###### MSci|Undergraduate

# Mathematics and Computer Science

**Entry year**

Academic Year 2024/25

**Entry requirements**

AAA/A*AB

**Duration**

4 years (Full Time)

**UCAS code**

GGK1

Mathematics is the universal language of science while computer science is the study of the hardware and algorithms that are used in modern computer systems. Since many of the early pioneers of computer science, for instance Alan Turing, were mathematicians it is not surprising that these two subjects are closely related. This is a four-year joint degree programme, in conjunction with the School of Electronics, Electrical Engineering and Computer Science, that combines the study of the two subjects at each level.

### Mathematics and Computer Science Degree highlights

In 2020, more than 90% of 1st and 2nd year Maths students expressed overall satisfaction with their course

#### 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 and computer science at work in the real world, and to enhance your career prospects at the same time. Several local financial services companies (including Clarus FT, Effex Capital and AquaQ Analytics) were founded by our graduates.

#### Career Development

- 87% of Maths students are in graduate employment or further study 15 months after graduation (11th in the UK)

#### World Class Facilities

- A new Teaching Centre for Mathematics and Physics opened in September 2016. This provides a dedicated space for teaching within the School. Facilities for mathematics include new lecture and group-study rooms, a new student social area and state-of-the-art computer facilities. Computer Science teaching takes place in the iconic Bernard Crossland building on the Malone Road, just a short walk from the Mathematics department. The building was recently refurbished at a cost of £14M, and houses standard computer and lecture rooms as well as laboratories and break-out spaces.

#### 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. In the 2021 REF peer-review exercise, Mathematics Research has the 11th highest impact in the UK.

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 introduced 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.
- School has the 3rd highest postgraduate research student satisfaction in the university.
- Placement Year

Students can take an optional placement year between years 2/3 or years 3/4 of their course. Completion of an approved placement will be acknowledged in your final degree certificate with the addition of the words "with placement year".

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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 | At Stage 1, students must take six compulsory modules. |

Stage 2 | At Stage 2, students must take modules totalling 120 units to be approved by an advisor of studies, with at least two modules from each Mathematics and Computer Sciences. |

Stage 3 | At Stage 3, students must take modules totalling 120 units to be approved by an advisor of studies, with at least two modules from Computer Sciences. |

Stage 4 | At Stage 4, modules open to MSci. students offer students the opportunity to study a selection of topics in greater depth than is possible in the BSc programme. 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. Students must select additional modules to be approved by an advisor of studies. |

#### People teaching you

Dr Thomas HuettemannAssociate Director of Education for Mathematics

**School of Maths and Physics**

Dr Huettemann is a Senior Lecturer in Mathematics with research interests in homological algebra, graded algebra and K-theory.

#### Contact Teaching Times

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

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. |

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

#### 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 and Computer Science 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. These span aspects of programming, computational tools and fundamental mathematical and algorithmic ideas.**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 students, as individuals or as part of a group, have submitted.
- Face to face comment. This may include occasions when students make use of the lecturers’ advertised “office hours” to help 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 students can review in their 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 students have reviewed their feedback, they are encouraged to identify and implement further improvements to the quality of their work.

PREV

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

24 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

24 weeks

Procedural Programming (20 credits)## Procedural Programming

### Overview

This module introduces the fundamentals of procedural programming. Using a problem-solving approach, real-world examples are explored to promote code literacy and good algorithm design. Students are introduced to the representation and management of primitive data, structures for program control and refinement techniques, which guide the development process from problem specification to code solution.

