# Mathematics and Statistics BSc (Hons)

• Offers study abroad opportunities
• Offers a Year in China

## Key information

• Course length: 3 years
• UCAS code: GG13
• Year of entry: 2019
• Typical offer: A-level : ABB / IB : 33 / BTEC : D*DD

### Programme Year One

#### Year One Compulsory Modules

• ##### Calculus I (MATH101)
Level 1 15 First Semester 80:20 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. 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 15 Second Semester 80:20 ·      To discuss local behaviour of functions using Taylor’s theorem. ·      To introduce multivariable calculus including partial differentiation, gradient, extremum values and double integrals. 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 15 First Semester 80:20 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. 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 15 First Semester 90:10 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. 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 15 Second Semester 80:20 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 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 15 Second Semester 80:20 ·         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. ​​ 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 15 Second Semester 80:20 To introduce topics in Statistics and to describe and discuss basic statistical methods. To describe the scope of the application of these methods. To know how to describe statistical data.​ To be able to use the Binomial, Poisson, Exponential and Normal distributions.To be able to ​perform simple goodness-of-fit tests.​ ​To be able to use an appropriate statistical software package to present data and to make statistical analysis.

#### Year One Optional Modules

• ##### Introduction to Programming (COMP101)
Level 1 15 First Semester 0:100 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. ​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 programDesign 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 15 First Semester 0:100 To acquire key mathematics-specific computer skills.To reinforce mathematics as a practical discipline by active experience and experimentation, using the computer as a tool. To illustrate and amplify mathematical concepts and techniques.To initiate and develop problem solving, group work and report writing skills.To initiate and develop modelling skills.  ​​To develop team work skills​. 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 have developed their team working skills​.

### Programme Year Two

Choose one module from:

• Introduction to Methods of Operational Research (MATH261)
• Operational Research: Probabilistic Models (MATH268)
• Financial Mathematics II (MATH262)

Choose three modules from:

• Introduction to the Methods of Applied Mathematics (MATH224)
• Vector Calculus With Applications in Fluid Mechanics (MATH225)
• Mathematical Models: Microeconomics and Population Dynamics (MATH227)
• Classical Mechanics (MATH228)
• Metric Spaces and Calculus (MATH241)
• Complex Functions (MATH243)
• Linear Algebra and Geometry (MATH244)
• Commutative Algebra (MATH247)
• Geometry of Curves (MATH248)
• Numerical Methods (MATH266)
• Financial Mathematics I (MATH267)

#### Year Two Compulsory Modules

• ##### Ordinary Differential Equations (MATH201)
Level 2 15 First Semester 75:25 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. To understand the basic properties of ODE, including main features of initial value problems and boundary value problems, such as existence and uniqueness of solutions.​​​​​​​​ To know the elementary techniques for the solution of ODEs.​To understand the idea of reducing a complex ODE to a simpler one.​To be able to solve linear ODE systems (homogeneous and non-homogeneous) with constant coefficients matrix.​To understand a range of applications of ODE.​
• ##### Statistical Theory and Methods I (MATH263)
Level 2 15 Second Semester 85:15 To introduce statistical methods with a strong emphasis on applying standard statistical techniques appropriately and with clear interpretation.  The emphasis is on applications. ​To have a conceptual and practical understanding of a range of commonly applied statistical procedures.​​​​ ​To have developed some familiarity with an appropriate statistical software package.​​
• ##### Statistical Theory and Methods II (MATH264)
Level 2 15 Second Semester 90:10 To introduce statistical distribution theory which forms the basis for all applications of statistics, and for further statistical theory. ​ To have an understanding ​​of basic probability calculus. ​To have an understanding ​of a range of techniques for solving real life problems of probabilistic nature.

