careers Service and external experts.

### Learning Outcomes

Identify gaps in personal employability skills. Plan a programme of work to result in a successful work placement application.

### Skills

Plan self-learning and improve performance, as the foundation for lifelong learning/CPD. Decide on action plans and implement them effectively. Clearly identify criteria for success and evaluate their own performance against them .

Coursework

100%

Examination

0%

Practical

0%

###### Stage/Level

2

###### Credits

0

###### Module Code

MTH2010

###### Teaching Period

Autumn

###### Duration

12 weeks

Classical Mechanics (20 credits)## Classical Mechanics

### Overview

Introduction to calculus of variations.

Recap of Newtonian mechanics.

Generalised coordinates. Lagrangian. Least action principle. Conservation laws (energy, momentum, angular momentum), symmetries and Noether’s theorem. Examples of integrable systems. D’Alembert’s principle. Motion in a central field. Scattering. Small oscillations and normal modes. Rigid body motion.

Legendre transformation. Canonical momentum. Hamiltonian. Hamilton’s equations. Liouville’s theorem. Canonical transformations. Poisson brackets.### Learning Outcomes

On completion of the module, the students are expected to be able to:

• Derive the Lagrangian and Hamiltonian formalisms;

• Demonstrate the link between symmetries of space and time and conservation laws;

• Construct Lagrangians and Hamiltonians for specific systems, and derive and solve the corresponding equations of motion;

• Analyse the motion of specific systems;

• Identify symmetries in a given system and find the corresponding constants of the motion;

• Apply canonical transformations and manipulate Poisson brackets.### Skills

• Proficiency in classical mechanics, including its modelling and problem-solving aspects.

• Assimilating abstract ideas.

• Using abstract ideas to formulate and solve specific problems.Coursework

25%

Examination

75%

Practical

0%

###### Stage/Level

2

###### Credits

20

###### Module Code

MTH2031

###### Teaching Period

Autumn

###### Duration

12 weeks

Methods of Operational Research (20 credits)## Methods of Operational Research

### Overview

Deterministic and stochastic inventory models; simple and adaptive forecasting; theory of replacement of equipment; quality control, acceptance sampling by attribute and variable; network planning including the use of PERT, LP, Gantt charts and resource smoothing; decision theory, including utility curves, decision trees and Bayesian statistics; simple heuristics.

### Learning Outcomes

On completion of the module, it is intended that students will be able to: demonstrate understanding of the Economic Order Quantity model and its use in determining minimum inventory costs; use Lagrange multipliers to obtain optimal batch sizes; use dynamic programming techniques to determine optimal replacement policies; determine both single and double sampling plans and understand how to decide which is the more appropriate in different circumstances; determine critical activities of a project and apply linear programming methods to determine the optimal duration; use decision tress to determine an optimal course of action; use a range of techniques based on past experience to forecast future sales.

### Skills

Formulating problems in accordance with simple mathematical models. Use of spreadsheets.

Coursework

20%

Examination

70%

Practical

10%

###### Stage/Level

2

###### Credits

20

###### Module Code

SOR2003

###### Teaching Period

Autumn

###### Duration

12 weeks

- Year 3
#### Core Modules

Investigations (20 credits)## Investigations

### Overview

Students conduct a short practice investigation, followed by two short investigations (in small groups and solo) in a range of problems in Applied Mathematics and Theoretical Physics. This is followed by a long investigation, which is a literature study of a Mathematical or Theoretical Physics topic not covered in the offered (or chosen) modules. The two short and the long investigation are typed up in reports and submitted for assessment.

### Learning Outcomes

On completion of the module, it is intended that students will be able to:

consider a problem or phenomenon and develop a mathematical model that describes it, stating any assumptions made;

solve the model or its simplified version and analyse the results;

suggest generalisations or extensions of the model to related problems or phenomena, and indicate possible ways of solving them;

communicate the results of an investigation in a written (typed) report, with mathematical equations, tables, etc. as required, and illustrated by diagrams;

investigate an unfamiliar topic using one or a number of literature sources, and write (type) a report that explains the topic in a logical manner, puts the topic in a wider context, uses equations, mathematical derivations, graphs and tables as necessary, and contains a bibliography list.### Skills

Research skills, presentational skills. Use of many sources of information.

Coursework

80%

Examination

0%

Practical

20%

###### Stage/Level

3

###### Credits

20

###### Module Code

AMA3020

###### Teaching Period

Spring

###### Duration

12 weeks

Measure and Integration (20 credits)## Measure and Integration

### Overview

- sigma-algebras, measure spaces, measurable functions

- Lebesgue integral, Fatou's lemma, monotone and dominated convergence theorems

- Fubini’s Theorem, change of variables theorem

- Integral inequalities and Lp spaces### Learning Outcomes

It is intended that students shall, on successful completion of the module, be able to: understand the concepts of an algebra and a sigma-algebra of sets, additive and sigma-additive functions on algebras of sets, measurability of a function with respect to a sigma-algebra of subsets of the domain, integrability, measure and Lp-convergence of sequences of measurable functions; demonstrate knowledge and confidence in applying the Caratheodory extension theorem, Fatou's lemma and the monotone convergence theorem, the Lebesgue dominated convergence theorem, the Riesz theorem, Fubini’s theorem, change of variable’s theorem and integral inequalities; proofs excepting those of the Caratheodory and Riesz theorems; understand similarities and differences between Riemann and Lebesgue integration of functions on an interval of the real line.

### Skills

Analytic argument skills, problem solving, analysis and construction of proofs.

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

MTH3011

###### Teaching Period

Autumn

###### Duration

12 weeks

Mathematical Investigations (20 credits)## Mathematical Investigations

### Overview

This module is concerned with the investigation processes of mathematics, including the construction of conjectures based on simple examples and the testing of these with further examples, aided by computers where appropriate. A variety of case studies will be used to illustrate these processes. A series of group and individual investigations will be made by students under supervision, an oral presentation will be made on one of these investigations. While some of the investigations require little more than GCSE as a background, students will be required to undertake at least one investigation which needs knowledge of Mathematics at Level 2 or Level 3 standard and/or some background reading.

### Learning Outcomes

It is intended that the students shall, on successful completion of the module, be able: to reformulate a complex problem in abstract language, and thereby to analyse and understand the given problem in mathematical terms; to find solutions of a given problem by mathematical analysis; to communicate the results in precise language, in particular specifying the assumptions made while solving the problem, and verifying the correctness of the solution; to work individually and as members of a team on a complex problem, combining information from various sources and validating their correctness.

### Skills

Research skills, presentational skills. Use of many sources of information.

Coursework

90%

Examination

0%

Practical

10%

###### Stage/Level

3

###### Credits

20

###### Module Code

PMA3013

###### Teaching Period

Spring

###### Duration

12 weeks

#### Optional Modules

Topological Data Analysis (20 credits)## Topological Data Analysis

### Overview

1. Simplicial complexes

2. PL functions

3. Simplicial homology

4. Filtrations and barcodes

5. Matrix reduction

6. The Mapper Algorithm

7. Learning with topological descriptors

8. Statistics with topological descriptors### Learning Outcomes

It is intended that students shall, on successful completion of the module, demonstrate knowledge and confidence in applying key ideas and concepts of topological data analysis, such as simplicial complexes, simplicial homology, barcodes, matrix reduction and the analysis of topological descriptors.

In addition, students should be able to use standard software (e.g. the freely available R package TDA) to analyse simple data sets.### Skills

Knowing and applying basic techniques of topological data analysis. In particular, this includes the analysis and interpretation of topological invariants of data sets; the production of graphical representations of such descriptors; and basic computational aspects of linear algebra.

