# Mathematics with Languages BSc (Hons)

- Course length: 3/4 years
- UCAS code: G19R
- Year of entry: 2022
- A-level requirements: ABB

## Honours Select

×This programme offers Honours Select combinations.

## Honours Select 100

×This programme is available through Honours Select as a Single Honours (100%).

## Honours Select 75

×This programme is available through Honours Select as a Major (75%).

## Honours Select 50

×This programme is available through Honours Select as a Joint Honours (50%).

## Honours Select 25

×This programme is available through Honours Select as a Minor (25%).

## Study abroad

×This programme offers study abroad opportunities.

## Year in China

×This programme offers the opportunity to spend a Year in China.

## Accredited

×This programme is accredited.

### Module details

**Due to the impact of COVID-19 we're changing how the course is delivered.**

### Programme Year One

Students must select 3 MATH modules and 1 Language Module each Semester. Students who combine a 75% Mathematics component with a 25% language component without a Year Abroad, and who follow the beginner's pathway or intermediate pathway within the language component, are not allowed to choose MATH1** modules in their second year or MATH2** modules in their final year.

#### Year One Compulsory Modules

##### Calculus I (MATH101)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**1. To introduce the basic ideas of differential and integral calculus, to develop the basic skills required to work with them and to apply these skills to a range of problems.

2. To introduce some of the fundamental concepts and techniques of real analysis, including limits and continuity.3. To introduce the notions of sequences and series and of their convergence.

**Learning Outcomes**(LO1) Understand the key definitions that underpin real analysis and interpret these in terms of straightforward examples.

(LO2) Apply the methods of calculus and real analysis to solve previously unseen problems (of a similar style to those covered in the course).

(LO3) Understand in interpret proofs in the context of real analysis and apply the theorems developed in the course to straightforward examples.

(LO4) Independently construct proofs of previously unseen mathematical results in real analysis (of a similar style to those demonstrated in the course).

(LO5) Differentiate and integrate a wide range of functions;

(LO6) Sketch graphs and solve problems involving optimisation and mensuration

(LO7) Understand the notions of sequence and series and apply a range of tests to determine if a series is convergent

(S1) Numeracy

##### Introduction to Linear Algebra (MATH103)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**45:55 **Aims**• To develop techniques of complex numbers and linear algebra, including equation solving, matrix arithmetic and the computation of eigenvalues and eigenvectors.

• To develop geometrical intuition in 2 and 3 dimensions.

• To introduce students to the concept of subspace in a concrete situation.

• To provide a foundation for the study of linear problems both within mathematics and in other subjects**Learning Outcomes**(LO1) Manipulate complex numbers and solve simple equations involving them, solve arbitrary systems of linear equations.

(LO2) Understand and use matrix arithmetic, including the computation of matrix inverses.

(LO3) Compute and use determinants.

(LO4) Understand and use vector methods in the geometry of 2 and 3 dimensions.

(LO5) Calculate eigenvalues and eigenvectors.

(S1) Numeracy

##### Mathematical It Skills (MATH111)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**•To acquire key mathematics-specific computer skills.

•To reinforce mathematics as a practical discipline by active experience and experimentation, using the computer as a tool.

•To illustrate and amplify mathematical concepts and techniques.

•To initiate and develop problem solving, group work and report writing skills.

•To initiate and develop modelling skills.

•To develop team work skills.**Learning Outcomes**(LO1) After completing the module, students should be able to tackle project work, including writing up of reports detailing their solutions to problems.

(LO2) After completing the module, students should be able to use computers to create documents containing formulae, tables, plots and references.

(LO3) After completing the module, students should be able to use mathematical software packages such as Maple and Matlab to manipulate mathematical expressions and to solve simple problems.

(LO4) After completing the module, students should be able to better understand the mathematical topics covered, through direct experimentation with the computer.

(S1) Problem solving skills

(S2) Numeracy

(S3) Communication skills

(S4) IT skills

(S5) Teamwork

(S6) Adaptability

(S7) Leadership

(S8) Mathematical modelling skills

##### Calculus II (MATH102)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**To discuss local behaviour of functions using Taylor’s theorem.

To introduce multivariable calculus including partial differentiation, gradient, extremum values and double integrals.**Learning Outcomes**(LO1) Use Taylor series to obtain local approximations to functions

(LO2) Obtain partial derivatives and use them in several applications such as, error analysis, stationary points change of variables.

(LO3) Evaluate double integrals using Cartesian and Polar Co-ordinates.

#### Year One Optional Modules

##### Newtonian Mechanics (MATH122)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To provide a basic understanding of the principles of Classical Mechanics and their application to simple dynamical systems. Learning Outcomes: After completing the module students should be able to analyse real world problems involving: - the motions of bodies under simple force systems - conservation laws for momentum and energy - rigid body dynamics using centre of mass, angular momentum and moments of inertia

**Learning Outcomes**(LO1) the motions of bodies under simple force systems

(LO2) conservation laws for momentum and energy

(LO3) rigid body dynamics using centre of mass, angular momentum and moments

(LO4) oscillation, vibration, resonance

(S1) Representing physical problems in a mathematical way

(S2) Problem Solving Skills

##### Numbers, Groups and Codes (MATH142)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**- To provide an introduction to rigorous reasoning in axiomatic systems exemplified by the framework of group theory.