#### Year Two Optional Modules

• ##### Introduction to Methods of Operational Research (MATH261)
Level 2 15 First Semester 70:30 ​After completing the module students should: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. ​​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. ​
• ##### Operational Research: Probabilistic Models (MATH268)
Level 2 15 Second Semester 90:10 To introduce a range of models and techniques for solving under uncertainty in Business, Industry, and Finance. 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.
• ##### Financial Mathematics (MATH262)
Level 2 15 Second Semester 100:0 To provide an understanding of basic theories in Financial Mathematics used in the study process of actuarial/financial interest.To provide an introduction to financial methods and derivative pricing financial instruments in discrete time set up.To gain understanding of some financial models (discrete time) with applications to financial/insurance industry.To prepare the students adequately and to develop their skills in order to be ready to sit the CM2 subject of the Institute and Faculty of Actuaries.​​ ​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 role of Filtrations, martingales and option pricing in discrete time. ​To be able to discuss the theories of financial market behaviour.​To be able to perform calculations using the risk measures.
• ##### Introduction to the Methods of Applied Mathematics (MATH224)
Level 2 15 Second Semester 90:10 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. 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.
• ##### Vector Calculus With Applications in Fluid Mechanics (MATH225)
Level 2 15 First Semester 85:15 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. 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 15 First Semester 90:10 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. 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.
• ##### Classical Mechanics (MATH228)
Level 2 15 Second Semester 90:10 To provide an understanding of the principles of Classical Mechanics and their application to dynamical systems. To understand the variational principles, Lagrangian mechanics, Hamiltonian mechanics. ​To be able to use Newtonian gravity and Kepler''s laws to perform the calculations of the orbits of satellites, comets and planetary motions.​To understand the motion relative to a rotating frame, Coriolis and centripetal forces, motion under gravity over the Earth''s surface.​To understand the connection between symmetry and conservation laws.​ To be able to work with inertial and non-inertial frames.
• ##### Metric Spaces and Calculus (MATH241)
Level 2 15 First Semester 90:10 To introduce the basic elements of the theories of metric spaces and calculus of several variables. ​To become familiar with a range of examples of metric spaces.​To develop an understanding of the notions of convergence and continuity.​To understand the contraction mapping theorem and appreciate some of its applications.​​To become familiar with the concept of the derivative of a vector valued function of several variables as a linear map.​ ​​To understand the inverse function and implicit function theorems and appreciate their importance.​​​To develop an appreciation of the role of proof and rigour in mathematics.​
• ##### Complex Functions (MATH243)
Level 2 15 First Semester 80:20 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. To understand the central role of complex numbers in mathematics;.​To develop the knowledge and understanding of all the classical holomorphic functions.​To be able to compute Taylor and Laurent series of standard holomorphic functions.​To understand various Cauchy formulae and theorems and their applications.​To be able to reduce a real definite integral to a contour integral.​To be competent at computing contour integrals.
• ##### Linear Algebra and Geometry (MATH244)
Level 2 15 First Semester 90:10 To introduce general concepts of linear algebra and its applications in geometry and other areas of mathematics. To understand the geometric meaning of linear algebraic ideas.​To know the concept of an abstract vector space and how it is used in different mathematical situations.To be able to ​apply a change of coordinates to simplify a linear map,.​To be able to work with matrix groups, in particular GL(n), O(n) and SO(n),.​To understand bilinear forms from a geometric point of view.​
• ##### Commutative Algebra (MATH247)
Level 2 15 Second Semester 90:10 To give an introduction to abstract commutative algebra and show how it both arises naturally, and is a useful tool, in number theory. 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 15 Second Semester 90:10 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. 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.
• ##### Numerical Methods (MATH266)
Level 2 15 Second Semester 90:10 To provide an introduction to the main topics in Numerical Analysis and their relation to other branches of Mathematics (LO1) To strengthen students’ knowledge of scientific programming, building on the ideas introduced in MATH111.(LO2) To provide an introduction to the foundations of numerical analysis and its relation to other branches of Mathematics.(LO3) To introduce students to theoretical concepts that underpin numerical methods, including fixed point iteration, interpolation, orthogonal polynomials and error estimates based on Taylor series.(LO4) To demonstrate how analysis can be combined with sound programming techniques to produce accurate, efficient programs for solving practical mathematical problems.(S1) Numeracy(S2) Problem solving skills
• ##### Theory of Interest (MATH267)
Level 2 15 First Semester 90:10 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. 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.