Coursework

25%

Examination

75%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

MTH4322

###### Teaching Period

Autumn

###### Duration

12 weeks

Financial Mathematics (20 credits)## Financial Mathematics

### Overview

Introduction to financial derivatives: forwards, futures, swaps and options; Future markets and prices; Option markets; Binomial methods and risk-free portfolio; Stochastic calculus and random walks; Ito's lemma; the Black-Scholes equation; Pricing models for European Options; Greeks; Credit Risk.

### Learning Outcomes

On completion of the module, it is intended that students will be able to: explain and use the basic terminology of the financial markets; calculate the time value of portfolios that include assets (bonds, stocks, commodities) and financial derivatives (futures, forwards, options and swaps); apply arbitrage-free arguments to derivative pricing; use the binomial model for option pricing; model the price of an asset as a stochastic process; define a Wiener process and derive its basic properties; obtain the basic properties of differentiation for stochastic calculus; derive and solve the Black-Scholes equation; modify the Black-Scholes equation for various types of underlying assets; price derivatives using risk-neutral expectation arguments; calculate Greeks and explain credit risk.

### Skills

Application of Mathematics to financial modelling. Apply a range of mathematical methods to solve problems in finance. Assimilating abstract ideas.

Coursework

20%

Examination

70%

Practical

10%

###### Stage/Level

3

###### Credits

20

###### Module Code

MTH3025

###### Teaching Period

Spring

###### Duration

12 weeks

Geometry of Optimisation (20 credits)## Geometry of Optimisation

### Overview

• Functionals on R^n, linear equations and inequalities; hyperplanes; half-spaces

• Convex polytopes; faces

• Specific examples: e.g., traveling salesman polytope, matching polytopes

• Linear optimisation problems; geometric interpretation; graphical solutions

• Simplex algorithm

• LP duality

• Further topics in optimisation, e.g., integer programming, ellipsoid method### Learning Outcomes

It is intended that students shall, on successful completion of the module, be able to:

• demonstrate understanding of the foundational geometry of convex polytopes;

• demonstrate understand of the geometric ideas behind linear optimisation;

• solve simple optimisation problems graphically;

• apply the simplex algorithm to concrete optimisation problems.### Skills

Knowing and applying basic techniques of polytope theory and optimisation.

Coursework

25%

Examination

75%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

MTH4323

###### Teaching Period

Autumn

###### Duration

12 weeks

Functional Analysis (20 credits)## Functional Analysis

### Overview

A characterisation of finite-dimensional normed spaces; the Hahn-Banach theorem with consequences; the bidual and reflexive spaces; Baire’s theorem, the open mapping theorem, the closed graph theorem, the uniform boundedness principle and the Banach-Steinhaus theorem; weak topologies and the Banach-Alaoglu theorem; spectral theory for bounded and compact linear operators.

### Learning Outcomes

It is intended that students shall, on successful completion of the module, be able to: recognise when a normed space is finite dimensional; determine when linear functionals on normed spaces are bounded and determine their norms; be familiar with the basic theorems of functional analysis (Hahn-Banach, Baire, open mapping, closed graph and Banach-Steinhaus theorems) and be able to apply them; understand dual spaces, recognise the duals of the standard Banach spaces and recognise which of the standard Banach spaces are reflexive; understand the relations between weak topologies on normed spaces and compactness properties; be familiar with the basic spectral theory of bounded and compact linear operators.

### Skills

Analysis of proof and development of mathematical techniques in linear infinite dimensional problems.

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

MTH4311

###### Teaching Period

Spring

###### Duration

12 weeks

Quantum Fields (20 credits)## Quantum Fields

### Overview

1. Review of fundamental quantum theory.

2. Relativistic Lagrangian Field Theory.

3. Symmetries and Conservation laws; Noether’s theorem

4. The Klein-Gordon field. Second quantisation.

5. The Dirac equation and the Dirac field. Spin-Statistics theorem.

6. Covariant electromagnetic field quantisation.

7. Interacting fields; the S-matrix; Wick’s theorem.

8. Fundamentals of Quantum Electrodynamics; Feynman diagrams; probabilities of basic processes.### Learning Outcomes

On successful completion of the module, it is intended that students will be able to:

1. Write relativistic Lagrangians for scalar, spinor and vector fields.

2. Understand the fundamental role of symmetries and conservation laws for quantum fields.

3. Describe the main features of the Klein-Gordon, Dirac and electromagnetic fields.

4. Develop the S-matrix formalism for interacting fields.

5. Apply the S-matrix to basic processes in quantum electrodynamics.

6. Associate Feynman diagrams with scattering amplitudes and calculate related probabilities.

7. Present complex ideas orally.### Skills

• Mathematical modelling

• Problem solving

• Abstract thinkingCoursework

20%

Examination

60%

Practical

20%

###### Stage/Level

3

###### Credits

20

###### Module Code

MTH4331

###### Teaching Period

Spring

###### Duration

12 weeks

Fourier Analysis and Applications to PDEs (20 credits)## Fourier Analysis and Applications to PDEs

### Overview

Introduction:

- Examples of important classical PDEs (e.g. heat equation, wave equation, Laplace’s equation)

- method of separation of variables

Fourier series:

- pointwise and L^2 convergence

- differentiation and integration of Fourier series; using Fourier series to solve PDEs

Distributions:

- basic concepts and examples (space of test functions and of distributions, distributional derivative, Dirac delta)

- convergence of Fourier series in distributions

- Schwartz space, tempered distributions, convolution

Fourier transform:

- Fourier transform in Schwartz space, L^1, L^2 and tempered distributions

- convolution theorem

- fundamental solutions (Green’s functions) of classical PDEs### Learning Outcomes

On completion of the module it is intended that students will be able to:

- use separation of variables to solve simple PDEs

- understand the concept of Fourier series and be able to justify their convergence in various senses

- find solutions of basic PDEs using Fourier series (including a justification of convergence)

- understand the concept of distributions and tempered distributions

- perform basic operations with distributions

- understand the concept of Fourier transform in various settings

- solve classical PDEs using Fourier transform (finding and using fundamental solutions)### Skills

Analytic argument skills, problem solving, use of generalized methods.

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

MTH4321

###### Teaching Period

Spring

###### Duration

12 weeks

Modelling and Simulation (20 credits)## Modelling and Simulation

### Overview

In this module, students will analyse real-life situations, build a mathematical model, solve it using analytical and/or numerical techniques, and analyse and interpret the results and the validity of the model by comparing to actual data. The emphasis will be on the construction and analysis of the model rather than on solution methods. Two group projects will fix the key ideas and incorporate the methodology. This will take 7-8 weeks of term and will be supported with seminars and workshops on the modelling process. Then students will focus on a solo project (relevant to their pathways) with real-life application and work individually on this for the remaining weeks of term. They will present their results in seminars with open discussion, and on a Webpage.

The starting group project will be focused, and offer a limited number of specific modelling problems. For the other projects, students will build on these initial problems by addressing a wider problem taken from, but not exclusively, the following areas: classical mechanics, biological models, finance, quantum mechanics, traffic flow, fluid dynamics, and agent-based models, including modelling linked to problems of relevance to the UN sustainable development goals. A pool of options will be offered, but students will also have the opportunity to propose a problem of their own choice.### Learning Outcomes

On successful completion of the module, it is intended that students will be able to:

1. Develop mathematical models of different kinds of systems using multiple kinds of appropriate abstractions

2. Explain basic relevant numerical approaches

3. Implement their models in Python and use analytical tools when appropriate

4. Apply their models to make predictions, interpret behaviour, and make decisions

5. Validate the predictions of their models against real data.### Skills

1. Creative mathematical thinking

2. Formulation of models, the modelling process and interpretation of results

3. Teamwork

4. Problem-solving

5. Effective verbal and written communication skillsCoursework

100%

Examination

0%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

MTH3024

###### Teaching Period

Spring

###### Duration

12 weeks

Statistical Mechanics (20 credits)## Statistical Mechanics

### Overview

Fundamentals of classical thermodynamics: systems, phases, thermodynamic variables, equilibrium, equations of state, distinction between intensive and extensive thermodynamic variables, work and heat, Carnot cycle, Gibbs phase rule, first and second laws of thermodynamics, definitions of thermodynamic potentials, derivation of Maxwell relations, response functions, thermodynamic stability and Ehrenfest classification of phase transitions.

