# Mathematics MMath

- Course length: 4 years
- UCAS code: G101
- Year of entry: 2019
- Typical offer: A-level : AAB / IB : 35 / BTEC : Applications considered

## 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

#### Year One Compulsory Modules

##### Calculus I (MATH101)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **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**differentiate and integrate a wide range of functions;

sketch graphs and solve problems involving optimisation and mensuration

understand the notions of sequence and series and apply a range of tests to determine if a series is convergent

##### Calculus II (MATH102)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **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**use Taylor series to obtain local approximations to functions;

obtain partial derivaties and use them in several applications such as, error analysis, stationary points change of variables

evaluate double integrals using Cartesian and Polar Co-ordinates

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

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **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**manipulate complex numbers and solve simple equations involving them

solve arbitrary systems of linear equations

understand and use matrix arithmetic, including the computation of matrix inverses

compute and use determinants

understand and use vector methods in the geometry of 2 and 3 dimensions

calculate eigenvalues and eigenvectors and, if time permits, apply these calculations to the geometry of conics and quadrics

##### Numbers and Sets (MATH105)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**1. To bridge the gap in language and philosophy between A-level and University mathematics.

2. To train students to think clearly and logically, and to appreciate the nature of definitions, theorems, and proofs.

3. To give an appreciation of the richness and importance of the structures of the integer, rational, real and complex number systems.

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

1. Use mathematical language and symbols accurately;

Understand the nature of a definition, and show that simple definitions are or are not satisfied by given examples;

Use theorems to draw logical conclusions from given information

Understand the logic of direct proofs and proofs by contradiction, and construct very simple proofs, including proofs by induction;

Interpret statements involving quantifiers, and negate statements with one or two quantifiers

Use the language of naive set theory

Understand the integer, rational, real and complex number systems and the relationship between them.

##### Newtonian Mechanics (MATH122)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **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**

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

oscillation, vibration, resonance

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

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **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** Use the division algorithm to construct the greatest common divisor of a pair of positive integers;

Solve linear congruences and find the inverse of an integer modulo a given integer;

Code and decode messages using the public-key method

Manipulate permutations with confidence

Decide when a given set is a group under a specified operation and give formal axiomatic proofs;

Understand the concept of a subgroup and use Lagrange''s theorem;

Understand the concept of a group action, an orbit and a stabiliser subgroup

Understand the concept of a group homomorphism and be able to show (in simple cases) that two groups are isomorphic;

##### Introduction to Statistics (MATH162)

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **Aims**To introduce topics in Statistics and to describe and discuss basic statistical methods.

To describe the scope of the application of these methods.

**Learning Outcomes**to describe statistical data;

to use the Binomial, Poisson, Exponential and Normal distributions;

to perform simple goodness-of-fit tests

to use the package Minitab to present data, and to make statistical analysis

#### Year One Optional Modules

##### Introduction to Programming (COMP101)

**Level**1 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**0:100 **Aims**- To introduce the concepts and principles of problem solving using computational thinking.
- To identify and employ algorithms in the solution of identified problems.
- To develop sound principles in designing programming solutions to identified problems using appropriate data structures.
- To introduce the concepts of implementing solutions in a high level programming language.

**Learning Outcomes**Identify the principles and practice of using high-level programming constructs to solve a problem

Use relevant data structures to solve problems

Produce documentation in support of a programmed solution

Use a suitable Integrated Development Environment to carry out Implementation, interpretation/compilation, testing and execution.

Identify appropriate design approaches to formulate a solution to a program

Design and apply effective test cases

Develop debugging skills to correct a program

Specific learning outcomes are listed above.

General learning outcomes:

- An understanding of the principles and practice of analysis and design in the construction of robust, maintainable programs which satisfy their requirements;
- A competence to design, write, compile, test and execute straightforward programs using a high-level language;
- An appreciate of the principles of procedural programming;
- An awareness of the need for a professional approach to design and the importance of good documentation to the finished programs.

##### 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 employability skills.

**Learning Outcomes**After completing the module, students should be able to

- tackle project work, including writing up of reports detailing their solutions to problems;

- use computers to create documents containing formulae, tables, plots and references;

- use mathematical software packages such as Maple and Matlab to manipulate mathematical expressions and to solve simple problems,

- better understand the mathematical topics covered, through direct experimentation with the computer.

After completing the module, students should be able to

- list skills required by recruiters of graduates in mathematical sciences;

- recognise what constitutes evidence for those skills;

- identify their own skills gaps and plan to develop their skills.

### Programme Year Two

In the second and subsequent years of study, there is a wide range of modules. For the programme that you choose there may be no compulsory modules (although you may have to choose a few from a subset such as Pure Mathematics). If you make a different choice, you will find that one or more modules have to be taken. Each year you will choose a number of optional modules. Please note that we regularly review our teaching so the choice of modules may change.

#### Year Two Compulsory Modules

##### Ordinary Differential Equations (MATH201)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**75:25 **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**After completing the module students should be:

- familiar with elementary techniques for the solution of ODE''s, and the idea of reducing a complex ODE to a simpler one;

- familiar with basic properties of ODE, including main features of initial value problems and boundary value problems, such as existence and uniqueness of solutions;

- well versed in the solution of linear ODE systems (homogeneous and non-homogeneous) with constant coefficients matrix;

- aware of a range of applications of ODE.

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

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**To introduce general concepts of linear algebra and its applications in geometry and other areas of mathematics.

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

• appreciate the geometric meaning of linear algebraic ideas,

• appreciate the concept of an abstract vector space and how it is used in different mathematical situations,

• apply a change of coordinates to simplify a linear map,

• manipulate matrix groups, in particular GL(n), O(n) and SO(n),

• understand bilinear forms from a geometric point of view.

#### Year Two Optional Modules

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

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**85:15 **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**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.

##### Mathematical Models: Microeconomics and Population Dynamics (MATH227)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**1. To provide an understanding of the techniques used in constructing, analysing, evaluating and interpreting mathematical models.

2. To do this in the context of two non-physical applications, namely microeconomics and population dynamics.

3. To use and develop mathematical skills introduced in Year 1 - particularly the calculus of functions of several variables and elementary differential equations.