- To give an appreciation of the utility and power of group theory as the study of symmetries.

- To introduce public-key cryptosystems as used in the transmission of confidential data, and also error-correcting codes used to ensure that transmission of data is accurate. Both of these ideas are illustrations of the application of algebraic techniques.

**Learning Outcomes**(LO1) Be able to apply the Euclidean algorithm to find the greatest common divisor of a pair of positive integers, and use this procedure to find the inverse of an integer modulo a given integer.

(LO2) Be able to solve linear congruences and apply appropriate techniques to solve systems of such congruences.

(LO3) Be able to perform a range of calculations and manipulations with permutations.

(LO4) Recall the definition of a group and a subgroup and be able to identify these in explicit examples.

(LO5) Be able to prove that a given mapping between groups is a homomorphism and identify isomorphic groups.

(LO6) To be able to apply group theoretic ideas to applications with error correcting codes.

(LO7) Engage in group project work to investigate applications of the theoretical material covered in the module.

##### Introduction to Statistics Using R (MATH163)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**1. Use software R to display and analyse data, perform tests and demonstrate basic statistical concepts.

2. Describe statistical data and display it using variety of plots and diagrams.

3. Understand basic laws of probability: law of total probability, independence, Bayes’ rule.

4. Be able to estimate mean and variance.

5. Be familiar with properties of some probability distributions and relations between them: Binomial, Poisson, Normal, t, Chi-squared.

6. To perform simple statistical tests: goodness-of-fit test, z-test, t-test.

7. To understand and be able to interpret p-values.

8. To be able to report finding of statistical outcomes to non-specialist audience.

9. Group work will help students to develop transferable skills such as communication, the ability to coordinate and prioritise tasks, time management and teamwork.

**Learning Outcomes**(LO1) An ability to apply statistical concepts and methods covered in the module's syllabus to well defined contexts and interpret results.

(LO2) An ability to understand, communicate, and solve straightforward problems related to the theory and derivation of statistical methods covered in the module's syllabus.

(LO3) An ability to understand, communicate, and solve straightforward theoretical and applied problems related to probability theory covered in the syllabus.

(LO4) Use the R programming language fluently in well-defined contexts. Students should be able to understand and correct given code; select appropriate code to solve given problems; select appropriate packages to solve given problems; and independently write small amounts of code.

### Programme Year Two

At most one of MATH122, MATH142 and MATH162 may be taken if not already taken in Year 1.

#### Year Two Optional Modules

##### Vector Calculus With Applications in Fluid Mechanics (MATH225)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**70:30 **Aims**To provide an understanding of the various vector integrals, the operators div, grad and curl and the relations between them. To give an appreciation of the many applications of vector calculus to physical situations. To provide an introduction to the subjects of fluid mechanics and electromagnetism.

**Learning Outcomes**(LO1) After completing the module students should be able to: - Work confidently with different coordinate systems. - Evaluate line, surface and volume integrals. - Appreciate the need for the operators div, grad and curl together with the associated theorems of Gauss and Stokes. - Recognise the many physical situations that involve the use of vector calculus. - Apply mathematical modelling methodology to formulate and solve simple problems in electromagnetism and inviscid fluid flow. All learning outcomes are assessed by both examination and course work.

##### Complex Functions (MATH243)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**To introduce the student to a surprising, very beautiful theory having intimate connections with other areas of mathematics and physical sciences, for instance ordinary and partial differential equations and potential theory.

**Learning Outcomes**(LO1) To understand the central role of complex numbers in mathematics;.

(LO2) To develop the knowledge and understanding of all the classical holomorphic functions.

(LO3) To be able to compute Taylor and Laurent series of standard holomorphic functions.

(LO4) To understand various Cauchy formulae and theorems and their applications.

(LO5) To be able to reduce a real definite integral to a contour integral.

(LO6) To be competent at computing contour integrals.

(S1) Problem solving skills

(S2) Numeracy

(S3) Adaptability

##### Linear Algebra and Geometry (MATH244)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:48 **Aims**To introduce general concepts of linear algebra and its applications in geometry and other areas of mathematics.

**Learning Outcomes**(LO1) To understand the geometric meaning of linear algebraic ideas.

(LO2) To know the concept of an abstract vector space and how it is used in different mathematical situations.

(LO3) To be able to apply a change of coordinates to simplify a linear map.

(LO4) To be able to work with matrix groups, in particular GL(n), O(n) and SO(n),.

(LO5) To understand bilinear forms from a geometric point of view.

(S1) Problem solving skills

(S2) Numeracy

(S3) Adaptability

##### Statistics and Probability I (MATH253)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**Use the R programming language fluently to analyse data, perform tests, ANOVA and SLR, and check assumptions.

Develop confidence to understand and use statistical methods to analyse and interpret data; check assumptions of these methods.

Develop an awareness of ethical issues related to the design of

studies.**Learning Outcomes**(LO1) An ability to apply advanced statistical concepts and methods covered in the module's syllabus to well defined contexts and interpret results.

(LO2) Use the R programming language fluently for a broad selection of statistical tests, in well-defined contexts.