Equilibrium statistical mechanics: microstates and phase space, role of information and connection with entropy, method of Lagrange multipliers, generalised partition function, microcanoncial, canonical, isothermal-isobaric and grand-canonical partition functions, ensemble averages and connection between fluctuations and response functions.

Computer simulation: importance sampling, Monte Carlo algorithm, molecular dynamics.### Learning Outcomes

On completion of the module students should be able to:

Explain how the terms system, state, equilibrium and phase are used within thermodynamics and state the 1st and 2nd laws. They should then be able to use these definitions, together with the theory of partial differential equations, to derive Maxwell relations and constraints on the values of the response functions.

Explain the meanings of the terms microstate and phase space and give an expression for the probability of being in a particular microstate for the microcanonical, canonical, isothermal-isobaric and grand-canonical ensembles.

Write a Monte Carlo program in Python that calculates ensemble averages and fluctuations for a simple model Hamiltonian. Students should be able to use this program to calculate response functions such as the heat capacity by finite differences and from the fluctuations.

Evaluate publications on recent developments in atomistic simulation critically by making comments on issues of model accuracy and sampling methodologies.### Skills

By the end of this module students will have developed the following skills:

An ability to write computer programs to calculate integrals by means of Monte Carlo simulation.

Improved presentation skills

Critical reading skillsCoursework

55%

Examination

45%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

MTH4332

###### Teaching Period

Spring

###### Duration

12 weeks

Quantum Theory (20 credits)## Quantum Theory

### Overview

• Overview of classical physics and the need for new theory.

• Basic principles: states and the superposition principle, amplitude and probability, linear operators, observables, commutators, uncertainty principle, time evolution (Schrödinger equation), wavefunctions and coordinate representation.

• Elementary applications: harmonic oscillator, angular momentum, spin.

• Motion in one dimension: free particle, square well, square barrier.

• Approximate methods: semiclassical approximation (Bohr-Sommerfeld quantisation), variational method, time-independent perturbation theory, perturbation theory for degenerate states (example: spin-spin interaction, singlet and triplet states).

• Motion in three dimensions: Schrödinger equation, orbital angular momentum, spherical harmonics, motion in a central field, hydrogen atom.

• Atoms: hydrogen-like systems, Pauli principle, structure of many-electron atoms and the Periodic Table.### Learning Outcomes

On the completion of this module, successful students will be able to

• Understand, manipulate and apply the basic principles of Quantum Theory involving states, superpositions, operators and commutators;

• Apply a variety of mathematical methods to solve a range of basic problems in Quantum Theory, including the finding of eigenstates, eigenvalues and wavefunctions;

• Use approximate methods to solve problems in Quantum Theory and identify the range of applicability of these methods;

• Understand the structure and classification of states of the hydrogen atom and explain the basic principles behind the structure of atoms and Periodic Table.### Skills

• Proficiency in quantum mechanics, including its modelling and problem-solving aspects.

• Assimilating abstract ideas.

• Using abstract ideas to formulate specific problems.

• Applying a range of mathematical methods to solving specific problems.Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

MTH3032

###### Teaching Period

Autumn

###### Duration

12 weeks

Statistical Data Mining with Machine Learning (20 credits)## Statistical Data Mining with Machine Learning

### Overview

Introduction to Data Mining; Exploratory Data Analysis; Cluster analysis; Classification including Probabilistic Modelling, Bayesian Networks, Decision tree analysis; Prediction including Regression trees, Random Forests, Neural nets.

### Learning Outcomes

On completion of the module, it is intended that students will be able to: demonstrate understanding of the field of data mining, how it has developed and the need for data mining techniques in today’s society; demonstrate knowledge familiarity with data warehouses, webhouses and data marts, the various forms of storing, managing and maintaining large amounts of data; employ exploratory data analysis techniques for univariate analyses, when one outcome variable is considered compared to bivariate, or multivariate analyses for more than one variable in terms of multivariate exploratory analysis of both quantitative and qualitative data and to apply and interpret the results of principal component analysis for multiple variables; demonstrate knowledge of classification and of classification methods including simple linear, nearest neighbour, decision tree models, Bayes classifying, neural networks and random forests; demonstrate knowledge of the purpose of clustering and to use hierarchical clustering and the non-hierarchical clustering methods of k means and nearest neighbour when applied to real data sets; understand and use association rules and their application on real data sets.

### Skills

Problem solving and computational skills.

Coursework

40%

Examination

60%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

SOR3008

###### Teaching Period

Spring

###### Duration

12 weeks

Stochastic Processes and Risk (20 credits)## Stochastic Processes and Risk

### Overview

Logic and Boolean algebra, counting and combinatorics, set algebra, inclustion-exclusion theorem, mutually exclusive events, De Morgan Laws.

Axioms of probability, events and probability spaces, sigma-field, random variables, conditional probability, and expectation, Bayes’ theorem, discrete and continuous random variables, moments and moment generating function. Laws of large numbers and central limit theorem.

Pairs of random variables, marginal probabilities, Cauchy Schwartz Inequality in statistics, correlation and covariance.

Discrete time Markov chains, Chapman Kolmogorov relation, limiting behaviour, transient, recurrent states and periodic states, limiting stationary distribution, hitting times and hitting probabilities.

Continuous time Markov chains, Kolmogorov forward equations, stationary distribution for continuous time Markov chains, Poisson process, MM1 Queue, inhomogeneous Poisson process and compound Poisson process.### Learning Outcomes

By the end of this module students will be able to:

- Calculate expectations and variances directly, using the moment generating function and by using the conditional expectation theorem. Students should also be able to explain what predictions can be made given the expectation and/or the variance.

- Recognize which type of random variable is appropriate for modeling a given phenomenon, to identify the assumptions that they have made in constructing this model and to critically assess its validity.

- Explain what it means when we state that a time dependent process has independent and stationary increments and how this differs from a Markov process. By using their understanding of this distinction students should be able to construct probabilistic models for time dependent phenomena, explain the assumptions that they have made in constructing these models and critically assess their validity.### Skills

Write computer programs that generate random variables as well as computer programs for evaluating sample means, histograms and confidence limits.

Discuss the results obtained by running the computer programs described in the previous point.Coursework

55%

Examination

45%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

SOR3012

###### Teaching Period

Autumn

###### Duration

12 weeks

Linear Models (20 credits)## Linear Models

### Overview

Linear regression. Non-singular case: analysis of variance, extra sum of squares principle, generalised least squares, residuals. Singular case: generalised inverse solution, estimable functions. Experimental designs: completely randomised, randomised block, factorial; contrasts, analysis of covariance; Generalised linear model (GLM): maximum likelihood and least squares; exponential family; Poisson and logistic models; model selection for GLM.