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

- Use techniques from several variable calculus in tackling problems in microeconomics.

- Use techniques from elementary differential equations in tackling problems in population dynamics.

- Apply mathematical modelling methodology in these subject areas.

All learning outcomes are assessed by both examination and course work.

##### Metric Spaces and Calculus (MATH241)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**To introduce the basic elements of the theories of metric spaces and calculus of several variables.

**Learning Outcomes**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.

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

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

Understand the inverse function and implicit function theorems and appreciate their importance.

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

##### Complex Functions (MATH243)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**80:20 **Aims**To introduce the student to a surprising, very beautiful theory which has intimate connections with other areas of mathematics and physical sciences, for instance ordinary and partial differential equations and potential theory.

**Learning Outcomes**After completing this module students should:

- appreciate the central role of complex numbers in mathematics;

- be familiar with all the classical holomorphic functions;

- be able to compute Taylor and Laurent series of such functions;

- understand the content and relevance of the various Cauchy formulae and theorems;

- be familiar with the reduction of real definite integrals to contour integrals;

- be competent at computing contour integrals.

##### Introduction to Methods of Operational Research (MATH261)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**- Appreciate the operational research approach.
- Be able to apply standard methods to a wide range of real-world problems as well as applications in other areas of mathematics.
- Appreciate the advantages and disadvantages of particular methods.
- Be able to derive methods and modify them to model real-world problems.
- Understand and be able to derive and apply the methods of sensitivity analysis.

**Learning Outcomes**Appreciate the operational research approach.Be able to apply standard methods to a wide range of real-world problems as well asapplications in other areas of mathematics.

Appreciate the advantages and disadvantages of particular methods.

Be able to derive methods and modify them to model real-world problems.Understand and be able to derive and apply the methods of sensitivity analysis. Appreciate the importance of sensitivity analysis.

##### Theory of Interest (MATH267)

**Level**2 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**This module aims to provide students with an understanding of the fundamental concepts of Financial Mathematics, and how the concepts above are applied in calculating present and accumulated values for various streams of cash flows. Students will also be given an introduction to financial instruments, such as derivatives and the concept of no-arbitrage.

**Learning Outcomes**To understand and calculate all kinds of rates of interest, find the future value and present value of a cash flow and to write the equation of value given a set of cash flows and an interest rate.

To derive formulae for all kinds of annuities.

To understand an annuity with level payments, immediate (or due), payable m-thly, (or payable continuously) and any three of present value, future value, interest rate, payment, and term of annuity as well as to calculate the remaining two items.

To calculate the outstanding balance at any point in time.

To calculate a schedule of repayments under a loan and identify the interest and capital components in a given payment.

To calculate a missing quantity, being given all but one quantities, in a sinking fund arrangement.

To calculate the present value of payments from a fixed interest security, bounds for the present value of a redeemable fixed interest security.

Given the price, to calculate the running yield and redemption yield from a fixed interest security.

To calculate the present value or real yield from an index-linked bond.

To calculate the price of, or yield from, a fixed interest security where the income tax and capital gains tax are implemented.

To calculate yield rate, the dollar-weighted and time weighted rate of return, the duration and convexity of a set of cash flows.

To describe the concept of a stochastic interest rate model and the fundamental distinction between this and a deterministic model.

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

**Level**1 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**80:20 **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** Use the division algorithm to construct the greatest common divisor of a pair of positive integers;

Solve linear congruences and find the inverse of an integer modulo a given integer;

Code and decode messages using the public-key method

Manipulate permutations with confidence

Decide when a given set is a group under a specified operation and give formal axiomatic proofs;

Understand the concept of a subgroup and use Lagrange''s theorem;

Understand the concept of a group action, an orbit and a stabiliser subgroup

Understand the concept of a group homomorphism and be able to show (in simple cases) that two groups are isomorphic;

##### Operational Research: Probabilistic Models (MATH268)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To introduce a range of models and techniques for solving under uncertainty in Business, Industry, and Finance.

**Learning Outcomes**The ability to understand and describe mathematically real-life optimization problems.

Understanding the basic methods of dynamical decision making.

Understanding the basics of forecasting and simulation.

The ability to analyse elementary queueing systems.

##### Group Project Module (MATH206)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**0:100 **Aims**· To give students experience of working effectively in small groups.

· To train students to write about mathematics.

· To give students practice in delivering presentations.

· To develop students’ ability to study independently.

· To prepare students for later individual project work.

· To enhance students’ appreciation of the connections between different areas of mathematics.

· To encourage students to discuss mathematics with each other.

**Learning Outcomes**Work effectively in groups, and delegate common tasks.

Write substantial mathematical documents in an accessible form.Give coherent verbal presentations of more advanced mathematical topics.

Appreciate how mathematical techniques can be applied in a variety of different contexts##### Introduction to the Methods of Applied Mathematics (MATH224)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To provide a grounding in elementary approaches to solution of some of the standard partial differential equations encountered in the applications of mathematics.

To introduce some of the basic tools (Fourier Series) used in the solution of differential equations and other applications of mathematics.

**Learning Outcomes**After completing the module students should:

- be fluent in the solution of basic ordinary differential equations, including systems of first order equations;

- be familiar with the concept of Fourier series and their potential application to the solution of both ordinary and partial differential equations;

- be familiar with the concept of Laplace transforms and their potential application to the solution of both ordinary and partial differential equations;

- be able to solve simple first order partial differential equations;

- be able to solve the basic boundary value problems for second order linear partial differential equations using the method of separation of variables.

##### Classical Mechanics (MATH228)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To provide an understanding of the principles of Classical Mechanics and their application to dynamical systems.

**Learning Outcomes**Understanding of variational principles, Lagrangian mechanics, Hamiltonian mechanics.

Newtonian gravity and Kepler''s laws, including calculations of the orbits of satellites, comets and planetary motions

Motion relative to a rotating frame, Coriolis and centripetal forces, motion under gravity over the Earth''s surface

Connection between symmetry and conservation laws

Inertial and non-inertial frames.

##### Commutative Algebra (MATH247)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **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**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.