(S1) Problem solving skills

(S2) Numeracy

(S3) IT skills

(S4) Communication skills

##### Newtonian Mechanics (MATH122)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To provide a basic understanding of the principles of Classical Mechanics and their application to simple dynamical systems. Learning Outcomes: After completing the module students should be able to analyse real world problems involving: - the motions of bodies under simple force systems - conservation laws for momentum and energy - rigid body dynamics using centre of mass, angular momentum and moments of inertia

**Learning Outcomes**(LO1) the motions of bodies under simple force systems

(LO2) conservation laws for momentum and energy

(LO3) rigid body dynamics using centre of mass, angular momentum and moments

(LO4) oscillation, vibration, resonance

(S1) Representing physical problems in a mathematical way

(S2) Problem Solving Skills

##### Numbers, Groups and Codes (MATH142)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**- To provide an introduction to rigorous reasoning in axiomatic systems exemplified by the framework of group theory.

- To give an appreciation of the utility and power of group theory as the study of symmetries.

- To introduce public-key cryptosystems as used in the transmission of confidential data, and also error-correcting codes used to ensure that transmission of data is accurate. Both of these ideas are illustrations of the application of algebraic techniques.

**Learning Outcomes**(LO1) Be able to apply the Euclidean algorithm to find the greatest common divisor of a pair of positive integers, and use this procedure to find the inverse of an integer modulo a given integer.

(LO2) Be able to solve linear congruences and apply appropriate techniques to solve systems of such congruences.

(LO3) Be able to perform a range of calculations and manipulations with permutations.

(LO4) Recall the definition of a group and a subgroup and be able to identify these in explicit examples.

(LO5) Be able to prove that a given mapping between groups is a homomorphism and identify isomorphic groups.

(LO6) To be able to apply group theoretic ideas to applications with error correcting codes.

(LO7) Engage in group project work to investigate applications of the theoretical material covered in the module.

##### Introduction to Statistics Using R (MATH163)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**1. Use software R to display and analyse data, perform tests and demonstrate basic statistical concepts.

2. Describe statistical data and display it using variety of plots and diagrams.

3. Understand basic laws of probability: law of total probability, independence, Bayes’ rule.

4. Be able to estimate mean and variance.

5. Be familiar with properties of some probability distributions and relations between them: Binomial, Poisson, Normal, t, Chi-squared.

6. To perform simple statistical tests: goodness-of-fit test, z-test, t-test.

7. To understand and be able to interpret p-values.

8. To be able to report finding of statistical outcomes to non-specialist audience.

9. Group work will help students to develop transferable skills such as communication, the ability to coordinate and prioritise tasks, time management and teamwork.

**Learning Outcomes**(LO1) An ability to apply statistical concepts and methods covered in the module's syllabus to well defined contexts and interpret results.

(LO2) An ability to understand, communicate, and solve straightforward problems related to the theory and derivation of statistical methods covered in the module's syllabus.

(LO3) An ability to understand, communicate, and solve straightforward theoretical and applied problems related to probability theory covered in the syllabus.

(LO4) Use the R programming language fluently in well-defined contexts. Students should be able to understand and correct given code; select appropriate code to solve given problems; select appropriate packages to solve given problems; and independently write small amounts of code.

##### Differential Equations (MATH221)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**•To familiarize students with basic ideas and fundamental techniques to solve ordinary differential equations.

•To illustrate the breadth of applications of ODEs and fundamental importance of related concepts.

**Learning Outcomes**(LO1) To understand the basic properties of ODE, including main features of initial value problems and boundary value problems, such as existence and uniqueness of solutions.

(LO2) To know the elementary techniques for the solution of ODEs.

(LO3) To understand the idea of reducing a complex ODE to a simpler one.

(LO4) To be able to solve linear ODE systems (homogeneous and non-homogeneous) with constant coefficients matrix.

(LO5) To understand a range of applications of ODE.

(S1) Problem solving skills

(S2) Numeracy

##### Numerical Methods for Applied Mathematics (MATH226)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**20:80 **Aims**To demonstrate how these ideas can be implemented using a high-level programming language, leading to accurate, efficient mathematical algorithms.

**Learning Outcomes**(LO1) To strengthen students’ knowledge of scientific programming, building on the ideas introduced in MATH111.

(LO2) To provide an introduction to the foundations of numerical analysis and its relation to other branches of Mathematics.

(LO3) To introduce students to theoretical concepts that underpin numerical methods, including fixed point iteration, interpolation, orthogonal polynomials and error estimates based on Taylor series.

(LO4) To demonstrate how analysis can be combined with sound programming techniques to produce accurate, efficient programs for solving practical mathematical problems.

(S1) Numeracy

(S2) Problem solving skills

##### Classical Mechanics (MATH228)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To provide an understanding of the principles of Classical Mechanics and their application to dynamical systems.

**Learning Outcomes**(LO1) To understand the variational principles, Lagrangian mechanics, Hamiltonian mechanics.

(LO2) To be able to use Newtonian gravity and Kepler's laws to perform the calculations of the orbits of satellites, comets and planetary motions.

(LO3) To understand the motion relative to a rotating frame, Coriolis and centripetal forces, motion under gravity over the Earth's surface.

