Modules

• 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

  Pre-requisite

  No

  Core/Optional

  Core

  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

  Pre-requisite

  No

  Core/Optional

  Core

  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

  Pre-requisite

  No

  Core/Optional

  Core

  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

  Pre-requisite

  No

  Core/Optional

  Core

  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

  Pre-requisite

  No

  Core/Optional

  Core

  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

  Pre-requisite

  No

  Core/Optional

  Core

• 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

  Pre-requisite

  Yes

  Core/Optional

  Core

  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

  Pre-requisite

  No

  Core/Optional

  Core

  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

  Pre-requisite

  Yes

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  Yes

  Core/Optional

  Optional

  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

  Pre-requisite

  Yes

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  Introduction to Artificial Intelligence and Machine Learning (20 credits)

  ×

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  20%

  Examination

  80%

  Practical

  0%

  Stage/Level

  2

  Credits

  20

  Module Code

  MTH2031

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  Yes

  Core/Optional

  Optional

• 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

  Pre-requisite

  No

  Core/Optional

  Core

  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

  Pre-requisite

  No

  Core/Optional

  Core

  Optional Modules

  Discrete Mathematics (20 credits)

  ×

  Discrete Mathematics

  Overview

  1. Intro Enumerative Combinatorics: basic counting, pigeonhole principle, inclusion-exclusion, recurrence relations, generating functions.
  2. Intro Elementary Number Theory: Divisibility and primes, Euclidean algorithm, linear congruences, Chinese Remainder Theorem.
  3. Intro Graph Theory: basic notions, trees, connectivity, matchings, graph colouring, planarity, basic Ramsey theory.
  4. Intro Algorithmics: analysis of algorithms, sorting, greedy and divide-and-conquer algorithms, basic graph and NT algorithms.

  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 discrete mathematics, such as generating functions, the inclusion-exclusion principle, factorisation of integers, the Chinese Remainder Theorem and basic Ramsey theory. Students should be able to use concepts of graph theory to give precise formulations of problems and solve basic graph colouring problems.
  In addition, students should understand basic algorithmic principles such as binary trees, greedy algorithms and divide-and-conquer; they should be able to analyse the run time of basic algorithms such as the classical sorting algorithms by applying the techniques discussed in the first half of the module.

  Skills

  Knowing and applying basic techniques of discrete mathematics; formally analysing the run time of classical algorithms.

  Coursework

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH3022

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  Yes

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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 skills

  Coursework

  100%

  Examination

  0%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH3024

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Geometry of Optimisation (20 credits)

  ×

  Geometry of Optimisation

  Overview

  • Functionals on R^n, linear equations and inequalities; hyperplanes; half-spaces
  • Convex polytopes; faces
  • Specific examples: e.g., traveling salesman polytope, matching polytopes
  • Linear optimisation problems; geometric interpretation; graphical solutions
  • Simplex algorithm
  • LP duality
  • Further topics in optimisation, e.g., integer programming, ellipsoid method

  Learning Outcomes

  It is intended that students shall, on successful completion of the module, be able to:
  • demonstrate understanding of the foundational geometry of convex polytopes;
  • demonstrate understand of the geometric ideas behind linear optimisation;
  • solve simple optimisation problems graphically;
  • apply the simplex algorithm to concrete optimisation problems.

  Skills

  Knowing and applying basic techniques of polytope theory and optimisation.

  Coursework

  20%

  Examination

  80%

  Practical

  0%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH4323

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  Yes

  Core/Optional

  Optional

  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 thinking

  Coursework

  20%

  Examination

  60%

  Practical

  20%

  Stage/Level

  3

  Credits

  20

  Module Code

  MTH4331

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  Yes

  Core/Optional

  Optional

  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

  Pre-requisite

  Yes

  Core/Optional

  Optional

  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

  Pre-requisite

  Yes

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  Yes

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  Yes

  Core/Optional

  Optional

• 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

  Pre-requisite

  No

  Core/Optional

  Core

  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

  20%

  Examination

  80%

  Practical

  0%

  Stage/Level

  4

  Credits

  20

  Module Code

  CSC4003

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  Yes

  Core/Optional

  Core

  Digital Transformation: Software Design, Management and Practical Implementation (20 credits)

  ×

  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

  Pre-requisite

  No

  Core/Optional

  Core

  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

  Pre-requisite

  No

  Core/Optional

  Core

  Optional Modules

  Applied Algebra and Cryptography (20 credits)

  ×

  Applied Algebra and Cryptography

  Overview

  - (finite) fields and rings of polynomials over them.
  - the division algorithm and splitting of polynomials.
  - ideals and quotient rings, (principal) ideal domains, with examples from rings of polynomials.
  - polynomials in several indeterminates, Hilbert’s basis theorem.
  - applications of algebra to cryptography (such as affine Hill ciphers, RSA, lattice cryptography, Diophantine equations).
  - optional topics may include Euclidean rings, unique factorisation domains, greatest common divisor domains.

  Learning Outcomes

  It is intended that students shall, on successful completion of the module, be able to:
  understand the concept of a ring of polynomials over a (finite field);
  apply the factorisation algorithm;
  understand ideals, quotient rings and the properties of quotient rings;
  understand how algebra can be applied to cryptography and be able to encrypt messages using methods from the module.

  Skills

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

  Coursework

  20%

  Examination

  80%

  Practical

  0%

  Stage/Level

  4

  Credits

  20

  Module Code

  MTH4021

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  Mathematical Methods for Quantum Information Processing (20 credits)

  ×

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  Geometry of Optimisation (20 credits)

  ×

  Geometry of Optimisation

  Overview

  • Functionals on R^n, linear equations and inequalities; hyperplanes; half-spaces
  • Convex polytopes; faces
  • Specific examples: e.g., traveling salesman polytope, matching polytopes
  • Linear optimisation problems; geometric interpretation; graphical solutions
  • Simplex algorithm
  • LP duality
  • Further topics in optimisation, e.g., integer programming, ellipsoid method

  Learning Outcomes

  It is intended that students shall, on successful completion of the module, be able to:
  • demonstrate understanding of the foundational geometry of convex polytopes;
  • demonstrate understand of the geometric ideas behind linear optimisation;
  • solve simple optimisation problems graphically;
  • apply the simplex algorithm to concrete optimisation problems.

  Skills

  Knowing and applying basic techniques of polytope theory and optimisation.

  Coursework

  20%

  Examination

  80%

  Practical

  0%

  Stage/Level

  4

  Credits

  20

  Module Code

  MTH4323

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  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 thinking

  Coursework

  20%

  Examination

  60%

  Practical

  20%

  Stage/Level

  4

  Credits

  20

  Module Code

  MTH4331

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  Pre-requisite

  No

  Core/Optional

  Optional