##### Geometry of Curves (MATH248)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To introduce geometric ideas and develop the basic skills in handling them.

To study the line, circle, ellipse, hyperbola, parabola, cubics and many other curves.

To study theoretical aspects of parametric, algebraic and projective curves.

To study and sketch curves using an appropriate computer package.

**Learning Outcomes**After completing this module students should be able to:

- use a computer package to study curves and their evolution in both parametric and algebraic forms.

-determine and work with tangents, inflexions, curvature, cusps, nodes, length and other features.

-calculate envelopes and evolutes.

- solve the position and shape of some algebraic curves including conics.

The first learning outcome is assessed by coursework, the others by both coursework and examination.

##### Financial Mathematics (MATH262)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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,
- to gain understanding of some financial models with applications to financial/insurance industry,
- to prepare the students adequately and to develop their skills in order to be ready to sit the CT1 & CT8 subject of the Institute of Actuaries (the module covers the material of CT8 and 20% of CT1).

**Learning Outcomes**To understand the assumptions of the Capital Asset Pricing Model (CAPM), to be able to explain the no riskless lending or borrowing and other lending and borrowing assumptions, to be able to use the formulas of CAPM, to be able to derive the capital market line and security market line.

To be able to describe the Arbitrage Theory Model (APT) and explain its assumptions as well as perform estimating and testing in APT

To be able to explain the terms long/short position, spot/delivery/forward price, understand the use of future contracts, describe what a call/put option (European/American) is as well as be able to create graphs and explain their payouts, describe the hedging for reducing the exposure to risk, to be able to explain arbitrage, understand the mechanism of short sales.

To be able to explain/describe what arbitrage is, what the risk neutral probability measure is, as well as to be able to use (and perform calculation) the binomial tree for European and American style options.To understand the probabilistic interpretation and the basic concept of the random walk of asset pricing.

To understand the concepts of replication, hedging, and delta hedging in continuous time.

To be able to use Ito''s formula, derive/use the Black‐Scholes formula, price contingent claims (in particular European/American style options and forward contracts), to be able to explain the properties of the Black‐Scholes formula and to be able to use the Normal distribution function in numerical examples of pricing,

To understand the role of Greeks, to be able to describe intuitively what Delta, Theta, Gamma is, and to calculate them in numerical examples.##### Statistical Theory and Methods I (MATH263)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**85:15 **Aims**To introduce statistical methods with a strong emphasis on applying standard statistical techniques appropriately and with clear interpretation. The emphasis is on applications.

**Learning Outcomes**After completing the module students should have a conceptual and practical understanding of a range of commonly applied statistical procedures. They should have also developed some familiarity with the statistical package MINITAB.

##### Statistical Theory and Methods II (MATH264)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To introduce statistical distribution theory which forms the basis for all applications of statistics, and for further statistical theory.

**Learning Outcomes**After completing the module students should understand basic probability calculus. They should be familiar with a range of techniques for solving real life problems of the probabilistic nature.

##### Numerical Methods (MATH266)

**Level**2 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **Aims**To provide an introduction to the main topics in Numerical Analysis and their relation to other branches of Mathematics

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

• write simple mathematical computer programs in Maple,

• understand the consequences of using fixed-precision arithmetic,

• analyse the efficiency and convergence rate of simple numerical methods,

• develop and implement algorithms for solving nonlinear equations,

• develop quadrature methods for numerical integration,

• apply numerical methods to solve systems of linear equations and to calculate eigenvalues and eigenvectors,

• solve boundary and initial value problems using finite difference methods.

##### Mathematics Education and Communication (MATH291)

**Level**2 **Credit level**15 **Semester**Whole Session **Exam:Coursework weighting**0:100 **Aims**Improving communication skills.

Exposing students to current pedagogical practice and issues related to child protection

Encouraging students to reflect on mathematics with which they are familiar in a teaching context.

**Learning Outcomes**Confidence in planning and presenting mathematics to school-age children.

Knowledge of current best pedagogical practice and child protection issues.

Ability to work in a team.

Understanding the role of outreach in mathematics education.

### Programme Year Three

Choose 5 modules from the options

- Further Methods of Applied Mathematics (MATH323)
- Cartesian Tensors and Mathematical Models of Solids and VIscous Fluids (MATH324)
- Quantum Mechanics (MATH325)
- Population Dynamics (MATH332)
- Group Theory (MATH343)
- Combinatorics (MATH344)
- Analysis and Number Theory (MATH351)
- Applied Probability (MATH362)
- Linear Statistical Models (MATH363)
- Measure Theory and Probability (MATH365)
- Networks in Theory and Practice (MATH367)
- Chaos and Dynamical Systems (MATH322)
- Relativity (MATH326)
- Mathematical Economics (MATH331)
- Riemann Surfaces (MATH340)
- Number Theory (MATH342)
- The Magic of Complex Numbers: Complex Dynamics, Chaos and the Mandelbrot Set (MATH345)
- Differential Geometry (MATH349)
- Applied Stochastic Models (MATH360)
- Theory of Statistical Inference (MATH361)
- Medical Statistics (MATH364)
- Mathematical Risk Theory (MATH366)
- Projects in Mathematics (MATH399)

Choose 3 modules from the options

- Linear Differential Operators in Mathematical Physics (MATH421)
- Quantum Field Theory (MATH425)
- Variational Calculus and Its Applications (MATH430)
- Introduction to String Theory (MATH423)
- Analytical & Computational Methods for Applied Mathematics (MATH424)
- Advanced Topics in Mathematical Biology (MATH426)
- Waves, Mathematical Modelling (MATH427)
- Asymptotic Methods for Differential Equations (MATH433)
- Stochastic Analysis and Its Applications (MATH483)
- Project for M.math (MATH499)

#### Year Three 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**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**100:0 **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**After completing the module, students should be able to understand and actively use the basic concepts of continuum mechanics such as stress, deformation and constitutive relations, and apply mathematical methods for analysis of problems involving the flow of viscous fluid or behaviour of solid elastic materials.