(LO4) To understand the connection between symmetry and conservation laws.

(LO5) To be able to work with inertial and non-inertial frames.

(S1) Applying mathematics to physical problems

(S2) Problem solving skills

##### Metric Spaces and Calculus (MATH242)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To introduce the basic elements of the theory of metric spaces and calculus of several variables.

**Learning Outcomes**(LO1) After completing the module students should: Be familiar with a range of examples of metric spaces. Have developed their understanding of the notions of convergence and continuity.

(LO2) Understand the contraction mapping theorem and appreciate some of its applications.

(LO3) Be familiar with the concept of the derivative of a vector valued function of several variables as a linear map.

(LO4) Understand the inverse function and implicit function theorems and appreciate their importance.

(LO5) Have developed their appreciation of the role of proof and rigour in mathematics.

(S1) problem solving skills

##### Commutative Algebra (MATH247)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To give an introduction to abstract commutative algebra and show how it both arises naturally, and is a useful tool, in number theory.

**Learning Outcomes**(LO1) After completing the module students should be able to: • Work confidently with the basic tools of algebra (sets, maps, binary operations and equivalence relations). • Recognise abelian groups, different kinds of rings (integral, Euclidean, principal ideal and unique factorisation domains) and fields. • Find greatest common divisors using the Euclidean algorithm in Euclidean domains. • Apply commutative algebra to solve simple number-theoretic problems.

##### Statistics and Probability II (MATH254)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To introduce statistical distribution theory which forms the basis for all applications of statistics, and for further statistical theory.

**Learning Outcomes**(LO1) To have an understanding of basic probability calculus.

(LO2) To have an understanding of a range of techniques for solving real life problems of probabilistic nature.

(S1) Problem solving skills

(S2) Numeracy

##### Financial Mathematics (MATH260)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To provide an understanding of basic theories in Financial Mathematics used in the study process of actuarial/financial interest.

To provide an introduction to financial methods and derivative pricing financial instruments in discrete time set up.

**Learning Outcomes**(LO1) Know how to optimise portfolios and calculating risks associated with investment.

(LO2) Demonstrate principles of markets.

(LO3) Assess risks and rewards of financial products.

(LO4) Understand mathematical principles used for describing financial markets.

##### Numerical Methods for Applied Mathematics (MATH266)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**20:80 **Aims**To demonstrate how these ideas can be implemented using a high-level programming language, leading to accurate, efficient mathematical algorithms.

**Learning Outcomes**(LO1) To strengthen students’ knowledge of scientific programming, building on the ideas introduced in MATH111.

(LO2) To provide an introduction to the foundations of numerical analysis and its relation to other branches of Mathematics.

(LO3) To introduce students to theoretical concepts that underpin numerical methods, including fixed point iteration, interpolation, orthogonal polynomials and error estimates based on Taylor series.

(LO4) To demonstrate how analysis can be combined with sound programming techniques to produce accurate, efficient programs for solving practical mathematical problems.

(S1) Numeracy

(S2) Problem solving skills

##### Operational Research (MATH269)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**The aims of the module are to develop an understanding of how mathematical modelling and operational research techniques are applied to real-world problems and to gain an understanding of linear and convex programming, multi-objective problems, inventory control and sensitivity analysis.

**Learning Outcomes**(LO1) To understand the operational research approach.

(LO2) To be able to apply standard methods of operational research to a wide range of real-world problems as well as to problems in other areas of mathematics.

(LO3) To understand the advantages and disadvantages of particular operational research methods.

(LO4) To be able to derive methods and modify them to model real-world problems.

(LO5) To understand and be able to derive and apply the methods of sensitivity analysis.

(LO6) To understand the importance of sensitivity analysis.

(S1) Adaptability

(S2) Problem solving skills

(S3) Numeracy

(S4) Self-management readiness to accept responsibility (i.e. leadership), flexibility, resilience, self-starting, initiative, integrity, willingness to take risks, appropriate assertiveness, time management, readiness to improve own performance based on feedback/reflective learning

### Programme Year Three

Optional year abroad https://www.liverpool.ac.uk/modern-languages-and-cultures/study/year-abroad/

### Programme Year Four

This is Year Three if you choose not to take a year abroad. Choose six optional modules plus 30 Credits of language modules.

MATH399 may be taken in either semester, but not in both.

#### Year Four Optional Modules

##### Further Methods of Applied Mathematics (MATH323)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**•To give an insight into some specific methods for solving important types of ordinary differential equations.

•To provide a basic understanding of the Calculus of Variations and to illustrate the techniques using simple examples in a variety of areas in mathematics and physics.

•To build on the students'' existing knowledge of partial differential equations of first and second order.

**Learning Outcomes**(LO1) After completing the module students should be able to:

- use the method of "Variation of Arbitrary Parameters" to find the solutions of some inhomogeneous ordinary differential equations.- solve simple integral extremal problems including cases with constraints;

- classify a system of simultaneous 1st-order linear partial differential equations, and to find the Riemann invariants and general or specific solutions in appropriate cases;

- classify 2nd-order linear partial differential equations and, in appropriate cases, find general or specific solutions.

