Modules

• Year 1

  Core Modules

  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 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

  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

  French 1 (40 credits)

  ×

  French 1

  Overview

  This module aims to consolidate and develop the students existing written and oral language skills and knowledge of French and Francophone culture, equip them with professional and employability skills and prepare them to go further in the study of French. It consists of four elements designed to provide a comprehensive consolidation of French language competence:
  
  1. Language Seminar (1hr per week)
  Seminar aims to develop students ability to understand, translate and compose French language materials in a range of forms: text, image, audio-visual. Language will be engaged in context, guided by themes such as University life, Culture and Identity and Culture and Communication. Linguistic competence will be developed through a range of methods that may include: group discussion, comprehension, translation, responsive and essay writing.
  
  2. Grammar Workshop (1hr per week)
  Workshop designed to consolidate and enrich students' knowledge and understanding of French grammar and syntax. All major areas of grammar will be encountered, laying the foundations for future study of the language and its nuances. It focuses particularly on developing competence in the key area of translation into French.
  
  3. Professional skills (1hr per week)
  The class focuses on language skills for special purposes and contains two strands: Language for Business and Language for Law. Both provide linguistic and socio-cultural knowledge important to work-related situations in different fields.
  
  4. Conversation class (1hr per week)
  Conversation class is led by a native speaker of French and compliments the content of the Language hour. Students will meet in small groups to discuss, debate and present on the main themes of the course.

  Learning Outcomes

  On successful completion of the modules students should:
  1. Be able to read French texts in a variety of forms and demonstrate a sensitivity to their detail and nuance in speech, writing and when translating.
  2. Be able to produce French texts appropriate to different requirements and registers.
  3. Be able to investigate, structure and present a complex argument in longer pieces of written work.
  4. Be able to communicate using more sophisticated grammatical and syntactical constructions with a good level of accuracy (without basic errors).

  Skills

  On successful completion of the modules students should have developed the following range of skills: comprehensive dexterity using French grammar; translation skills; text analysis; comprehension; essay writing; lexicographical skills; report writing skills; IT skills; presentation skills; spoken language skills

  Coursework

  35%

  Examination

  40%

  Practical

  25%

  Stage/Level

  1

  Credits

  40

  Module Code

  FRH1101

  Teaching Period

  Full Year

  Duration

  24 weeks

  Pre-requisite

  No

  Core/Optional

  Core

• Year 2

  Core Modules

  French 2 (40 credits)

  ×

  French 2

  Overview

  Course contents: Building on skills acquired at Level 1, this module aims to consolidate productive (writing and speaking) and receptive (reading and listening) skills in French language. Key components are: comprehension, translation into English and into French, résumé, grammar, CV preparation. The oral French component includes presentations and preparation for job interviews. Languages for special purposes strands equip students in law or business with skills for legal and professional contexts.
  This module will contain the following elements:
  1.Written language (2 hrs per week)
  This component will focus on enhancing ability in written French through engagement with a range of journalistic and literary written texts at appropriate level. A variety of topics will be covered, dealing with current themes in society and topical issues. Written language tasks include translation (from and into French), résumé, comprehension and grammar exercises.
  2.Oral language (1 hr per week)
  This component will focus on enhancing ability in oral French. A variety of topics and themes are covered, which aim to develop knowledge of issues in present-day France, prepare students for the year abroad and for job interviews in the target language. Stimulus materials from a range of media (textual, visual, audio, video) are used.
  3.Contextual Study (filière; 1 hr per week)
  This component will raise awareness of cultural and linguistic issues in French and allow students to deepen their perspective of the field, as well as preparing students for a residence in a French-speaking country.

  Learning Outcomes

  Learning Outcomes: On successful completion of the modules students should:
  1) be able to demonstrate fluency, accuracy and spontaneity in spoken and written French, with a broad range of vocabulary and expression, so as to be able to discuss a variety of complex issues;
  2) be able to read wide variety of French texts and identify important information and ideas within them;
  3) be able to translate a range of texts into and from French;
  4) have developed a detailed critical understanding of representative textual and other material;
  5) be able to engage in complex problem-solving exercises.

  Skills

  On successful completion of the modules students should have developed the following range of skills:
  Skills in written and oral expression; critical awareness and problem-solving; close textual analysis; translation; comprehension; presentation; IT skills; employability skills, such as interview technique and cv preparation.

  Coursework

  35%

  Examination

  40%

  Practical

  25%

  Stage/Level

  2

  Credits

  40

  Module Code

  FRH2101

  Teaching Period

  Full Year

  Duration

  24 weeks

  Pre-requisite

  Yes

  Core/Optional

  Core

  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

  No

  Core/Optional

  Core

  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

  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

  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

  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

  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

• Year 3

  Core Modules

  International Placement – Year Abroad (120 credits)

  ×

  International Placement – Year Abroad

  Overview

  Students complete a work, volunteer or study placement in fulfilment of the residence abroad requirements associated with their chosen Mathematics degree.

  Learning Outcomes

  On successful completion of this module students should be able to demonstrate:
  - Advanced linguistic skills (where appropriate)
  - Enhanced cultural and intercultural awareness
  - An understanding of the work environment and professional skills OR an understanding of a different university system and enhanced academic skills

  Skills

  Students undertaking the placement will develop their skills in the following areas: linguistic skills (reading, writing, listening, speaking); professional or academic skills; cultural and intercultural awareness; personal development.