##### Quantum Mechanics (MATH325)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**The development of Quantum Mechanics, requiring as it did revolutionary changes in our understanding of the nature of reality, was arguably the greatest conceptual achievement of all time. The aim of the module is to lead the student to an understanding of the way that relatively simple mathemactics (in modern terms) led Bohr, Einstein, Heisenberg and others to a radical change and improvement in our understanding of the microscopic world.

**Learning Outcomes**After completing the module students should be able to solve Schrodinger''s equation for simple systems, and have some intuitive understanding of the significance of quantum mechanics for both elementary systems and the behaviour of matter.

##### Population Dynamics (MATH332)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**- To provide a theoretical basis for the understanding of population ecology

- To explore the classical models of population dynamics

- To learn basic techniques of qualitative analysis of mathematical models

**Learning Outcomes**The ability to relate the predictions of the mathematical models to experimental results obtained in the field.The ability to recognise the limitations of mathematical modelling in understanding the mechanics of complex biological systems.The ability to use analytical and graphical methods to investigate population growth and the stability of equilibrium states for continuous-time and discrete-time models of ecological systems. ##### Group Theory (MATH343)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**To introduce the basic techniques of finite group theory with the objective of explaining the ideas needed to solve classification results.

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

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

The understanding of connections of the subject with other areas of Mathematics.

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

##### Combinatorics (MATH344)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **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**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.

##### Analysis and Number Theory (MATH351)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To provide an introduction to uniform distribution modulo one as a meeting ground for other branches of pure mathematics such as topology, analysis and number theory.

To give an idea of how these subjects work together in this context to shape modern analysis by focusing on one particular rich context.

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

understand completions and irrationality;

understand diophantine approximation and its relation to uniform distribution;

appreciate that analysis has a complex unity and to have a feel for basic computations in analysis.

calculate rational approximations to real and p adic numbers and to put this to use in number theoretic situations.

calculate approximations to functions from families of simpler functions e.g. continuous functions by polynomials and step functions by trigonometric sums.

work with basic tools from analysis, like Fourier series and continuous functions to prove distributional properties of sequences of numbers.

##### Applied Probability (MATH362)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **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 occuring over time. To familiarise students with an important area of probability modelling.

**Learning Outcomes**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**100:0 **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 familiarity with the computer package SPSS.

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

understand the rationale and assumptions of linear regression and analysis of variance.

· understand the rationale and assumptions of generalized linear models.

· recognise the correct analysis for a given experiment.

· carry out and interpret linear regressions and analyses of variance, and derive appropriate theoretical results.

· carry out and interpret analyses involving generalised linear models and derive appropriate theoretical results.

· perform linear regression, analysis of variance and generalised linear model analysis using the SPSS computer package.

##### 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**After completing the module students should be ableto:

master the basic results about measures and measurable functions;

master the basic results about Lebesgue integrals and their properties;to understand deeply the rigorous foundations ofprobability theory;

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

##### 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**After completing the module students should

. be able to model problems in terms of networks.

· be able to apply effectively a range of exact and heuristic optimisation techniques.

##### Chaos and Dynamical Systems (MATH322)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To develop expertise in dynamical systems in general and study particular systems in detail.

**Learning Outcomes**After completing the module students will be able to understand the possible behaviour of dynamical systems with particular attention to chaotic motion;

After completing the module students will be familiar with techniques for extracting fixed points and exploring the behaviour near such fixed points;

After completing the module students will understand how fractal sets arise and how to characterise them.

##### Relativity (MATH326)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To impart

(i) a firm grasp of the physical principles behind Special and General Relativity and their main consequences;

(ii) technical competence in the mathematical framework of the subjects - Lorentz transformation, coordinate transformations and geodesics in Riemann space;

(iii) knowledge of some of the classical tests of General Relativity - perihelion shift, gravitational deflection of light;

(iv) basic concepts of black holes and (if time) relativistic cosmology.

**Learning Outcomes**After completing this module students should

(i) understand why space-time forms a non-Euclidean four-dimensional manifold;

(ii) be proficient at calculations involving Lorentz transformations, energy-momentum conservation, and the Christoffel symbols.

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

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

##### Mathematical Economics (MATH331)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**After completing the module students should:

· Have further extended their appreciation of the role of mathematics in modelling in Economics and the Social Sciences.

· Be able to formulate, in game-theoretic terms, situations of conflict and cooperation.

· Be able to solve mathematically a variety of standard problems in the theory of games.

· To understand the relevance of such solutions in real situations.

##### Riemann Surfaces (MATH340)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To introduce to a beautiful theory at the core of modern mathematics. Students will learn how to handle some abstract geometric notions from an elementary point of view that relies on the theory of holomorphic functions. This will provide those who aim to continue their studies in mathematics with an invaluable source of examples, and those who plan to leave the subject with the example of a modern axiomatic mathematical theory.

**Learning Outcomes**Students should be familiar with themost basic examples of Riemann surfaces: the Riemann sphere, hyperelliptic Riemann surfaces, and smooth plane algebraic curves.

Students should understand and be able to use the abstract notions used to build the theory: holomorphic maps, meromorphic differentials, residues and integrals, Euler characteristic and genus.

##### Number Theory (MATH342)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**After completing this module students should be able to understand and solve a wide range of problems about the integers, and have a better understanding of the properties of prime numbers.

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

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**will understand the compactification of the complex plane to the Riemann sphere, and be able to use spherical distances and derivatives

will be able to use Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems

will be able to state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties

will be able to determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set

will know how to apply advanced results from complex analysis in a dynamical setting

will be able to determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not

##### Differential Geometry (MATH349)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**85:15 **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**1. Knowledge and understanding

After the module, students should have a basic understanding of

a) invariants used to describe the shape of explicitly given curves and surfaces,

b) special curves on surfaces,

c) the difference between extrinsically defined properties and those which depend only on the surface metric,

d) understanding the passage from local to global properties exemplified by the Gauss-Bonnet Theorem.

2. Intellectual abilities

After the module, students should be able to

a) use differential calculus to discover geometric properties of explicitly given curves and surface,

b) understand the role played by special curves on surfaces.

3. Subject-based practical skills

Students should learn to

a) compute invariants of curves and surfaces,

b) interpret the invariants of curves and surfaces as indicators of their geometrical properties.