### Learning Outcomes

On completion of the module, it is intended that students will be able to: understand and use linear models and multiple linear regression for modelling a measured response as a function of explanatory variables using the least squares approach, and so perform model selection and diagnostics expanding their knowledge to the weighted least squares model; understand ANOVA as a method of analysis for experimentally designed data using non-singular and singular cases; apply the extra sum of squares principle to analyse and interpret residuals, the generalized inverse solution, and assess whether a function is estimable and the hypotheses testable; recognise, apply and interpret the results of analysis of variance for the completely randomized, randomized block, and factorial designs; demonstrate familiarity with using contrasts and apply analysis of covariance; upon developing a full understanding of linear models, extend this to Generalized Linear Models and apply them and model selection to discrete recorded responses using maximum likelihood and least square estimation for distributions from the exponential family, Poisson and logistic; build on their ability to use SAS for the development and selection of linear models.

### Skills

Use of appropriate statistical software in applying linear and generalised linear models.

Coursework

20%

Examination

70%

Practical

10%

###### Stage/Level

3

###### Credits

20

###### Module Code

SOR3004

###### Teaching Period

Autumn

###### Duration

12 weeks

Rings and Modules (20 credits)## Rings and Modules

### Overview

Rings, subrings, prime and maximal ideals, quotient rings, homomorphisms, isomorphism theorems, integral domains, principal ideal domains, modules, submodules and quotient modules, module maps, isomorphism theorems, chain conditions (Noetherian and Artinian), direct sums and products of modules, simple and semisimple modules.

### Learning Outcomes

It is intended that students shall, on successful completion of the module, be able to: understand, apply and check the definitions of ring and module; subring/submodule and ideal against concrete examples; understand and apply the isomorphism theorems; understand and check the concepts of integral domain, principal ideal domain and simple ring; understand and be able to produce the proof of several statements regarding the structure of rings and modules; master the concept of Noetherian and Artinian Modules and rings.

### Skills

Numeracy and analytic argument skills, problem solving, analysis and construction of proofs.

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

MTH3012

###### Teaching Period

Autumn

###### Duration

12 weeks

Classical Fields (20 credits)## Classical Fields

### Overview

• Recapping of the least action principle in Classical Mechanics. Lagrangians for continuous systems (e.g., a string), and derivation of the wave equation from Lagrange’s equation.

• 4-dimensional space time, interval, 4-vectors, tensors, Lorentz covariance.

• Action and Lagrangian for a particle, energy and momentum.

• 4-potential and the Lagrangian for a charged particle in an electromagnetic field, relativistic equation of motion and Lorentz’s force, electric and magnetic fields.

• Lagrangian of the electromagnetic field, Maxwell’s equations in covariant form, charge density and current density, continuity equation, Maxwell’s equations in conventional (3+1) form, and in integral form.

• Electrostatics (general ideas, Coulomb’s law, fields of various charge distributions, electric dipole moment).

• Magnetostatics (general ideas, Biot-Savart-Laplace law, fields of systems of currents, magnetic dipole moment).

• Electromagnetic waves, plane wave, polarisation, monochromatic wave.

• Electromagnetic radiation: retarded potentials, dipole radiation (electric, magnetic), Larmor formula.### Learning Outcomes

On the completion of this module, successful students will be able to

• Understand the basic notions of 4-space, 4-vectors and covariance;

• Derive the equations of the motions for the particle and for the fields (Maxwell equations) and solve them in a number of simple settings;

• Derive the equations of electrostatics and magnetostatics from Maxwell’s equations and find their solutions for basic systems;

•Understand the origins and nature of electromagnetic waves and determine the radiation by charges.### Skills

• Proficiency in classical fields, including its modelling and problem-solving aspects.

• Assimilating abstract ideas.

• Using abstract ideas and mathematical methods to formulate and solve specific problems.Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

MTH3031

###### Teaching Period

Autumn

###### Duration

12 weeks

Numerical Analysis (20 credits)## Numerical Analysis

### Overview

• Introduction and basic properties of errors: Introduction; Review of basic calculus; Taylor's theorem and truncation error; Storage of non-integers; Round-off error; Machine accuracy; Absolute and relative errors; Richardson's extrapolation.

• Solution of equations in one variable: Bisection method; False-position method; Secant method; Newton-Raphson method; Fixed point and one-point iteration; Aitken's "delta-squared" process; Roots of polynomials.

• Solution of linear equations: LU decomposition; Pivoting strategies; Calculating the inverse; Norms; Condition number; Ill-conditioned linear equations; Iterative refinement; Iterative methods.

• Interpolation and polynomial approximation: Why use polynomials? Lagrangian interpolation; Neville's algorithm; Other methods.

• Approximation theory: Norms; Least-squares approximation; Linear least-squares; Orthogonal polynomials; Error term; Discrete least-squares; Generating orthogonal polynomials.

• Numerical quadrature: Newton-Cotes formulae; Composite quadrature; Romberg integration; Adaptive quadrature; Gaussian quadrature (Gauss-Legendre, Gauss-Laguerre, Gauss-Hermite, Gauss-Chebyshev).

• Numerical solution of ordinary differential equations: Boundary-value problems; Finite-difference formulae for first and second derivatives; Initial-value problems; Errors; Taylor-series methods; Runge-Kutta methods.### Learning Outcomes

On completion of the module, it is intended that students should: appreciate the importance of numerical methods in mathematical modelling; be familiar with, and understand the mathematical basis of, the numerical methods employed in the solution of a wide variety of problems;

through the computing practicals and project, have gained experience of scientific computing and of report-writing using a mathematically-enabled word-processor.### Skills

Problem solving skills; computational skills; presentation skills.

Coursework

50%

Examination

50%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

MTH3023

###### Teaching Period

Autumn

###### Duration

12 weeks

Dynamical Systems (20 credits)## Dynamical Systems

### Overview

Continuous dynamical systems

- Fundamental theory: existence, uniqueness and parameter dependence of solutions;

- Linear systems: constant coefficient systems and the matrix exponential; nonautonomous linear systems; periodic linear systems.

- Topological dynamics: invariant sets; limit sets; Lyapunov stability.

- Grobman-Hartman theorem.

- Stable, unstable and centre manifolds.

- Periodic orbits: Poincare-Bendixson theorem.

- Bifurcations

- Applications: the Van der Pol oscillator; the SIR compartmental model; the Lorenz system.

Discrete dynamical systems

- One-dimensional dynamics: the discrete logistic model; chaos; the Cantor middle-third set.### Learning Outcomes

It is intended that students shall, on successful completion of the module: have a good understanding and ability to use the basics of dynamical systems; be able to perform computations pertaining to problems in these areas; have reached a good level of skill in manipulating basic and complex questions within this framework, and be able to reproduce, evaluate and extend logical arguments; be able to select suitable tools to solve a problem, and to communicate the mathematical reasoning accurately and confidently.

### Skills

Analytic argument skills, computation, manipulation, problem solving, understanding of logical arguments.

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

MTH3021

###### Teaching Period

Spring

###### Duration

12 weeks

- Year 4
#### Core Modules

Project (40 credits)## Project

### Overview

A substantial investigation of a research problem incorporating literature survey, development of appropriate theoretical models and when necessary the construction of computer programs to solve specific stages of the problem, presentation of the work in the form of a technical report, a sequence of oral presentations culminating in a 30-minute presentation which is assessed.

### Learning Outcomes

On completion of this two-semester module, it is intended that students will be able to: undertake a substantial research project in which they increasingly take ownership of the planning and development of the work; work independently, under supervision; survey and use existing literature as a basis for their work; develop mathematical theory of models relevant to the project and where appropriate use or develop computer programs to advance the work and draw conclusions; give a coherent written account of the work undertaken, of its significance and of the outcomes of the research, in a technical report which is accessible to a range of interested readers; make a substantial oral presentation of the work undertaken, the results obtained and the conclusions drawn, to an audience not all of whom will be experts in the field of study.

### Skills

Independent working. Oral and written presentational skills.