[This might involve a practical understanding of a variety of mathematics tools; e.g. conformal mapping and Fourier transforms.]##### Cartesian Tensors and Mathematical Models of Solids and VIscous Fluids (MATH324)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**To provide an introduction to the mathematical theory of viscous fluid flows and solid elastic materials. Cartesian tensors are first introduced. This is followed by modelling of the mechanics of continuous media. The module includes particular examples of the flow of a viscous fluid as well as a variety of problems of linear elasticity.

**Learning Outcomes**(LO1) To understand and actively use the basic concepts of continuum mechanics such as stress, deformation and constitutive relations.

(LO2) To apply mathematical methods for analysis of problems involving the flow of viscous fluid or behaviour of solid elastic materials.

(S1) Problem solving skills

(S2) Numeracy

(S3) Adaptability

##### Quantum Mechanics (MATH325)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**The aim of the module is to lead the student to an understanding of the way that relatively simple mathematics (in modern terms) led Bohr, Einstein, Heisenberg and others to a radical change and improvement in our understanding of the microscopic world.

**Learning Outcomes**(LO1) To be able to solve Schrodinger's equation for simple systems.

(LO2) To have an understanding of the significance of quantum mechanics for both elementary systems and the behaviour of matter.

(S1) Problem solving skills

(S2) Numeracy

##### Relativity (MATH326)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**(i) To introduce the physical principles behind Special and General Relativity and their main consequences;

(ii) To develop the competence in the mathematical framework of the subjects - Lorentz transformation and Minkowski space-time, semi-Riemannian geometry and curved space-time, symmetries and conservation laws, Variational principles.

(iii) To develop the understanding of the dynamics of particles and of the Maxwell field in Minkowski space-time, and of particles in curved space-time

(iv) To develop the knowledge of tests of General Relativity, including the classical tests (perihelion shift, gravitational deflection of light)

(v) To understand the basic concepts of black holes and (time permitting) relativistic cosmology and gravitational waves.

**Learning Outcomes**(LO1) To be proficient at calculations involving Lorentz transformations, the kinematical and dynamical quantities associated to particles in Minkowski space-times, and the application of the conservation law for the four-momentum to scattering processes.

(LO2) To know the relativistically covariant form of the Maxwell equations .

(LO3) To know the action principles for relativistic particles, the Maxwell field and the gravitational field.

(LO4) To be proficient at calculations in semi-Riemannian geometry as far as needed for General Relativity, including calculations involving general coordinate transformations, tensor fields, covariant derivatives, parallel transport, geodesics and curvature.

(LO5) To understand the arguments leading to the Einstein's field equations and how Newton's law of gravity arises as a limiting

(LO6) To be able to calculate the trajectories of bodies in a Schwarzschild space-time.

(S1) problem solving skills

(S2) numeracy

##### Number Theory (MATH342)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**35:65 **Aims**To give an account of elementary number theory with use of certain algebraic methods and to apply the concepts to problem solving.

**Learning Outcomes**(LO1) To understand and solve a wide range of problems about integers numbers.

(LO2) To have a better understanding of the properties of prime numbers.

(S1) Problem solving skills

(S2) Numeracy

(S3) Communication skills

##### Group Theory (MATH343)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To introduce the basic techniques of finite group theory with the objective of explaining the ideas needed to solve classification results.

**Learning Outcomes**(LO1) Understanding of abstract algebraic systems (groups) by concrete, explicit realisations (permutations, matrices, Mobius transformations).

(LO2) The ability to understand and explain classification results to users of group theory.

(LO3) The understanding of connections of the subject with other areas of Mathematics.

(LO4) To have a general understanding of the origins and history of the subject.

(S1) Problem solving skills

(S2) Logical reasoning

##### Differential Geometry (MATH349)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**This module is designed to provide an introduction to the methods of differential geometry, applied in concrete situations to the study of curves and surfaces in euclidean 3-space. While forming a self-contained whole, it will also provide a basis for further study of differential geometry, including Riemannian geometry and applications to science and engineering.

**Learning Outcomes**(LO1) 1a. Knowledge and understanding: Students will have a reasonable understanding of invariants used to describe the shape of explicitly given curves and surfaces.

(LO2) 1b. Knowledge and understanding: Students will have a reasonable understanding of special curves on surfaces.

(LO3) 1c. Knowledge and understanding: Students will have a reasonable understanding of the difference between extrinsically defined properties and those which depend only on the surface metric.

(LO4) 1d. Knowledge and understanding: Students will have a reasonable understanding of the passage from local to global properties exemplified by the Gauss-Bonnet Theorem.

(LO5) 2a. Intellectual abilities: Students will be able to use differential calculus to discover geometric properties of explicitly given curves and surfaces.

(LO6) 2b. Intellectual abilities: Students will be able to understand the role played by special curves on surfaces.

(LO7) 3a. Subject-based practical skills: Students will learn to compute invariants of curves and surfaces.

(LO8) 3b. Subject-based practical skills: Students will learn to interpret the invariants of curves and surfaces as indicators of their geometrical properties.

(LO9) 4a. General transferable skills: Students will improve their ability to think logically about abstract concepts,

(LO10) 4b. General transferable skills: Students will improve their ability to combine theory with examples in a meaningful way.