4. General transferable skills

Students will improve their ability to

a) think logically about abstract concepts,

b) combine theory with examples in a meaningful way.

##### Applied Stochastic Models (MATH360)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**After completing the module students should have a grounding in the theory of continuous-time Markov chains and diffusion processes. They should be able to solve corresponding problems arising in epidemiology, mathematical biology, financial mathematics, etc.

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

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **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**After completing the module students should have a good understanding of the classical approach to, and especially the likelihood methods for, statistical inference.

The students should also gain an appreciation of the blossoming area of Bayesian approach to inference

##### Medical Statistics (MATH364)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**identify the types of problems encountered in medical statistics

demonstrate the advantages and disadvantages of different epidemiological study designs

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

explain and apply statistical techniques used in survival analysis

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

discuss statistical issues related to systematic review and apply appropriate methods of meta-analysisapply Bayesian methods to simple medical problems.

##### Mathematical Risk Theory (MATH366)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**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 the

R(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).

##### Projects in Mathematics (MATH399)

**Level**3 **Credit level**15 **Semester**Whole Session **Exam:Coursework weighting**0:100 **Aims**a) To study in depth an area of pure mathematics and report on it; or

b) To construct and study mathematical models of a chosen problem; or

c) To demonstrate a critical understanding and historical appreciation of some branch of mathematics by means of directed reading and preparation of a report.; or

d) To study in depth a particular problem in statistics, probability or operational research.

**Learning Outcomes**a) (Pure Maths)

After completing the report with suitable guidance the student should have

· gained a greater understanding of the chosen mathematical topic

· gained experience in applying his/her mathematical skills

· had experience in consulting relevant literature

· learned how to construct a written project report

· had experience in making an oral presentation

b) (Applied Mathematics)

After completing the project with suitable guidance the students should have:

- learned strategies for simple model building

- gained experience in choosing and using appropriate mathematics

- understood the nature of approximations used

- made critical appraisal of results

- had experience in consulting related relevant literature

- learned how to construct a written project report

- had experience in making an oral presentation.

c) (Applied Maths/Theoretical Physics)

After researching and preparing the mathematical essay the student should have:

· gained a greater understanding of the chosen mathematical topic

· gained an appreciation of the historical context

· learned how to abstract mathematical concepts and explain them

· had experience in consulting related relevant literature

· learned how to construct a written project report

· had experience in making an oral presentation.

d) (Statistics, Probability and Operational Research)

After completing the project the student should have:

· gained an in-depth understanding of the chosen topic

· had experience in consulting relevant literature

· learned how to construct a written project report;

· had experience in making an oral presentation.

##### Linear Differential Operators in Mathematical Physics (MATH421)

**Level**M **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**This module provides a comprehensive introduction to the theory of partial differential equations, and it provides illustrative applications and practical examples in the theory of elliptic boundary value problems, wave propagation and diffusion problems.

**Learning Outcomes**This module will enable students to understand and actively use the basic concepts of mathematical physics, such as generalised functions, fundamental solutions and Green''s functions, and apply powerful mathematical methods to problems of electromagnetism, elasticity, heat conduction and wave propagation.

##### Quantum Field Theory (MATH425)

**Level**M **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To provide a broad understanding of the essentials of quantum field theory.

**Learning Outcomes**After the course the students should understand the important features of the mathematical tools necessary for particle physics. In particular they should

· be able to compute simple Feynman diagrams,

· understand the basic principles of regularisation and renormalisation

· be able to calculate elementary scattering cross-sections.

##### Variational Calculus and Its Applications (MATH430)

**Level**M **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**This module provides a comprehensive introduction to the theory of the calculus of variations, providing illuminating applications and examples along the way.

**Learning Outcomes**Students will posses a solid understanding of the fundamentals of variational calculus

Students will be confident in their ability to apply the calculus of variations to range of physical problems

Students will also have the ability to solve a wide class of non-physical problems using variational methods

Students will develop an understanding of Hamiltonian formalism and have the ability to apply this framework to solve physical and non-physical problems

Students will be confident in their ability to analyse variational symmetries and generate the associated conservation laws

##### Introduction to String Theory (MATH423)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To provide a broad understanding of string theory, and its utilization as a theory that unifies all of the known fundamental matter and interactions.

**Learning Outcomes**After completing the module the students should:

- be familiar with the properties of the classical string.

be familiar with the basic structure of modern particle physics and how it may arise from string theory.

be familiar with the basic properties of first quantized string and the implications for space-time dimensions.

be familiar with string toroidal compactifications and T-duality.##### Analytical & Computational Methods for Applied Mathematics (MATH424)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To provide an introduction to a range of analytical and numerical methods for partial differential equations arising in many areas of applied mathematics.

To provide a focus on advanced analytical techniques for solution of both elliptic and parabolic partial differential equations, and then on numerical discretisation methods of finite differences and finite elements.

To provide the algorithms for solving the linear equations arising from the above discretisation techniques.

**Learning Outcomes**Apply a range of standard numerical methods for solution of PDEs and should have an understanding of relevant practical issues.

Obtain solutions to certain important PDEs using a variety of analytical techniques and should be familiar with important properties of the solution.

Understand and be able to apply standard approaches for the numerical solution of linear equations

Have a basic understanding of the variation approach to inverse problems.

##### Advanced Topics in Mathematical Biology (MATH426)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**- To introduce some hot problems of contemporary mathematical biology, including analysis of developmental processes, networks and biological mechanics.
To further develop mathematical skills in the areas of difference equations and ordinary and partial differential equations.

To explore biological applications of fluid dynamics in the limit of low

and high Reynolds number.**Learning Outcomes**To familiarise with mathematical modelling methodology used in contemporary mathematical biology. Be able to use techniques from difference equations and ordinary and partial differential equations in tackling problems in biology.

##### Waves, Mathematical Modelling (MATH427)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**This module gives an introduction to the mathematical theory of linear and non-linear waves. Illustrative applications involve problems of acoustics, gas dynamics and examples of solitary waves.