### Learning Outcomes

Students must be able to:

• Demonstrate knowledge, understanding and the application of the principles of procedural programming, including:

o Primitive data types (including storage requirements)

o Program control structures: Sequencing, selection and iteration

o Functions/methods and data scope

o Simple abstract data structures, i.e. strings and arrays

o File I/O and error handling

o Pseudocode and algorithm definition/refinement

• Apply good programming standards in compliance with the relevant codes of practice e.g. naming conventions, comments and indentation

• Analyse real-world challenges in combination with programming concepts to write code in an effective way to solve the problem.### Skills

KNOWLEDGE & UNDERSTANDING: Understand fundamental theories of procedural programming

INTELLECTUAL AND PRACTICAL:

• Be able to design and develop small programs, which meet simple functional requirements expressed in English.

• Programs designed, developed and tested will contain a combination of some or all of the features as within the Knowledge and Understanding learning outcomes.Coursework

60%

Examination

0%

Practical

40%

###### Stage/Level

1

###### Credits

20

###### Module Code

CSC1025

###### Teaching Period

Autumn

###### Duration

12 weeks

Object Oriented Programming (20 credits)## Object Oriented Programming

### Overview

This module introduces the fundamentals of object-oriented programming. Real-world problems and exemplar code solutions are examined to encourage effective data modelling, code reuse and good algorithm design. Fundamental OO programming concepts including abstraction, encapsulation, inheritance and polymorphism are practically reviewed through case studies, with an emphasis on testing and the use of code repositories for better management of code version control.

### Learning Outcomes

Students must be able to:

• Demonstrate knowledge, understanding and the application of the principles and application of object-oriented design, to include:

o Abstraction, encapsulation, inheritance and polymorphism

• Demonstrate knowledge of static data modelling techniques (through UML)

• Demonstrate knowledge, understanding and the application of the principles and application of object extensibility and object reuse.

• Demonstrate knowledge, understanding and the application of more advanced programming concepts, to include:

o Recursion

o Searching and sorting

o Basic data structures

• Demonstrate knowledge, understanding and the application of testing, in particular, unit and integration testing.

• Apply good programming standards in compliance with the relevant codes of practice and versioning tools being employed e.g. naming conventions, comments and indentation

• Analyse real-world challenges in combination with OO programming concepts to write code in an effective way to solve the problem.### Skills

KNOWLEDGE & UNDERSTANDING: Understand fundamental theories of object-oriented programming

INTELLECTUAL AND PRACTICAL:

• Be able to design, develop and test programs, which meet functional requirements expressed in English.

• Programs designed, developed and tested will contain a combination of some or all of the features as within the Knowledge and Understanding learning outcomes.Coursework

50%

Examination

20%

Practical

30%

###### Stage/Level

1

###### Credits

20

###### Module Code

CSC1029

###### Teaching Period

Spring

###### Duration

12 weeks

- Year 2
#### Core Modules

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

Data Structures and Algorithms (20 credits)## Data Structures and Algorithms

### Overview

• Data structures: Stacks, Lists, Queues, Trees, Hash tables, Graphs, Sets and Maps

• Algorithms: Searching, Sorting, Recursion (with trees, graphs, hash tables etc.)

• Asymptotic analysis of algorithms

• Programming languages representation and implementation### Learning Outcomes

• Demonstrate understanding of the operation and implementation of common data structures and algorithmic processes (including stacks, lists, queues, trees, hash tables, graphs, sets and maps, alongside searching, sorting and recursion algorithms).

• Select, implement and use data structures and searching, sorting and recursive algorithms to model and solve problems.

• Perform asymptotic analysis of simple algorithms.

• Demonstrate understanding of the fundamentals of programming languages representation, implementation and execution.### Skills

Problem solving by analysis, solution design and application of techniques (e.g. suitable data structures, algorithms, and implementation in C++). Precision and conciseness of expression. Rigour in thought.

Coursework

50%

Examination

0%

Practical

50%

###### Stage/Level

2

###### Credits

20

###### Module Code

CSC2059

###### Teaching Period

Autumn

###### Duration

12 weeks

#### Optional Modules

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

Professional and Transferrable Skills (20 credits)## Professional and Transferrable Skills

### Overview

This module will prepare students for employment by developing an awareness of the business environment and the issues involved in successful career management combined with the development of key transferrable skills such as problem solving, communication and team working. Students will build their professional practice and ability to critically self-reflect to improve their performance.