(S1) Problem solving skills

(S2) Numeracy

##### Applied Probability (MATH362)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of probabilistic model building for ‘‘dynamic" events occurring over time. To familiarise students with an important area of probability modelling.

**Learning Outcomes**(LO1) 1. Knowledge and Understanding After the module, students should have a basic understanding of:

(a) some basic models in discrete and continuous time Markov chains such as random walk and Poisson processes

(b) important subjects like transition matrix, equilibrium distribution, limiting behaviour etc. of Markov chain

(c) special properties of the simple finite state discrete time Markov chain and Poisson processes, and perform calculations using these.

2. Intellectual Abilities After the module, students should be able to:

(a) formulate appropriate situations as probability models: random processes

(b) demonstrate knowledge of standard models (c) demonstrate understanding of the theory underpinning simple dynamical systems

3. General Transferable Skills

(a) numeracy through manipulation and interpretation of datasets

(b) communication through presentation of written work and preparation of diagrams

(c) problem solving through tasks set in tutorials

(d) time management in the completion of practicals and the submission of assessed work

(e) choosing, applying and interpreting results of probability techniques for a range of different problems.##### Linear Statistical Models (MATH363)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**40:60 **Aims**- To understand how regression methods for continuous data extend to include multiple continuous and categorical predictors, and categorical response variables.

- To provide an understanding of how this class of models forms the basis for the analysis of experimental and also observational studies.

- To understand generalized linear models.

- To develop skills in using an appropriate statistical software package.

**Learning Outcomes**(LO1) Be able to understand the rationale and assumptions of linear regression and analysis of variance.

(LO2) Be able to understand the rationale and assumptions of generalized linear models.

(LO3) Be able to recognise the correct analysis for a given experiment.

(LO4) Be able to carry out and interpret linear regressions and analyses of variance, and derive appropriate theoretical results.

(LO5) Be able to carry out and interpret analyses involving generalised linear models and derive appropriate theoretical results.

(LO6) Be able to perform linear regression, analysis of variance and generalised linear model analysis using an appropriate statistical software package.

##### Statistical Physics (MATH327)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**1. To develop an understanding of the foundations of Statistical Physics normally including statistical ensembles and related extensive and intrinsic quantities.

2. To develop an understanding of the properties of classical and quantum gases and an appreciation of their applications to concepts such as the classical equation of state or the statistical

theory of photons.

3. To obtain a reasonable level of skill in using computer simulations for describing diffusion and transport in terms of stochastic processes.

4. To know the laws of thermodynamics and thermodynamical cycles.

5. To obtain a reasonable understanding of interacting statistical systems and related phenomenons such as phase transitions.**Learning Outcomes**(LO1) Demonstrate understanding of the microcanonical, canonical and grand canonical ensembles, their relation and the derived concepts of entropy, temperature and particle number

density.(LO2) Understand the derivation of the equation-of-state for non-interacting classical or quantum gases.

(LO3) Demonstrate numerical skills to understand diffusion from an underlying stochastic process.

(LO4) Know the laws of thermodynamics and demonstrate their application to thermodynamic cycles.

(LO5) Be aware of the effect of interactions including an understanding of the origin of phase transitions.

(S1) Problem solving skills

(S2) Numeracy

(S3) Adaptability

(S4) Communication skills

(S5) IT skills

(S6) Organisational skills

(S7) Teamwork

##### Game Theory (MATH331)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To explore, from a game-theoretic point of view, models which have been used to understand phenomena in which conflict and cooperation occur. To see the relevance of the theory not only to parlour games but also to situations involving human relationships, economic bargaining (between trade union and employer, etc), threats, formation of coalitions, war, etc. To treat fully a number of specific games including the famous examples of "The Prisoners' Dilemma" and "The Battle of the Sexes". To treat in detail two-person zero-sum and non-zero-sum games. To give a brief review of n-person games. In microeconomics, to look at exchanges in the absence of money, i.e. bartering, in which two individuals or two groups are involved.To see how the Prisoner's Dilemma arises in the context of public goods.

**Learning Outcomes**(LO1) To extend the appreciation of the role of mathematics in modelling in Economics and the Social Sciences.

(LO2) To be able to formulate, in game-theoretic terms, situations of conflict and cooperation.

(LO3) To be able to solve mathematically a variety of standard problems in the theory of games and to understand the relevance of such solutions in real situations.

##### Numerical Methods for Ordinary and Partial Differential Equations (MATH336)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**Many real-world systems in mathematics, physics and engineering can be described by differential equations. In rare cases these can be solved exactly by purely analytical methods, but much more often we can only solve the equations numerically, by reducing the problem to an iterative scheme that requires hundreds of steps. We will learn efficient methods for solving ODEs and PDEs on a computer.

**Learning Outcomes**(LO1) Demonstrate an advanced knowledge of the analysis of ODEs and PDEs underpinning the scientific programming within our context.

(LO2) Demonstrate an extended understanding of scientific programming and its application to numerical analysis and to other branches of Mathematics.

(LO3) Continuous engagement with putting practical problems into mathematical language.