**Learning Outcomes**Students will learn essential modelling techniques in problems of wave propagation. They will also understand that mathematical models of the same type can be successfully used to describe different physical phenomena. Students will also study background mathematical theory in models of acoustics, gas dynamics and water waves.

##### Asymptotic Methods for Differential Equations (MATH433)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**This module provides an introduction into the perturbation theory for partial differential equations. We consider singularly and regularly perturbed problems and applications in electro-magnetism, elasticity, heat conduction and propagation of waves.

**Learning Outcomes**The ability to make appropriate use of asymptotic approximations.

The ability to analyse boundary layer effects.

The ability to use the method of compound asymptotic expansions in the analysis of singularly perturbed problems.

##### Stochastic Analysis and Its Applications (MATH483)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**This module aims to demonstrate the advanced mathematical techniques underlying financial markets and the practical use of financial derivative products to analyse various problems arising in financial markets. Emphases are on the stochastic techniques, probability theory, Markov processes and stochastic calculus, together with the related applications

**Learning Outcomes**At the end of the module students should be able to do the following things.

1. A critical awareness of current problems and research issues in the fields of probability and stochastic processes, stochastic analysis and financial mathematics.

2. The ability to formulate stochastic cuclulus for the purpose of modelling particular financial questions.

3. The ability to use appropriate mathematical tools and techniques in the context of a particular financial model.

4. The ability to read, understand and communicate research literature in the fields of probability, stochastic analysis and financial mathematics.

5. The ability to recognise potential research opportunities and research directions.

##### Project for M.math (MATH499)

**Level**M **Credit level**15 **Semester**Whole Session **Exam:Coursework weighting**0:100 **Aims**To demonstrate a critical understanding and historical appreciation of some branch of mathematics by means of directed reading and preparation of a report.

**Learning Outcomes**After researching and preparing the report the student should have:

- gained a greater understanding of the chosen mathematical topic.

- gained an appreciation of the historical context.

- learned how to abstract mathematical concepts and explain them.

- had experience in consulting related relevant literature.

- learned how to construct a written project report.

- had experience in making an oral presentation.

- gained familiarity with a scientific word-processing package such as LaTeX or TeX.

### Programme Year Four

There is a large set of modules available, some of which are taught in alternate years. MMath/MPhys students will take at least seven of these during Years Three and Four. There is also a compulsory project.

Choose 3 Modules (NOT taken in Year 3) from:

- Further Methods Of Applied Mathematics (MATH323)
- Cartesian Tensors and Mathematical Models of Solids and Viscous Fluids (MATH324)
- Quantum Mechanics (MATH325)
- Population Dynamics (MATH332)
- Group Theory (MATH343)
- Combinatorics (MATH344)
- Analysis and Number Theory (MATH351)
- Applied Probability (MATH362)
- Linear Statistical Models (MATH363)
- Measure Theory and Probability (MATH365)
- Networks In Theory And Practice (MATH367)
- Chaos and Dynamical Systems (MATH322)
- Relativity (MATH326)
- Mathematical Economics (MATH331)
- Riemann Surfaces (MATH340)
- Number Theory (MATH342)
- The Magic of Complex Numbers (MATH345)
- Differential Geometry (MATH349)
- Applied Stochastic Models (MATH360)
- Theory Of Statistical Inference (MATH361)
- Medical Statistics (MATH364)
- Mathematical Risk Theory (MATH366)

Choose 3 (NOT taken in Year 3) from:

- Linear Differential Operators In Mathematical Physics (MATH421)
- Analytical and Computational Methods for Applied Mathematics (MATH424)
- Quantum Field Theory (MATH425)
- Variational Calculus & it's Applications (MATH430)
- Asymptotic Methods for differential Equations (MATH433)
- Introduction to String Theory (MATH423)
- Advanced Topics in Mathematical Biology (MATH426)
- Waves. Mathematical Modelling (MATH427)
- Stochastic Analysis and its Applications (MATH483)
- Project For M.Math. (MATH490)
- Project For M.Math. (MATH499)

#### Year Four Compulsory 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**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**100:0 **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**After completing the module, students should be able to understand and actively use the basic concepts of continuum mechanics such as stress, deformation and constitutive relations, and apply mathematical methods for analysis of problems involving the flow of viscous fluid or behaviour of solid elastic materials.

##### Quantum Mechanics (MATH325)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**The development of Quantum Mechanics, requiring as it did revolutionary changes in our understanding of the nature of reality, was arguably the greatest conceptual achievement of all time. The aim of the module is to lead the student to an understanding of the way that relatively simple mathemactics (in modern terms) led Bohr, Einstein, Heisenberg and others to a radical change and improvement in our understanding of the microscopic world.

**Learning Outcomes**After completing the module students should be able to solve Schrodinger''s equation for simple systems, and have some intuitive understanding of the significance of quantum mechanics for both elementary systems and the behaviour of matter.

##### Population Dynamics (MATH332)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**- To provide a theoretical basis for the understanding of population ecology

- To explore the classical models of population dynamics

- To learn basic techniques of qualitative analysis of mathematical models

**Learning Outcomes**The ability to relate the predictions of the mathematical models to experimental results obtained in the field.The ability to recognise the limitations of mathematical modelling in understanding the mechanics of complex biological systems.The ability to use analytical and graphical methods to investigate population growth and the stability of equilibrium states for continuous-time and discrete-time models of ecological systems. ##### Group Theory (MATH343)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**To introduce the basic techniques of finite group theory with the objective of explaining the ideas needed to solve classification results.

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

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

The understanding of connections of the subject with other areas of Mathematics.

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

##### Combinatorics (MATH344)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **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**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.

##### Analysis and Number Theory (MATH351)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To provide an introduction to uniform distribution modulo one as a meeting ground for other branches of pure mathematics such as topology, analysis and number theory.

To give an idea of how these subjects work together in this context to shape modern analysis by focusing on one particular rich context.

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

understand completions and irrationality;

understand diophantine approximation and its relation to uniform distribution;

appreciate that analysis has a complex unity and to have a feel for basic computations in analysis.

calculate rational approximations to real and p adic numbers and to put this to use in number theoretic situations.

calculate approximations to functions from families of simpler functions e.g. continuous functions by polynomials and step functions by trigonometric sums.

work with basic tools from analysis, like Fourier series and continuous functions to prove distributional properties of sequences of numbers.