Key elements will explode legal, social, ethical and professional issues (LSEPIs) including intellectual property, computer-aided crime, data protection and privacy including GDPR, security, net neutrality, communication through technology, cultural sensitivity and gender neutrality. The British Computer Society (BCS) code of conduct will be exploded and understood.### Learning Outcomes

To prepare students for employment in industry and research through developing an awareness of the business environment and key skills.

To develop and demonstrate a range of transferrable skills including communication skills, presentation, group working and problem solving.

To develop skills in critical reflection of self and others feeding into improvements.

To explore legal, social, ethical and professional issues (LSEPIs). Examples of areas to be explored will relate to: Intellectual Property, Computer Crime, Work Quality, Challenges of On-line content Quality, Digital Divide including Net Neutrality, Privacy including GDPR, Security, Globalisation, Communication through effective use of technology, Cultural Sensitivity, Gender Neutrality. British Computer Society (BCS) Code of Conduct will be explored covering Public Interest, Professional Competence and Integrity, Duty to Relevant Authority and Duty to the Profession.### Skills

Problem synthesis and resolution as an individual and as a team. Development and use of suitable communication mechanisms. Business and professional awareness.

Coursework

100%

Examination

0%

Practical

0%

###### Stage/Level

2

###### Credits

20

###### Module Code

CSC2065

###### Teaching Period

Autumn

###### Duration

12 weeks

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

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

Theory of Computation (20 credits)## Theory of Computation

### Overview

• Automata and Formal Languages

• Computability Theory (Turing Machines etc) and Decidability Theory (Halting Problem, etc)

• Complexity Theory### Learning Outcomes

• Explain how computation can occur using automata such as finite state machines and Turing machines.

• Reason about algorithmic complexity and determine what problems can/cannot be solved by computers.

• Describe the correspondence amongst Languages and Automata etc.

• Use proof techniques to construct simple proofs.### Skills

Problem analysis, Problem solving. Precision and conciseness of expression. Rigour in thought. Constructing logical arguments and proofs.

Coursework

40%

Examination

60%

Practical

0%

###### Stage/Level

2

###### Credits

20

###### Module Code

CSC2060

###### Teaching Period

Spring

###### Duration

12 weeks

## Introduction to Artificial Intelligence and Machine Learning

### Overview

• Concepts of artificial intelligence and machine learning.

• Fundamentals of supervised and unsupervised learning

• Fundamentals of experimental settings and hypothesis evaluation

• The concept of feature selection

• Evaluation in machine learning

o Type I and Type II errors

o Confusion matrices

o ROC and CMC curves

o Cross validation

• Linear and non-linear function fitting

o Linear Regression

o Kernels

• Classification models:

o Nearest Neighbour

o Naïve Bayes

o Decision Trees

• Clustering models:

o k-Means

o hierarchical clustering

o Anomaly detection### Learning Outcomes

• Knowledge and understanding of techniques and selected software relevant to the field of artificial intelligence.

• Ability to identify techniques relevant to particular problems in artificial intelligence and data analysis.

• Ability to discuss and provide reasonable argumentation using artificial intelligence and machine learning concepts.

• Ability to identify opportunities for software solutions in artificial intelligence and data analysis.

• Ability to solve specific data analysis problems using techniques of artificial intelligence and machine learning.### Skills

Problem and data analyses, design of logical and statistical models, application of computational techniques, understanding results.