(S1) Numeracy

(S2) Problem solving skills

(S3) Programming skills

##### Combinatorics (MATH344)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**50:50 **Aims**To provide an introduction to the problems and methods of Combinatorics, particularly to those areas of the subject with the widest applications such as pairings problems, the inclusion-exclusion principle, recurrence relations, partitions and the elementary theory of symmetric functions.

**Learning Outcomes**(LO1) After completing the module students should be able to: understand of the type of problem to which the methods of Combinatorics apply, and model these problems; solve counting and arrangement problems; solve general recurrence relations using the generating function method; appreciate the elementary theory of partitions and its application to the study of symmetric functions.

##### The Magic of Complex Numbers: Complex Dynamics, Chaos and the Mandelbrot Set (MATH345)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**1. To introduce students to the theory of the iteration of functions of one complex variable, and its fundamental objects;

2. To introduce students to some topics of current and recent research in the field;

3. To study various advanced results from complex analysis, and show how to apply these in a dynamical setting;

4. To illustrate that many results in complex analysis are "magic", in that there is no reason to expect them in a real-variable context, and the implications of this in complex dynamics;

5. To explain how complex-variable methods have been instrumental in questions purely about real-valued one-dimensional dynamical systems, such as the logistic family.

6. To deepen students' appreciations for formal reasoning and proof. After completing the module, students should be able to:

1. understand the compactification of the complex plane to the Riemann sphere, and use spherical distances and derivatives.

2. use Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems.

3. state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties.

4. determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set.

5. apply advanced results from complex analysis in the setting of complex dynamics.

6. determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not.**Learning Outcomes**(LO1) To understand the compactification of the complex plane to the Riemann sphere, and be able to use spherical distances and derivatives.

(LO2) To be able to use Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems.

(LO3) To be able to state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties.

(LO4) To be able to determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set.

(LO5) To know how to apply advanced results from complex analysis in a dynamical setting.

(LO6) To be able to determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not.

(S1) Problem solving/ critical thinking/ creativity analysing facts and situations and applying creative thinking to develop appropriate solutions.

(S2) Problem solving skills

##### Topology (MATH346)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**1. To introduce students to the mathematical notions of space and continuity.

2. To develop students’ ability to reason in an axiomatic framework.

3. To provide students with a foundation for further study in the area of topology and geometry, both within their degree and subsequently.

4. To introduce students to some basic constructions in topological data analysis.

5. To enhance students’ understanding of mathematics met elsewhere within their degree (in particular real and complex analysis, partial orders, groups) by placing it within a broader context.

6. To deepen students’ understanding of mathematical objects commonly discussed in popular and recreational mathematics (e.g. Cantor sets, space-filling curves, real surfaces).**Learning Outcomes**(BH1) An understanding of the ubiquity of topological spaces within mathematics.

(BH2) Knowledge of a wide range of examples of topological spaces, and of their basic properties.

(BH3) The ability to construct proofs of, or counter-examples to, simple statements about topological spaces and continuous maps.

(BH4) The ability to decide if a (simple) space is connected and/or compact.

(BH5) The ability to construct the Cech and Vietoris-Rips complexes of a point set in Euclidean spac. e

(BH6) The ability to compute the fundamental group of a (simple) space, and to use it to distinguish spaces.

##### Applied Stochastic Models (MATH360)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To give examples of empirical phenomena for which stochastic processes provide suitable mathematical models. To provide an introduction to the methods of stochastic model building for 'dynamic' events occurring over time or space. To enable further study of the theory of stochastic processes by using this course as a base.

**Learning Outcomes**(LO1) To understand the theory of continuous-time Markov chains.

(LO2) To understand the theory of diffusion processes.

(LO3) To be able to solve problems arising in epidemiology, mathematical biology, financial mathematics, etc. using the theory of continuous-time Markov chains and diffusion processes.

(LO4) To acquire an understanding of the standard concepts and methods of stochastic modelling.

(S1) Problem solving skills

(S2) Numeracy

##### Theory of Statistical Inference (MATH361)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**To introduce some of the concepts and principles which provide theoretical underpinning for the various statistical methods, and, thus, to consolidate the theory behind the other second year and third year statistics options.

**Learning Outcomes**(LO1) To acquire a good understanding of the classical approach to, and especially the likelihood methods for, statistical inference.

(LO2) To acquire an understanding of the blossoming area of Bayesian approach to inference.

(S1) Problem solving skills

(S2) Numeracy

##### Medical Statistics (MATH364)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**The aims of this module are to: demonstrate the purpose of medical statistics and the role it plays in the control of disease and promotion of health explore different epidemiological concepts and study designs apply statistical methods learnt in other programmes, and some new concepts, to medical problems and practical epidemiological research enable further study of the theory of medical statistics by using this module as a base.

**Learning Outcomes**(LO1) identify the types of problems encountered in medical statistics

(LO2) demonstrate the advantages and disadvantages of different epidemiological study designs

(LO3) apply appropriate statistical methods to problems arising in epidemiology and interpret results

(LO4) explain and apply statistical techniques used in survival analysis

(LO5) critically evaluate statistical issues in the design and analysis of clinical trials

(LO6) discuss statistical issues related to systematic review and apply appropriate methods of meta-analysis

(LO7) apply Bayesian methods to simple medical problems.