Coursework

80%

Examination

0%

Practical

20%

###### Stage/Level

4

###### Credits

40

###### Module Code

AMA4005

###### Teaching Period

Full Year

###### Duration

24 weeks

Project (40 credits)## Project

### Overview

An extended project based on the research interests of members of staff. Attendance at a sequence of presentations on projects offered in Level 4.

### Learning Outcomes

The intention of this module is to give students experience of the more sustained type of mathematical work such as is generally undertaken by people working in industry, commerce or academic research, rather than the relatively short tasks that they are required to undertake in most lecture-based modules. The project is meant to be one third of the Level 4 work, and so should occupy on average about 13 hours per week. Students know how to use LaTeX (a mathematical typesetting language) and how to use mathematical databases in their work. It is intended that students shall, on successful completion of the module, have gained experience and confidence in transferable skills: among them, the ability to work in groups, enhanced ability to manage time, written presentation skills, oral presentation skills and leadership skills. This is achieved by organising three presentations during the year and a thesis that students write on the work they have carried out during the year.

### Skills

Independent work; presentational (oral and written) skills.

Coursework

80%

Examination

0%

Practical

20%

###### Stage/Level

4

###### Credits

40

###### Module Code

PMA4001

###### Teaching Period

Full Year

###### Duration

24 weeks

#### Optional Modules

Applied Algebra and Cryptography (20 credits)## Applied Algebra and Cryptography

### Overview

- (finite) fields and rings of polynomials over them.

- the division algorithm and splitting of polynomials.

- ideals and quotient rings, (principal) ideal domains, with examples from rings of polynomials.

- polynomials in several indeterminates, Hilbert’s basis theorem.

- applications of algebra to cryptography (such as affine Hill ciphers, RSA, lattice cryptography, Diophantine equations).

- optional topics may include Euclidean rings, unique factorisation domains, greatest common divisor domains.### Learning Outcomes

It is intended that students shall, on successful completion of the module, be able to:

understand the concept of a ring of polynomials over a (finite field);

apply the factorisation algorithm;

understand ideals, quotient rings and the properties of quotient rings;

understand how algebra can be applied to cryptography and be able to encrypt messages using methods from the module.### Skills

Analytic argument skills, problem solving, analysis and construction of proofs.

Coursework

20%

Examination

80%

Practical

0%

###### Stage/Level

4

###### Credits

20

###### Module Code

MTH4021

###### Teaching Period

Spring

###### Duration

12 weeks

## Mathematical Methods for Quantum Information Processing

### Overview

1. Operatorial quantum mechanics: review of linear algebra in Dirac notation; basics of quantum mechanics for pure states.

2. Density matrix and mixed states; Bloch sphere; generalised measurements.

3. Maps and operations: complete positive maps; Kraus operators.

4. Quantum Communication protocols: quantum cryptography; cloning; teleportation; dense coding.

5. Quantum computing: review of classical circuits and logic gates; quantum circuits and algorithms; implementation of quantum circuits on small prototypes of quantum computers (IBM Quantum Experience); examples of physical Hamiltonians implementing quantum gates.

6. Theory of entanglement: basic notions and pure-state entanglement manipulation; detection of entanglement; measures of entanglement; entanglement and non-locality, Bell's inequality; multipartite entanglement.### Learning Outcomes

On completion of the module, it is intended that students will be able to:

1. Express linear operators in terms of the Dirac notation; derive both the matrix and outer-product representation of linear operators in Dirac notation; recognise Hermitian, normal, positive and unitary operators, and put in use their respective basic properties; construct Kronecker products and functions of operators.

2. Comprehend and express the postulates of quantum mechanics in Dirac notation; define projective measurements and calculate their outcome probabilities and output states; give examples of destructive and non-destructive projective measurements; prove the uncertainty principle for arbitrary linear operators; define positive-operator-valued measurement and use their properties to discriminate between non-orthogonal states; prove the no-cloning theorem for generic pure states.

3. Explain the necessity of using the mixed-state description of quantum systems; define the density operator associated with an ensemble of pure states; express the postulate of quantum mechanics with the density operator formalism; distinguish pure and mixed states; describe two-level system in the Bloch sphere; geometrically describe generic n-level systems; calculate the partial trace and the reduced density operator of a tensor-product system.

4. Describe the dynamics of a non-isolated quantum system with the formalism of completely-positive and trace preserving (CPTP) dynamical maps; derive the operator-sum representation of a CPTP map; give examples of CPTP maps.

5. Demonstrate the most relevant communication protocols for pure states using the Dirac notation for states and linear operators: super-dense coding, quantum teleportation and quantum key distribution.

6. Describe the basic properties of classical circuit for classical computing in terms of elementary logic gates; define the main model of quantum computation in terms of quantum circuits and gates; comprehend and construct basic quantum algorithms: Grover, Deutsch-Josza and Shor algorithms; construct, implement, and test small quantum circuits on prototypes of quantum computers (IBM Quantum Experience).

7. Define quantum entanglement for pure and mixed states; identify entangled states; manipulate pure entangled state via LOCC operations; calculate the amount of entanglement in simple quantum systems; define Bell inequalities and calculate their violation; define the entanglement in multiple composite systems.### Skills

• Mathematical modelling of quantum systems, including problem solving aspects in the context of quantum technologies.

• Assimilating abstract ideas.

• Using abstract ideas to formulate specific problems.

• Applying a range of mathematical methods to solving specific problems.Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

4

###### Credits

20

###### Module Code

MTH4023

###### Teaching Period

Spring

###### Duration

12 weeks

Quantum Fields (20 credits)## Quantum Fields

### Overview

1. Review of fundamental quantum theory.

2. Relativistic Lagrangian Field Theory.

3. Symmetries and Conservation laws; Noether’s theorem

4. The Klein-Gordon field. Second quantisation.

5. The Dirac equation and the Dirac field. Spin-Statistics theorem.

6. Covariant electromagnetic field quantisation.

7. Interacting fields; the S-matrix; Wick’s theorem.

8. Fundamentals of Quantum Electrodynamics; Feynman diagrams; probabilities of basic processes.### Learning Outcomes

On successful completion of the module, it is intended that students will be able to:

1. Write relativistic Lagrangians for scalar, spinor and vector fields.

2. Understand the fundamental role of symmetries and conservation laws for quantum fields.

3. Describe the main features of the Klein-Gordon, Dirac and electromagnetic fields.

4. Develop the S-matrix formalism for interacting fields.

5. Apply the S-matrix to basic processes in quantum electrodynamics.

6. Associate Feynman diagrams with scattering amplitudes and calculate related probabilities.

7. Present complex ideas orally.### Skills

• Mathematical modelling

• Problem solving

• Abstract thinkingCoursework

20%

Examination

60%

Practical

20%

###### Stage/Level

4

###### Credits

20

###### Module Code

MTH4331

###### Teaching Period

Spring

###### Duration

12 weeks

Bayesian Statistics (20 credits)## Bayesian Statistics

### Overview

Within the last decade, the demand for Bayesian statistics has grown dramatically in both industry and research. Bayesian estimation is a collection of inferential methods based on the use of Bayes’ Theorem, providing the means of incorporating prior beliefs when estimating unknown parameters. This module will cover an introduction to Bayesian analysis, models for both discrete and continuous data, the concept of alternative priors and non-conjugate models and estimation utilising Markov Chain Monte Carlo. The theory will be applied to common statistical models and include model checking procedures such as the use of trace plots.

### Learning Outcomes

On successful completion of the module, it is intended that students will be able to:

1. Grasp the fundamental concepts within Bayesian analysis.

2. Apply the theory to distributional examples for both discrete and continuous data.

3. Understand the assumptions behind the choice of prior distributions.

4. Demonstrate knowledge and utilisation of key estimation techniques for Bayesian analysis such as Markov chain Monte Carlo estimation.