Coursework

60%

Examination

40%

Practical

0%

###### Stage/Level

2

###### Credits

20

###### Module Code

CSC2062

###### 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

30%

Examination

70%

Practical

0%

###### Stage/Level

2

###### Credits

20

###### Module Code

MTH2031

###### 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

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

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

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

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

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

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 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

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

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

Deep Learning (20 credits)## Deep Learning

### Overview

• Overview of generic machine learning pipelines

• Deep learning

o Feedforward neural networks

o Regularisation for deep learning

o Optimisation for training deep models

o Convolutional networks

o Auto-encoders

o Recurrent Networks

o Siamese Neural Network

• Evolving learned models

o Active Learning

o Transfer Learning

o Incremental Learning

• Applications of deep learning### Learning Outcomes

Be able to:

• Explain when and how machine learning is useful in industry, public institutions and research.

• Know and apply state-of-art deep learning techniques.

• Demonstrate the ability to understand and describe the underlying mathematical framework behind these operations.

• Design and develop original deep learning pipelines applied to a variety of problems

• Formulate and evaluate novel hypothesis

• Analyse an application problem, considering its suitability for applying deep learning, and propose a sensible solution

• Evaluate the performance of proposed deep learning solutions through rigorous experimentation

• Analyse quantitative results and use them to refine initial solutions

• Communicate finding effectively and in a convincing manner based on data, and compare proposed systems against existing state-of-art solutions### Skills

Problem solving. Self and independent learning. Research. Working with others and organisational skills. Critical analysis. Quantitative evaluation. Mathematical and logical thinking.

Coursework

60%

Examination

40%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

CSC3066

###### Teaching Period

Spring

###### Duration

12 weeks

Formal Methods (20 credits)## Formal Methods

### Overview

A rigorous approach to software development. Logical foundations. Specification of data types. Implicit and direct specification of functions and operations. Reasoning about specifications, refinement, axiomatic semantics.

### Learning Outcomes

To present a scientific approach to the construction of software systems.

### Skills

Precision and conciseness of expression. Rigour in thought.

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

CSC3001

###### Teaching Period

Spring

###### Duration

12 weeks

Concurrent Programming (20 credits)## Concurrent Programming

### Overview

Concurrent Programming Abstraction and Java Threads, the Mutual Exclusion Problem, Semaphores, Models of Concurrency, Deadlock, Safety and Liveness Properties. Notions are exemplified through a selection of concurrent objects such as Linked Lists, Queues and Hash Maps. Principles of graph analytics, experimental performance evaluation, application of concurrent programming to graph analytics.

### Learning Outcomes

To understand the problems that are specific to concurrent programs and the means by which such problems can be avoided or overcome.

### Skills

To model and to reason rigorously about the properties of concurrent programs; to analyse and construct concurrent programs in Java; to quantitatively analyse the performance of concurrent programs.

Coursework

100%

Examination

0%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

CSC3021

###### Teaching Period

Autumn

###### Duration

12 weeks

Video Analytics and Machine Learning (20 credits)## Video Analytics and Machine Learning

### Overview

• Overview of imaging and video systems and generic machine learning pipelines

• Pattern recognition problems: Verification, detection and identification

• Data pre-processing:

o Image enhancement: Normalisation. Point Operations, Brightness and contrast.

o Filtering and Noise reduction. Convolution

• Classification

o Support Vector Machines (SVM).

o Boosting and ensemble of classifiers

o RF

o Neural networks.

o Deep Learning.

• Vision-specific Feature extraction:

o Simple features

o Gradients and Edge extraction

o Colour Extraction and colour histograms

o SIFT

o Histogram of Gradients HoG

• Unsupervised learning:

o Clustering and Bag of Words for vision

o Self-organised maps

• Segmentation, tracking and post processing

o Brightness segmentation

o Motion detection; Background modelling and subtraction; Optical Flow

o Template Matching

o Tracking: Kalman Filter, Particle Filter and tracking by detection

o Introduction to time series analysis

• Dimensionality reduction techniques and latent spaces.

o The curse of dimensionality

o Principal component analysis (PCA).

o Linear discriminant analysis (LDA).

• Introduction to Deep Learning

• GPU acceleration for video processing.