##### Applied Probability (MATH362)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **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 occuring over time. To familiarise students with an important area of probability modelling.

**Learning Outcomes**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**100:0 **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 familiarity with the computer package SPSS.

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

understand the rationale and assumptions of linear regression and analysis of variance.

· understand the rationale and assumptions of generalized linear models.

· recognise the correct analysis for a given experiment.

· carry out and interpret linear regressions and analyses of variance, and derive appropriate theoretical results.

· carry out and interpret analyses involving generalised linear models and derive appropriate theoretical results.

· perform linear regression, analysis of variance and generalised linear model analysis using the SPSS computer package.

##### 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**After completing the module students should be ableto:

master the basic results about measures and measurable functions;

master the basic results about Lebesgue integrals and their properties;to understand deeply the rigorous foundations ofprobability theory;

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

##### 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**After completing the module students should

. be able to model problems in terms of networks.

· be able to apply effectively a range of exact and heuristic optimisation techniques.

##### Chaos and Dynamical Systems (MATH322)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To develop expertise in dynamical systems in general and study particular systems in detail.

**Learning Outcomes**After completing the module students will be able to understand the possible behaviour of dynamical systems with particular attention to chaotic motion;

After completing the module students will be familiar with techniques for extracting fixed points and exploring the behaviour near such fixed points;

After completing the module students will understand how fractal sets arise and how to characterise them.

##### Relativity (MATH326)

**Level**3 **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To impart

(i) a firm grasp of the physical principles behind Special and General Relativity and their main consequences;

(ii) technical competence in the mathematical framework of the subjects - Lorentz transformation, coordinate transformations and geodesics in Riemann space;

(iii) knowledge of some of the classical tests of General Relativity - perihelion shift, gravitational deflection of light;

(iv) basic concepts of black holes and (if time) relativistic cosmology.

**Learning Outcomes**After completing this module students should

(i) understand why space-time forms a non-Euclidean four-dimensional manifold;

(ii) be proficient at calculations involving Lorentz transformations, energy-momentum conservation, and the Christoffel symbols.

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

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

##### Mathematical Economics (MATH331)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**After completing the module students should:

· Have further extended their appreciation of the role of mathematics in modelling in Economics and the Social Sciences.

· Be able to formulate, in game-theoretic terms, situations of conflict and cooperation.

· Be able to solve mathematically a variety of standard problems in the theory of games.

· To understand the relevance of such solutions in real situations.

##### Riemann Surfaces (MATH340)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To introduce to a beautiful theory at the core of modern mathematics. Students will learn how to handle some abstract geometric notions from an elementary point of view that relies on the theory of holomorphic functions. This will provide those who aim to continue their studies in mathematics with an invaluable source of examples, and those who plan to leave the subject with the example of a modern axiomatic mathematical theory.

**Learning Outcomes**Students should be familiar with themost basic examples of Riemann surfaces: the Riemann sphere, hyperelliptic Riemann surfaces, and smooth plane algebraic curves.

Students should understand and be able to use the abstract notions used to build the theory: holomorphic maps, meromorphic differentials, residues and integrals, Euler characteristic and genus.

##### Number Theory (MATH342)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**After completing this module students should be able to understand and solve a wide range of problems about the integers, and have a better understanding of the properties of prime numbers.

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

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**will understand the compactification of the complex plane to the Riemann sphere, and be able to use spherical distances and derivatives

will be able to use Möbius transformations to transform the Riemann sphere and to normalise complex dynamical systems

will be able to state and apply the definitions of Julia and Fatou sets of polynomials, and understand their basic properties

will be able to determine whether points with simple orbits, such as certain periodic points, belong to the Julia set or the Fatou set

will know how to apply advanced results from complex analysis in a dynamical setting

will be able to determine whether certain types of quadratic polynomials belong to the Mandelbrot set or not

##### Differential Geometry (MATH349)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**85:15 **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**1. Knowledge and understanding

After the module, students should have a basic understanding of

a) invariants used to describe the shape of explicitly given curves and surfaces,

b) special curves on surfaces,

c) the difference between extrinsically defined properties and those which depend only on the surface metric,

d) understanding the passage from local to global properties exemplified by the Gauss-Bonnet Theorem.

2. Intellectual abilities

After the module, students should be able to

a) use differential calculus to discover geometric properties of explicitly given curves and surface,

b) understand the role played by special curves on surfaces.

3. Subject-based practical skills

Students should learn to

a) compute invariants of curves and surfaces,

b) interpret the invariants of curves and surfaces as indicators of their geometrical properties.

4. General transferable skills

Students will improve their ability to

a) think logically about abstract concepts,

b) combine theory with examples in a meaningful way.

##### Applied Stochastic Models (MATH360)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**After completing the module students should have a grounding in the theory of continuous-time Markov chains and diffusion processes. They should be able to solve corresponding problems arising in epidemiology, mathematical biology, financial mathematics, etc.

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

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**90:10 **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**After completing the module students should have a good understanding of the classical approach to, and especially the likelihood methods for, statistical inference.

The students should also gain an appreciation of the blossoming area of Bayesian approach to inference

##### Medical Statistics (MATH364)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**identify the types of problems encountered in medical statistics

demonstrate the advantages and disadvantages of different epidemiological study designs

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

explain and apply statistical techniques used in survival analysis

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

discuss statistical issues related to systematic review and apply appropriate methods of meta-analysisapply Bayesian methods to simple medical problems.

##### Mathematical Risk Theory (MATH366)

**Level**3 **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **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**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 the

R(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).

##### Linear Differential Operators in Mathematical Physics (MATH421)

**Level**M **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**This module provides a comprehensive introduction to the theory of partial differential equations, and it provides illustrative applications and practical examples in the theory of elliptic boundary value problems, wave propagation and diffusion problems.

**Learning Outcomes**This module will enable students to understand and actively use the basic concepts of mathematical physics, such as generalised functions, fundamental solutions and Green''s functions, and apply powerful mathematical methods to problems of electromagnetism, elasticity, heat conduction and wave propagation.