5. Utilise statistical software to implement the theory and build common statistical models using Bayesian estimation.

6. Understand the importance of and how to implement model checking procedures.### Skills

Students will develop problem solving skills, the ability to construct common statistical models and the effective use of appropriate statistical software to analyse data using Bayesian statistics.

Coursework

20%

Examination

70%

Practical

10%

###### Stage/Level

4

###### Credits

20

###### Module Code

SOR4008

###### Teaching Period

Spring

###### Duration

12 weeks

Geometry of Optimisation (20 credits)## Geometry of Optimisation

### Overview

• Functionals on R^n, linear equations and inequalities; hyperplanes; half-spaces

• Convex polytopes; faces

• Specific examples: e.g., traveling salesman polytope, matching polytopes

• Linear optimisation problems; geometric interpretation; graphical solutions

• Simplex algorithm

• LP duality

• Further topics in optimisation, e.g., integer programming, ellipsoid method### Learning Outcomes

It is intended that students shall, on successful completion of the module, be able to:

• demonstrate understanding of the foundational geometry of convex polytopes;

• demonstrate understand of the geometric ideas behind linear optimisation;

• solve simple optimisation problems graphically;

• apply the simplex algorithm to concrete optimisation problems.### Skills

Knowing and applying basic techniques of polytope theory and optimisation.

Coursework

20%

Examination

80%

Practical

0%

###### Stage/Level

4

###### Credits

20

###### Module Code

MTH4323

###### Teaching Period

Autumn

###### Duration

12 weeks

Fourier Analysis and Applications to PDEs (20 credits)## Fourier Analysis and Applications to PDEs

### Overview

Introduction:

- Examples of important classical PDEs (e.g. heat equation, wave equation, Laplace’s equation)

- method of separation of variables

Fourier series:

- pointwise and L^2 convergence

- differentiation and integration of Fourier series; using Fourier series to solve PDEs

Distributions:

- basic concepts and examples (space of test functions and of distributions, distributional derivative, Dirac delta)

- convergence of Fourier series in distributions

- Schwartz space, tempered distributions, convolution

Fourier transform:

- Fourier transform in Schwartz space, L^1, L^2 and tempered distributions

- convolution theorem

- fundamental solutions (Green’s functions) of classical PDEs### Learning Outcomes

On completion of the module it is intended that students will be able to:

- use separation of variables to solve simple PDEs

- understand the concept of Fourier series and be able to justify their convergence in various senses

- find solutions of basic PDEs using Fourier series (including a justification of convergence)

- understand the concept of distributions and tempered distributions

- perform basic operations with distributions

- understand the concept of Fourier transform in various settings

- solve classical PDEs using Fourier transform (finding and using fundamental solutions)### Skills

Analytic argument skills, problem solving, use of generalized methods.

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

4

###### Credits

20

###### Module Code

MTH4321

###### Teaching Period

Spring

###### Duration

12 weeks

Practical Methods for Partial Differential Equations (20 credits)## Practical Methods for Partial Differential Equations

### Overview

Basics: solving first order ordinary differential equations, partial derivatives, surface, volume and line integrals, the Gauss theorem, Stokes' Theorem.

Partial differential equations (PDE) and their relation to physical problems: heat conduction, flow of a liquid, wave propagation, Brownian motion.

First order PDE in two variables: the method of characteristics, the transversality condition, quasilinear equations and shock waves, conservation laws, the entropy condition, applications to traffic flows.

Second order linear PDEs: classification and canonical forms.

The wave equation: d`Alembert’s solution, the Cauchy problem, graphical methods.

The method of separation of variables: the wave and the heat equations.

Numerical methods: finite differences, stability, explicit and implicit schemes, the Crank-Nicolson scheme, a stable explicit scheme for the wave equation.

Practical: the students are offered to solve a heat and a wave equation using the method of separation of variables and a finite difference scheme.

The Sturm-Liouville problem: a theoretical justification for the method of separation of variables. Simple properties of the Sturmian eigenvalues and eigenfunctions.

Elliptic equations: the Laplace and Poisson equations, maximum principles for harmonic functions, separation of variables for Laplace equation on a rectangle.

Green's functions: their definition and possible applications, Green’s functions for the Poisson equation, the heat kernel.### Learning Outcomes

On completion of the module, it is intended that students will be able to:

understand the origin of PDEs which occur in mathematical physics, solve linear and quasilinear first order PDEs using the method of characteristics, classify and convert to a canonical form second order linear PDEs, solve numerically and using different methods the wave and the heat equations, as well as second order linear PDEs of a more general type, solve a Sturm-Liouville problem associated with a linear PDE and use the eigenfunctions to expand and evaluate its solution, understand the type of boundary conditions required by an elliptic PDE and solve it using the method of separation of variables, construct the Green's function for simple PDEs and use them to evaluate the solution.### Skills

Upon completion the student will have theoretical and practical skills for solving problems described by partial differential equations

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

4

###### Credits

20

###### Module Code

MTH4024

###### Teaching Period

Autumn

###### Duration

12 weeks

Integration Theory (20 credits)## Integration Theory

### Overview

Sigma-algebras, measure spaces, measurable functions. Lebesgue integral, Fatou's lemma, monotone and dominated convergence theorems. Integral inequalities. Lp-spaces. Orthogonal sequences, Fourier series.

### Learning Outcomes

It is intended that students shall, on successful completion of the module, be able to: understand the concepts of an algebra and a sigma-algebra of sets, additive and sigma-additive functions on algebras of sets, measurability of a function with respect to a sigma-algebra of subsets of the domain, integrability, measure and Lp-convergence of sequences of measurable functions; demonstrate knowledge and confidence in applying the Caratheodory extension theorem, Fatou's lemma and the monotone convergence theorem, the Lebesgue dominated convergence theorem, the Riesz theorem, integrability criteria and basic properties of Fourier series in L2 (including the proofs excepting those of the Caratheodory and Riesz theorems); understand similarities and differences between Riemann and Lebesgue integration of functions on an interval of the real line.

### Skills

Critical analysis of proof.

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

4

###### Credits

20

###### Module Code

PMA4004

###### Teaching Period

Autumn

###### Duration

12 weeks

Survival Analysis (20 credits)## Survival Analysis

### Overview

Survival data, survivor and hazard functions. Nonparametric method: estimating median and percentile survival and confidence intervals. Comparing two groups of survival data, the log-rank and Wilcoxon tests. Comparison of k-groups. The Cox proportional hazard model, baseline hazard, hazard ratio, including variates and factors, maximum likelihood, treatment of ties. Confidence intervals for the Cox model regression parameters and hypothesis testing. Estimating the baseline hazard. Model building, Wald tests, likelihood ratio tests and nested models.

### Learning Outcomes

On completion of the module, it is intended that students will be able to: demonstrate an understanding of survival analysis, the special features of survival data, skewed survival distribution and censoring and how to handle censored real data; demonstrate an understanding for the survival, hazard and cumulative hazard functions and how they relate and use the nonparametric procedures of life-tables and Kaplan-Meier to estimate the survival curve, hazard and survival percentiles, the survival median and confidence intervals; demonstrate how to treat more than one group of survival data and use log-rank and Wilcoxon tests for comparing up to k groups of survival data; demonstrate an understanding of the Cox proportional hazards model, using the baseline hazard function and hazard ratio, along with considering variates and factors and using maximum likelihood for the Cox model; calculate confidence intervals for the Cox model regression parameters, to implement hypothesis testing, to deal with ties in the data, to estimate the baseline hazard, use Wald and likelihood ratio tests to build models and formulate nested models; demonstrate an awareness of parametric models, time-dependent variables, non-proportional hazards, and accelerated-failure-time models.