• Applications:

o Video Surveillance

o Cyber-physical security

o Medical imaging

o Secure corridors.

o Pose estimation.

o Biometrics

o Face detection

o Human behaviour analysis.### Learning Outcomes

Be able to:

• Explain when and how machine learning and computer vision is useful in industry, public institutions and research.

• Know and apply a range of basic computer vision and machine learning techniques.

• Demonstrate the ability to understand and describe the underlying mathematical framework behind these operations.

• Design and develop machine learning pipelines applied to computer vision applications

• Formulate and evaluate hypothesis

• Evaluate the performance of proposed machine learning solutions through rigorous experimentation

• Analyse quantitative results and use them to refine initial solutions

• Communicate finding effectively and in a convincing manner based on data, and compare proposed systems against existing solutions### Skills

Problem solving. Self and independent learning. Research. Working with others and organisational skills. Critical analysis. Quantitative evaluation. Mathematical and logical thinking.

Coursework

40%

Examination

60%

Practical

0%

###### Stage/Level

3

###### Credits

20

###### Module Code

CSC3067

###### Teaching Period

Autumn

###### 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

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

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

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

- 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

Algorithms: Analysis and Application (20 credits)## Algorithms: Analysis and Application

### Overview

Analysis and design of algorithms, complexity, n-p completeness; algorithms for searching, sorting; algorithms which operate on trees, graphs, strings. Database algorithms, B-tree and hashing, disk access, algorithms. Applications of algorithms

### Learning Outcomes

To understand some of the principal algorithms used in Computer Science; to be able to analyse and design efficient algorithms to suit particular applications.

### Skills

Analysis, design and implementation of efficient algorithms.

Coursework

30%

Examination

70%

Practical

0%

###### Stage/Level

4

###### Credits

20

###### Module Code

CSC4003

###### Teaching Period

Spring

###### Duration

12 weeks

## Digital Transformation: Software Design, Management and Practical Implementation

### Overview

Opportunity Analysis, Entrepreneurship and Innovation, Business Planning, Modelling and documenting software design; Software Design principles and patterns; Software Architecture; Modern approaches to software design; Legal Social and Ethical considerations, Software Project and Team Management

### Learning Outcomes

Students will

i) Have a good knowledge of market evaluation, opportunity scoping, background research and software design related to a modern commercial setting.

ii) Gain the ability to evaluate systems in terms of architecture, general quality attributes and possible trade-offs presented within the given problem.

iii) Gain knowledge of the commercial and economic context of the development use and maintenance of computer-based systems.

iv) Be able to frame the opportunity within an innovative business model outlining the overall requirements i.e. model and analyse the extent to which a computer-based system meets the criteria defined for its current need, use and future development.

v) Recognise the legal, social, ethical and professional issues involved in the exploitation of 36 computer technology and be guided by the adoption of appropriate professional, ethical and legal practices.

vi) Be able to apply analytical skills within a team to a practical commercial opportunity.

vii) Understand the realisation of software requirements as software designs.

viii) Appreciate how to operate and contribute as part of a team, understanding the different ways of organising teams and the roles within a team in the development and delivery of an end-to-end software solution.

ix) Appreciation of risk management within the development process from an end user, commercial, team and individual perspective.

x) Deploy effectively suitable tools for the construction and documentation of computer applications and to use and apply information from technical literature### Skills

Knowledge of opportunity analyses, business modelling, and commercial delivery of software against a created set of requirements