(S1) Problem solving skills

##### Measure Theory and Probability (MATH365)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**The main aim is to provide a sufficiently deepintroduction to measure theory and to the Lebesgue theory of integration. Inparticular, this module aims to provide a solid background for the modernprobability theory, which is essential for Financial Mathematics.

**Learning Outcomes**(LO1) After completing the module students should be ableto:

(LO2) master the basic results about measures and measurable functions;

(LO3) master the basic results about Lebesgue integrals and their properties;

(LO4) to understand deeply the rigorous foundations ofprobability theory;

(LO5) to know certain applications of measure theoryto probability, random processes, and financial mathematics.

(S1) Problem solving skills

(S2) Logical reasoning

##### Mathematical Risk Theory (MATH366)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**50:50 **Aims**•To provide an understanding of the mathematical risk theory used in the study process of actuarial interest

• To provide an introduction to mathematical methods for managing the risk in insurance and finance (calculation of risk measures/quantities)

• To develop skills of calculating the ruin probability and the total claim amount distribution in some non‐life actuarial risk models with applications to insurance industry

• To prepare the students adequately and to develop their skills in order to be ready to sit for the exams of CT6 subject of the Institute of Actuaries (MATH366 covers 50% of CT6 in much more depth).

**Learning Outcomes**(LO1) After completing the module students should be able to:

(a) Define the loss/risk function and explain intuitively the meaning of it, describe and determine optimal strategies of game theory, apply the decision criteria's, be able to decide a model due to certain model selection criterion, describe and perform calculations with Minimax and Bayes rules.

(b) Understand the concept (and the mathematical assumptions) of the sums of independent random variables, derive the distribution function and the moment generating function of finite sums of independent random variables.

(c) Define and explain the compound Poisson risk model, the compound binomial risk model, the compound geometric risk model and be able to derive the distribution function, the probability function, the mean, the variance, the moment generating function and the probability generating function for exponential/mixture of exponential severities and gamma (Erlang) severities, be able to calculate the distribution of sums of independent compound Poisson random variables.

(d) Understand the use of convolutions and compute the distribution function and the probability function of the compound risk model for aggregate claims using convolutions and recursion relationships.

(e) Define the stop‐loss reinsurance and calculate the (mean) stop‐loss premium for exponential and mixtures of exponential severities, be able to compare the original premium and the stoploss premium in numerical examples.

(f) Understand and be able to use Panjer's equation when the number of claims belongs to theR(a, b, 0) class of distributions, use the Panjer's recursion in order to derive/evaluate the probability function for the total aggregate claims.

(g) Explain intuitively the individual risk model, be able to calculate the expected losses (as well as the variance) of group life/non‐life insurance policies when the benefits of the each person of the group are assumed to have deterministic variables.

(h) Derive a compound Poisson approximations for a group of insurance policies (individual risk model as approximation),

(i) Understand/describe the classical surplus process ruin model and calculate probabilities of the number of the risks appearing in a specific time period, under the assumption of the Poisson process.

(j) Derive the moment generating function of the classical compound Poisson surplus process, calculate and explain the importance of the adjustment coefficient, also be able to make use of Lundberg's inequality for exponential and mixtures of exponential claim severities.

(k) Derive the analytic solutions for the probability of ruin, psi(u), by solving the corresponding integro‐differential equation for exponential and mixtures of exponential claim amount severities,

(l) Define the discrete time surplus process and be able to calculate the infinite ruin probability, psi(u,t) in numerical examples (using convolutions).

(m) Derive Lundberg's equation and explain the importance of the adjustment coefficient under the consideration of reinsurance schemes.

(n) Understand the concept of delayed claims and the need for reserving, present claim data as a triangle (most commonly used method), be able to fill in the lower triangle by comparing present data with past (experience) data.

(o) Explain the difference and adjust the chain ladder method, when inflation is considered.

(p) Describe the average cost per claim method and project ultimate claims, calculate the required reserve (by using the claims of the data table).

(q) Use loss ratios to estimate the eventual loss and hence outstanding claims.

(r) Describe the Bornjuetter‐Ferguson method (be able to understand the combination of the estimated loss ratios with a projection method). Use the aforementioned method to calculate the revised ultimate losses (by making use of the credibility factor).##### Networks in Theory and Practice (MATH367)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**•To develop an appreciation of network models for real world problems.

•To describe optimisation methods to solve them.

•To study a range of classical problems and techniques related to network models.

**Learning Outcomes**(LO1) After completing the module students should be able to model problems in terms of networks and be able to apply effectively a range of exact and heuristic optimisation techniques.

The programme detail and modules listed are illustrative only and subject to change.

#### Teaching and Learning

Your learning activities will consist of lectures, tutorials, practical classes, problem classes, private study and supervised project work. In Year One, lectures are supplemented by a thorough system of group tutorials and computing work is carried out in supervised practical classes. Key study skills, presentation skills and group work start in first-year tutorials and are developed later in the programme. The emphasis in most modules is on the development of problem solving skills, which are regarded very highly by employers. Project supervision is on a one-to-one basis, apart from group projects in Year Two.

#### Assessment

Most modules are assessed by a two and a half hour examination in January or May, but many have an element of coursework assessment. This might be through homework, class tests, mini-project work or key skills exercises.