  Coursework

  0%

  Examination

  0%

  Practical

  100%

  Stage/Level

  3

  Credits

  120

  Module Code

  MTH3999

  Teaching Period

  Full Year

  Duration

  30 weeks

  Pre-requisite

  No

  Core/Optional

  Core

• Year 4

  Core Modules

  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

  4

  Credits

  20

  Module Code

  MTH3023

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  Yes

  Core/Optional

  Core

  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

  4

  Credits

  20

  Module Code

  MTH3024

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  Applied Mathematics Project (20 credits)

  ×

  Applied Mathematics Project

  Overview

  Self-study of an advanced mathematical topic under the supervision of a member of staff. Students will be offered a choice of subjects, which can span the entire range of applied mathematics, including theoretical physics. The study concludes with a written report and a poster presentation.

  Learning Outcomes

  At the end of the module students should be able to:
  • Demonstrate knowledge and understanding of an advanced topic in mathematics
  • Demonstrate the project management skills to carry out a mathematical investigation
  • Explain an extended study of mathematics in both written and oral presentations
  • Apply a range of mathematical techniques to loosely defined problems
  • Use and implement various project management skills
  • Use and implement various oral and written presentation skills

  Skills

  The project module is geared towards the development of a wide range of skills.
  1. Mathematics skills
  a. Consolidation of obtained knowledge
  b. Independence as a mathematician
  2. Project management skills
  a. Project planning
  b. Time management
  c. Adherence to deadlines
  3. Presentation skills.
  a. Report-writing skills
  b. Oral presentation skills

  Coursework

  80%

  Examination

  0%

  Practical

  20%

  Stage/Level

  4

  Credits

  20

  Module Code

  AMA3011

  Teaching Period

  Both

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  French 3 (40 credits)

  ×

  French 3

  Overview

  Building on skills acquired at level 2, this module aims to develop the skills and understanding required to deal with a broad variety of language tasks. Linguistic, sociolinguistic and cultural awareness will be consolidated and deepened. The module will contain the following elements:
  
  1. Written Language Skills (2 hours per week) which will offer students an opportunity to enrich their linguistic skills, consolidate grammatical awareness and develop facility in handling the structures of standard, modern French, across a variety of genres, by means of practical engagement with a range of texts carefully selected for both their linguistic interest (varying in style and register) and the insights they offer into aspects of contemporary France and the Francophone world. Emphasis is placed on accuracy, fluent and idiomatic expression, and linguistic flair. A variety of language acquisition and development methods will be employed: grammar practice, editing work, essay-writing, translation into English and into French.
  
  2. Spoken Language (1 hour per week), which will focus on aspects of contemporary France and the Francophone world, with the aim of training students to speak accurately and fluently in French, to express a range of different ideas and opinions, and to organise material logically and coherently when presenting. This component of the module includes a presentation and extended discussion.
  
  3. Contextual Study (1hr per week). This component, which will vary across the two semesters, will deepen and contextualise the other elements of the module by placing them in a broader cultural context and will include, for example, literary texts, films, art and linguistics. A specific languages for special purposes strands equip students in law or business with skills for legal and professional contexts. This element includes an essay in the target language.

  Learning Outcomes

  Learning Outcomes: On successful completion of the modules students should:
  1) be able to demonstrate a high level of fluency, accuracy and spontaneity in written and oral French, including the use of a broad variety of linguistic structures and vocabulary;
  2) be able to deal with a broad variety of material in the target language, including material which is complex and abstract, and which involves a variety of genres and registers; 3) be able to demonstrate an advanced knowledge of the structures of the language and their broader linguistic context and the ability to use appropriate reference works effectively;
  4) be able to structure and present arguments at a high level in a range of formats and registers.

  Skills

  On successful completion of the modules students should have developed the following range of skills: Communication skills; translation skills; textual analysis; essay writing; lexicographical skills; IT skills; presentation skills; employability skills, such as report writing and editing skills; problem solving and critical thinking.

  Coursework

  35%

  Examination

  40%

  Practical

  25%

  Stage/Level

  4

  Credits

  40

  Module Code

  FRH3101

  Teaching Period

  Full Year

  Duration

  24 weeks

  Pre-requisite

  Yes

  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

  4

  Credits

  20

  Module Code

  PMA3013

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Core

  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

  4

  Credits

  20

  Module Code

  MTH3025

  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

  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

  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

  30%

  Examination

  70%

  Practical

  0%

  Stage/Level

  4

  Credits

  20

  Module Code

  MTH4323

  Teaching Period

  Autumn

  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

  4

  Credits

  20

  Module Code

  MTH3032

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  No

  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

  4

  Credits

  20

  Module Code

  MTH3011

  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

  4

  Credits

  20

  Module Code

  MTH3012

  Teaching Period

  Autumn

  Duration

  12 weeks

  Pre-requisite

  Yes

  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

  4

  Credits

  20

  Module Code

  MTH3031

  Teaching Period

  Autumn

  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

  4

  Credits

  20

  Module Code

  MTH3021

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  No

  Core/Optional

  Optional

  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

  4

  Credits

  20

  Module Code

  MTH3022

  Teaching Period

  Spring

  Duration

  12 weeks

  Pre-requisite

  Yes

  Core/Optional

  Optional