Coursework

100%

Examination

0%

Practical

0%

###### Stage/Level

4

###### Credits

20

###### Module Code

CSC4008

###### Teaching Period

Autumn

###### Duration

12 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

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

4

###### Credits

20

###### Module Code

MTH4332

###### 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

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

4

###### Credits

20

###### Module Code

MTH4311

###### 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

4

###### Credits

20

###### Module Code

MTH4321

###### Teaching Period

Spring

###### Duration

12 weeks

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

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

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

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

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

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

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

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

4

###### Credits

20

###### Module Code

MTH4322

###### Teaching Period

Autumn

###### 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

4

###### Credits

20

###### Module Code

MTH4323

###### Teaching Period

Autumn

###### Duration

12 weeks

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Entry Requirements

#### Entrance requirements

A level requirementsAAA (Mathematics) OR A* (Mathematics) AB A maximum of one BTEC/OCR Single Award or AQA Extended Certificate will be accepted as part of an applicant's portfolio of qualifications with a Distinction* being equated to a grade A at A-level and a Distinction being equated to a grade B at A-level. |

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

Access CourseNot considered. Applicants should apply for the BSc Mathematics and Computer Science 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. |

NoteAll applicants must have GCSE English Language grade C/4 or an equivalent qualification acceptable to the University. |

#### 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 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.

For applicants offering the Irish Leaving Certificate, please note that performance at Irish Junior Certificate (IJC) is taken into account. For last year’s entry, applicants for this degree must have had a minimum of five IJC grades at C/Merit. The Selector also checks that any specific entry requirements in terms of Leaving Certificate subjects can be satisfied.

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 Mathematics and Computer Science 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 the 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 a list 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)

In line with Queen's membership to the Russell group (UK research-intensive universities), the School offers a variety of PhD programmes at the cutting-edge of internationally leading research. Many of our MSci students successfully embark in these post-graduate programmes.

**Employment after the Course**

The School has links with over 500 IT companies both here and abroad. We benefit from the fact that there are more software companies located in N Ireland than any other part of the UK, outside of London. This offers benefits on many levels for our students, from industrial input to the content of our courses, through to year long and summer placements,as well as activities such as competitions organised by the companies etc.

The Prospects website provides further information regarding the types of jobs that attract Computer Science Graduates.

Further study is also an option open to Computer Science graduates. Students can choose from a wide range of Masters programmes as well as a comprehensive list of research topics, see the School website www.qub.ac.uk/eeecs for more information.

Northern Ireland has an excellent international reputation for the quality and supply of its software engineers. Indeed many companies, both national and international, have opted for Northern Ireland as a base for their computing divisions in recognition of the high quality of graduates produced by the local universities.

Given this situation, it is not surprising that our graduates have had unparalleled job opportunities over the years, both locally and internationally. Because of the achievements of Queen's graduates already in the software engineering profession, a Computer Science degree from Queen's is a highly respected qualification. A good Honours degree in Computer Science from Queen's is of great benefit in seeking the best jobs.

Employers, from large multinational firms to small local organisations, actively target our students, recognising that Queen's Computer Science graduates are equipped with the skills they need. On graduating the majority of graduates take up posts associated with software design and implementation. Opportunities exist in fields as diverse as finance, games, pharmaceuticals, healthcare, research, consumer products, and public services - virtually all areas of business. Some of the employers include BT, Liberty IT, Kainos, Asidua, Autonomy, Accenture, Citi, NYSE

Statistics highlight that over 90% of recent graduates were pursuing their chosen pathway within six months of graduation. As the IT market has recovered, current industry analysis indicates that there is a shortage of IT graduates and this trend is forecast to continue. The types of career open to Computer Science graduates include: Software Engineer; Systems Analyst; Web Designer; Games Developer; Systems Developer; IT Consultant; Project Manager.

www.prospects.ac.uk

#### 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/Future Ready 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/Future Ready Award. It's what makes studying at Queen's University Belfast special.

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Entry requirements

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Fees and Funding

#### Tuition Fees

Northern Ireland (NI) ^{1} |
£4,750 |

Republic of Ireland (ROI) ^{2} |
£4,750 |

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

EU Other ^{3} |
£20,800 |

International | £20,800 |

^{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 and Computer Science 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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Careers

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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 2024 from 1 September 2023.

Advisory closing date: 31 January 2024 (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 2024) 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 2024. If you apply for 2024 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 2024.**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