### Skills

The effective use of appropriate statistical software to analyse survival data.

Coursework

20%

Examination

70%

Practical

10%

###### Stage/Level

4

###### Credits

20

###### Module Code

SOR4007

###### Teaching Period

Autumn

###### Duration

12 weeks

Topology (20 credits)## Topology

### Overview

- Definition and examples (natural, geometric and pathological)

- Continuity and homeomorphisms

- Compact, Connected, Hausdorff

- Subspaces and product spaces

- Introduction to homotopy, calculations and applications### Learning Outcomes

It is intended that students shall, on successful completion of this module, be able to: use effectively the notions of topological space, continuous function and homeomorphism and give examples thereof; state and use the basic properties of the product and subspace topologies; apply effectively the properties of connectedness, compactness, and Hausdorffness; understand the relation between metric and topological spaces; understand how topological maps are related via homotopy and apply homotopical calculations to examples.

### Skills

Analytic argument skills, problem solving, analysis and construction of proofs.

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

4

###### Credits

20

###### Module Code

MTH4011

###### Teaching Period

Autumn

###### Duration

12 weeks

Information Theory (20 credits)## Information Theory

### Overview

Introduction to information theory. Basic modular arithmetic and factoring. Finite-field arithmetic. Random variables and some concepts of probabilities. RSA cryptography and factorisation. Uniquely decipherable and instantaneous codes. Optimal codes and Huffman coding. Code extensions. Entropy, conditional entropy, joint entropy and mutual information. Shannon noiseless coding theorem. Noisy information channels. Binary symmetric channel. Decision rules. The fundamental theorem of information theory. Basic coding theory. Linear codes. A brief introduction to low-density parity-check codes.

### Learning Outcomes

On completion of the module, it is intended that students will be able to: explain the security of and put in use the RSA protocol; understand how to quantify information and mutual information; motivate the use of uniquely decipherable and instantaneous codes; use Huffman encoding scheme for optical coding; use source extension to improve coding efficiency; prove Shannon noiseless coding theorem; understand the relation between mutual information and channel capacity; calculate the capacity of some basic channels; use basic error correction techniques for reliable transmission over noisy channels.

### Skills

Problem solving skills; report writing skills; computing skills

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

4

###### Credits

20

###### Module Code

MTH4022

###### Teaching Period

Spring

###### Duration

12 weeks

Advanced Quantum Theory (20 credits)## Advanced Quantum Theory

### Overview

1. Review of fundamental quantum theory (Postulates of quantum mechanics; Dirac notation; Schrödinger equation; spin-1/2 systems; stationary perturbation theory).

2. Coupled angular momenta: spin-1/2 coupling; singlet and triplet subspaces for two coupled spin-1/2 particles; Coupling of general angular momenta;

3. Spin-orbit coupling; fine and hyperfine structures of the hydrogen atom.

4. Time-dependent perturbation theory.

5. Elements of collisions and scattering in quantum mechanics.

6. Identical particles and second quantisation; operators representation.

7. Basics of electromagnetic field quantisation.

8. Systems of interacting bosons: Bose-Einstein condensation and superfluidity.### Learning Outcomes

On successful completion of the module, it is intended that students will be able to:

1. Use the rules for the construction of a basis for coupled angular momenta.

2. Grasp the fundamental features of the fine and hyperfine structures of the hydrogen atom.

3. Understand the techniques for dealing with time-dependent perturbation theory.

4. Apply the theory of scattering to simple quantum mechanical problems.

5. Describe systems of identical particles in quantum mechanics and write the second quantisation representation of operators.

6. Apply the formalism of second quantisation to the electromagnetic field and systems of interacting bosons.### Skills

Mathematical modelling. Problem solving. Abstract thinking.

Coursework

0%

Examination

80%

Practical

20%

###### Stage/Level

4

###### Credits

20

###### Module Code

MTH4031

###### Teaching Period

Autumn

###### Duration

12 weeks

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Entry Requirements

#### Entrance requirements

A level requirementsAAA including Mathematics OR A* (Mathematics) AB |

Irish leaving certificate requirementsH2H2H3H3H3H3 including Higher Level grade H2 in Mathematics |

Access CourseNot considered. Applicants should apply for the BSc Mathematics degree. |

International Baccalaureate Diploma36 points overall including 6,6,6 at Higher Level including Mathematics. If not offered at Higher Level/GCSE then Standard Level grade 4 in English would be accepted |

GraduateA minimum of a 2:2 Honours Degree, provided any subject requirement is also met. |

#### Selection Criteria

In addition, to the entrance requirements above, it is essential that you read our guidance below on 'How we choose our students' prior to submitting your UCAS application.

Applications are dealt with centrally by the Admissions and Access Service rather than by the School of Mathematics and Physics. Once your on-line form has been processed by UCAS and forwarded to Queen's, an acknowledgement is normally sent within two weeks of its receipt at the University.

Selection is on the basis of the information provided on your UCAS form. Decisions are made on an ongoing basis and will be notified to you via UCAS.

For entry last year, applicants for programmes in the School of Mathematics and Physics offering A-level/BTEC Level 3 qualifications must have had, or been able to achieve, a minimum of five GCSE passes at grade C/4 or better (to include English Language and Mathematics), though this profile may change from year to year depending on the demand for places. The Selector also checks that any specific entry requirements in terms of GCSE and/or A-level subjects can be fulfilled.

Offers are normally made on the basis of three A-levels. The offer for repeat candidates may be one grade higher than for first time applicants. Grades may be held from the previous year.

Applicants offering two A-levels and one BTEC Subsidiary Diploma/National Extended Certificate (or equivalent qualification), or one A-level and a BTEC Diploma/National Diploma (or equivalent qualification) will also be considered. Offers will be made in terms of the overall BTEC grade(s) awarded. Please note that a maximum of one BTEC Subsidiary Diploma/National Extended Certificate (or equivalent) will be counted as part of an applicant’s portfolio of qualifications. The normal GCSE profile will be expected.

Applicants offering other qualifications will also be considered. The same GCSE (or equivalent) profile is usually expected of those candidates offering other qualifications.

The information provided in the personal statement section and the academic reference together with predicted grades are noted but, in the case of degree courses in the School of Mathematics and Physics, these are not the final deciding factors in whether or not a conditional offer can be made. However, they may be reconsidered in a tie break situation in August.

A-level General Studies and A-level Critical Thinking would not normally be considered as part of a three A-level offer and, although they may be excluded where an applicant is taking four A-level subjects, the grade achieved could be taken into account if necessary in August/September.

Candidates are not normally asked to attend for interview.

If you are made an offer then you may be invited to a Faculty/School Visit Day, which is usually held in the second semester. This will allow you the opportunity to visit the University and to find out more about the degree programme of your choice and the facilities on offer. It also gives you a flavour of the academic and social life at Queen's.

If you cannot find the information you need here, please contact the University Admissions and Access Service (admissions@qub.ac.uk), giving full details of your qualifications and educational background.

#### International Students

Our country/region pages include information on entry requirements, tuition fees, scholarships, student profiles, upcoming events and contacts for your country/region. Use the dropdown list below for specific information for your country/region.

#### English Language Requirements

An IELTS score of 6.0 with a minimum of 5.5 in each test component or an equivalent acceptable qualification, details of which are available at: http://go.qub.ac.uk/EnglishLanguageReqs

If you need to improve your English language skills before you enter this degree programme, INTO Queen's University Belfast offers a range of English language courses. These intensive and flexible courses are designed to improve your English ability for admission to this degree.

- Academic English: an intensive English language and study skills course for successful university study at degree level
- Pre-sessional English: a short intensive academic English course for students starting a degree programme at Queen's University Belfast and who need to improve their English.