##### Analytical & Computational Methods for Applied Mathematics (MATH424)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To provide an introduction to a range of analytical and numerical methods for partial differential equations arising in many areas of applied mathematics.

To provide a focus on advanced analytical techniques for solution of both elliptic and parabolic partial differential equations, and then on numerical discretisation methods of finite differences and finite elements.

To provide the algorithms for solving the linear equations arising from the above discretisation techniques.

**Learning Outcomes**Apply a range of standard numerical methods for solution of PDEs and should have an understanding of relevant practical issues.

Obtain solutions to certain important PDEs using a variety of analytical techniques and should be familiar with important properties of the solution.

Understand and be able to apply standard approaches for the numerical solution of linear equations

Have a basic understanding of the variation approach to inverse problems.

##### Quantum Field Theory (MATH425)

**Level**M **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**100:0 **Aims**To provide a broad understanding of the essentials of quantum field theory.

**Learning Outcomes**After the course the students should understand the important features of the mathematical tools necessary for particle physics. In particular they should

· be able to compute simple Feynman diagrams,

· understand the basic principles of regularisation and renormalisation

· be able to calculate elementary scattering cross-sections.

##### Variational Calculus and Its Applications (MATH430)

**Level**M **Credit level**15 **Semester**First Semester **Exam:Coursework weighting**90:10 **Aims**This module provides a comprehensive introduction to the theory of the calculus of variations, providing illuminating applications and examples along the way.

**Learning Outcomes**Students will posses a solid understanding of the fundamentals of variational calculus

Students will be confident in their ability to apply the calculus of variations to range of physical problems

Students will also have the ability to solve a wide class of non-physical problems using variational methods

Students will develop an understanding of Hamiltonian formalism and have the ability to apply this framework to solve physical and non-physical problems

Students will be confident in their ability to analyse variational symmetries and generate the associated conservation laws

##### Asymptotic Methods for Differential Equations (MATH433)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**This module provides an introduction into the perturbation theory for partial differential equations. We consider singularly and regularly perturbed problems and applications in electro-magnetism, elasticity, heat conduction and propagation of waves.

**Learning Outcomes**The ability to make appropriate use of asymptotic approximations.

The ability to analyse boundary layer effects.

The ability to use the method of compound asymptotic expansions in the analysis of singularly perturbed problems.

##### Introduction to String Theory (MATH423)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**To provide a broad understanding of string theory, and its utilization as a theory that unifies all of the known fundamental matter and interactions.

**Learning Outcomes**After completing the module the students should:

- be familiar with the properties of the classical string.

be familiar with the basic structure of modern particle physics and how it may arise from string theory.

be familiar with the basic properties of first quantized string and the implications for space-time dimensions.

be familiar with string toroidal compactifications and T-duality.##### Advanced Topics in Mathematical Biology (MATH426)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**- To introduce some hot problems of contemporary mathematical biology, including analysis of developmental processes, networks and biological mechanics.
To further develop mathematical skills in the areas of difference equations and ordinary and partial differential equations.

To explore biological applications of fluid dynamics in the limit of low

and high Reynolds number.**Learning Outcomes**To familiarise with mathematical modelling methodology used in contemporary mathematical biology. Be able to use techniques from difference equations and ordinary and partial differential equations in tackling problems in biology.

##### Waves, Mathematical Modelling (MATH427)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**This module gives an introduction to the mathematical theory of linear and non-linear waves. Illustrative applications involve problems of acoustics, gas dynamics and examples of solitary waves.

**Learning Outcomes**Students will learn essential modelling techniques in problems of wave propagation. They will also understand that mathematical models of the same type can be successfully used to describe different physical phenomena. Students will also study background mathematical theory in models of acoustics, gas dynamics and water waves.

##### Stochastic Analysis and Its Applications (MATH483)

**Level**M **Credit level**15 **Semester**Second Semester **Exam:Coursework weighting**100:0 **Aims**This module aims to demonstrate the advanced mathematical techniques underlying financial markets and the practical use of financial derivative products to analyse various problems arising in financial markets. Emphases are on the stochastic techniques, probability theory, Markov processes and stochastic calculus, together with the related applications

**Learning Outcomes**At the end of the module students should be able to do the following things.

1. A critical awareness of current problems and research issues in the fields of probability and stochastic processes, stochastic analysis and financial mathematics.

2. The ability to formulate stochastic cuclulus for the purpose of modelling particular financial questions.

3. The ability to use appropriate mathematical tools and techniques in the context of a particular financial model.

4. The ability to read, understand and communicate research literature in the fields of probability, stochastic analysis and financial mathematics.

5. The ability to recognise potential research opportunities and research directions.

##### Project for M.math (MATH490)

**Level**M **Credit level**30 **Semester**Whole Session **Exam:Coursework weighting**0:100 **Aims**To demonstrate a critical understanding and historical appreciation of some branch of mathematics by means of directed reading and preparation of a report.

The MATH490 project should treat its subject at a more advanced level and in greater depth than the MATH399 or MATH499 projects. Working on this year-long project can provide a good base to continue mathematical studies through PhD.

**Learning Outcomes**After researching and preparing the report the student should have:

- gained a greater understanding of the chosen mathematical topic.

- gained an appreciation of the historical context.

- learned how to abstract mathematical concepts and explain them.

- had experience in consulting related relevant literature.

- learned how to construct a written project report.

- had experience in making an oral presentation.

- gained familiarity with a scientific word-processing pakage such as LaTeX or TeX.

##### Project for M.math (MATH499)

**Level**M **Credit level**15 **Semester**Whole Session **Exam:Coursework weighting**0:100 **Aims**To demonstrate a critical understanding and historical appreciation of some branch of mathematics by means of directed reading and preparation of a report.

**Learning Outcomes**After researching and preparing the report the student should have:

- gained a greater understanding of the chosen mathematical topic.

- gained an appreciation of the historical context.

- learned how to abstract mathematical concepts and explain them.

- had experience in consulting related relevant literature.

- learned how to construct a written project report.

- had experience in making an oral presentation.

- gained familiarity with a scientific word-processing package such as LaTeX or TeX.

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.