#### International Students - Foundation and International Year One Programmes

INTO Queen's offers a range of academic and English language programmes to help prepare international students for undergraduate study at Queen's University. You will learn from experienced teachers in a dedicated international study centre on campus, and will have full access to the University's world-class facilities.

These programmes are designed for international students who do not meet the required academic and English language requirements for direct entry.

**Foundation**

The INTO progression course suited to this programme is

http://www.intostudy.com/en-gb/universities/queens-university-belfast/courses/international-foundation-in-engineering-and-science.

INTO - English Language Course(QSIS ELEMENT IS EMPTY)

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Careers

#### Career Prospects

**Introduction**

Studying for a Maths degree at Queen’s will assist students in developing the core skills and employment-related experiences that are valued by employers, professional organisations and academic institutions. Graduates from this degree at Queen’s are well regarded by many employers (local, national and international) and over half of all graduate jobs are now open to graduates of any discipline, including mathematics.

According to the Institute for Fiscal Studies, 5 years after graduation, Maths graduates earn 20 per cent more on average than other graduates (IFS 2018) and are the 3rd highest earners compared to all other subjects.

Although many of our graduates are interested in pursuing careers in teaching, banking and finance, significant numbers develop careers in a wide range of other sectors. The following is just a small selection of the major career sectors that have attracted our graduates in recent years:

Management Consultancy

Export Marketing (NI Programme)

Fast Stream Civil Service

Varied graduate programmes (Times Top 100 Graduate Recruiters/AGR, Association of Graduate Recruiters UK)

**Employment after the Course**

Typical career destinations of graduates include:

• Teaching

• Finance and Banking (Financial Analyst, Predictive Modelling, Quantitative Analyst)

• Management (Consultancy, Risk Analyst, Insurance)

• Engineering and Information Technology (Data Scientist, Software and Process Engineer)

• Statistics, Market and Operational Research

• Research (academia and industry)

• Government and Defence

• Medical Science

• Export Marketing (NI Programme)

• Varied graduate programmes (Times Top 100 Graduate Recruiters/AGR, Association of Graduate Recruiters UK)

Companies working in the sectors above that often employ our graduates include: AquaQ Analytics, Civil Service Fast Stream, Citi, First Derivatives, AllState, Liberty Insurance, PwC, Santander, Clarus Financial Technologies, Kainos, Teach First.

#### What employers say

At Citi, we value diverse thinking and we encourage Maths students to join our Graduate Programmes each year. The Maths graduates that we have hired have been extremely successful in their careers within Citi, in particular our Software Engineering opportunities. They excel in their profession with us as they are able to transfer their analytical, numerical and problem solving skills to their day to day responsibilities.

Aislinn Wilson, Applications Development Senior Manager, Citi group

Additional Awards Gained(QSIS ELEMENT IS EMPTY)

#### Prizes and Awards

Top performing students are eligible for a number of prizes within the School.

#### Degree plus award for extra-curricular skills

In addition to your degree programme, at Queen's you can have the opportunity to gain wider life, academic and employability skills. For example, placements, voluntary work, clubs, societies, sports and lots more. So not only do you graduate with a degree recognised from a world leading university, you'll have practical national and international experience plus a wider exposure to life overall. We call this Degree Plus. It's what makes studying at Queen's University Belfast special.

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Fees and Funding

#### Tuition Fees

Northern Ireland (NI) ^{1} |
£4,710 |

Republic of Ireland (ROI) ^{2} |
£4,710 |

England, Scotland or Wales (GB) ^{1} |
£9,250 |

EU Other ^{3} |
£18,800 |

International | £18,800 |

^{1} EU citizens in the EU Settlement Scheme, with settled status, will be charged the NI or GB tuition fee based on where they are ordinarily resident. Students who are ROI nationals resident in GB will be charged the GB fee.

^{2} EU students who are ROI nationals resident in ROI are eligible for NI tuition fees.

^{3} EU Other students (excludes Republic of Ireland nationals living in GB, NI or ROI) are charged tuition fees in line with international fees.

All tuition fees quoted relate to a single year of study and will be subject to an annual inflationary increase, unless explicitly stated otherwise.

Tuition fee rates are calculated based on a student’s tuition fee status and generally increase annually by inflation. How tuition fees are determined is set out in the Student Finance Framework.

#### Additional course costs

**All Students**

Depending on the programme of study, there may be extra costs which are not covered by tuition fees, which students will need to consider when planning their studies.

Students can borrow books and access online learning resources from any Queen's library.

If students wish to purchase recommended texts, rather than borrow them from the University Library, prices per text can range from £30 to £100. A programme may have up to 6 modules per year, each with a recommended text.

Students should also budget between £30 to £75 per year for photocopying, memory sticks and printing charges.

Students undertaking a period of work placement or study abroad, as either a compulsory or optional part of their programme, should be aware that they will have to fund additional travel and living costs.

If a final year includes a major project or dissertation, there may be costs associated with transport, accommodation and/or materials. The amount will depend on the project chosen. There may also be additional costs for printing and binding.

Students may wish to consider purchasing an electronic device; costs will vary depending on the specification of the model chosen.

There are also additional charges for graduation ceremonies, examination resits and library fines.

**Mathematics costs**

All essential software will be provided by the University, for use on University facilities, however for some software, students may choose to buy a version for home use.

#### How do I fund my study?

There are different tuition fee and student financial support arrangements for students from Northern Ireland, those from England, Scotland and Wales (Great Britain), and those from the rest of the European Union.

Information on funding options and financial assistance for undergraduate students is available at www.qub.ac.uk/Study/Undergraduate/Fees-and-scholarships/.

#### Scholarships

Each year, we offer a range of scholarships and prizes for new students. Information on scholarships available.

#### International Scholarships

Information on scholarships for international students, is available at www.qub.ac.uk/Study/international-students/international-scholarships/.

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Apply

**How and when to Apply**

##### How to Apply

Application for admission to full-time undergraduate and sandwich courses at the University should normally be made through the Universities and Colleges Admissions Service (UCAS). Full information can be obtained from the UCAS website at: www.ucas.com/students.

##### When to Apply

UCAS will start processing applications for entry in autumn 2023 from 1 September 2022.

Advisory closing date: 25 January 2023 (18:00). This is the 'equal consideration' deadline for this course.

Applications from **UK and EU (Republic of Ireland)** students after this date are, in practice, considered by Queen’s for entry to this course throughout the remainder of the application cycle (30 June 2023) subject to the availability of places.

Applications from **International and EU (Other) students** are normally considered by Queen’s for entry to this course until 30 June 2023. If you apply for 2023 entry after this deadline, you will automatically be entered into Clearing.

Applicants are encouraged to apply as early as is consistent with having made a careful and considered choice of institutions and courses.

The Institution code name for Queen's is QBELF and the institution code is Q75.

Further information on applying to study at Queen's is available at: www.qub.ac.uk/Study/Undergraduate/How-to-apply/

#### Terms and Conditions

The terms and conditions that apply when you accept an offer of a place at the University on a taught programme of study. Queen's University Belfast Terms and Conditions.

#### Additional Information for International (non-EU) Students

**Applying through UCAS**

Most students make their applications through UCAS (Universities and Colleges Admissions Service) for full-time undergraduate degree programmes at Queen's. The UCAS application deadline for international students is 30 June 2023.**Applying direct**

The Direct Entry Application form is to be used by international applicants who wish to apply directly, and only, to Queen's or who have been asked to provide information in advance of submitting a formal UCAS application. Find out more.**Applying through agents and partners**

The University’s in-country representatives can assist you to submit a UCAS application or a direct application. Please consult the Agent List to find an agent in your country who will help you with your application to Queen’s University.

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Fees and Funding