### Programme Year Three

Choose two modules from:

• Applied Probability (MATH362)
• Measure Theory and Probability (MATH365)
• Theory of Statistical Inference (MATH361)
• Applied Stochastic Models (MATH360)
• Medical Statistics (MATH364)
• Mathematical Risk Theory (MATH366)
• Projects in Mathematics (MATH399)

Choose at least one module 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)
• Networks in Theory and Practice (MATH367)
• Chaos and Dynamical Systems (MATH322)
• Relativity (MATH326)
• Mathematical Economics (MATH331)
• Riemann Surfaces (MATH340)
• Number Theory (MATH342)
• Differential Geometry (MATH349)

#### Year Three Compulsory Modules

• ##### Linear Statistical Models (MATH363)
Level 3 15 First Semester 100:0 ·      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. 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.

#### Year Three Optional Modules

• ##### Applied Probability (MATH362)
Level 3 15 First Semester 100:0 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. ​​​​​ 1. Knowledge and Understanding:After completing 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;(d) understanding of Markov jump processes.​​2. Intellectual Abilities:After completing 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 Skill:(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.​
• ##### Measure Theory and Probability (MATH365)
Level 3 15 First Semester 90:10 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. ​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.
• ##### Theory of Statistical Inference (MATH361)
Level 3 15 Second Semester 90:10 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. To acquire a good understanding of the classical approach to, and especially the likelihood methods for, statistical inference.  To acquire an understanding of the blossoming area of Bayesian approach to inference.
• ##### Applied Stochastic Models (MATH360)
Level 3 15 Second Semester 100:0 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. ​​​ To understans the theory of continuous-time Markov chains. ​To understans the theory of diffusion processes. ​​To be able to solve problems arising in epidemiology, mathematical biology, financial mathematics, etc. using the theory of continuous-time Markov chains and diffusion processes. ​​​​ ​To acquire an undertanding of the standard concepts and methods of stochastic modelling.
• ##### Medical Statistics (MATH364)
Level 3 15 Second Semester 100:0 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. 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-analysis ​apply Bayesian methods to simple medical problems.
• ##### Mathematical Risk Theory (MATH366)
Level 3 15 Second Semester 100:0 To provide an understanding of the mathematical risk theory used in the study process of actuarial interest,Tto provide an introduction to mathematical methods for managing the risk in insurance and finance (calculation of risk measures/quantities),Tto 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 CM2 and CS2 of the Institute and Faculty of Actuaries.​ (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).​​(s) Recognise extreme value distributions and calculate various measures of tail weight​
• ##### Projects in Mathematics (MATH399)
Level 3 15 Whole Session 0:100 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. 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.
• ##### Further Methods of Applied Mathematics (MATH323)
Level 3 15 First Semester 100:0 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. 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 15 First Semester 100:0 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. ​ To understand and actively use the basic concepts of continuum mechanics such as stress, deformation and constitutive relations.​To apply mathematical methods for analysis of problems involving the flow of viscous fluid or behaviour of solid elastic materials.​
• ##### Quantum Mechanics (MATH325)
Level 3 15 First Semester 100:0 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. ​ To be able to solve Schrodinger''s equation for simple systems.​To have an understanding of the significance of quantum mechanics for both elementary systems and the behaviour of matter.
• ##### Population Dynamics (MATH332)
Level 3 15 Second Semester 100:0 - 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 ​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 15 First Semester 90:10 To introduce the basic techniques of finite group theory with the objective of explaining the ideas needed to solve classification results. 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 15 First Semester 90:10 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. 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 15 First Semester 100:0 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. 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.
• ##### Networks in Theory and Practice (MATH367)
Level 3 15 First Semester 100:0 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. 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 15 First Semester 100:0 To develop expertise in dynamical systems in general and study particular systems in detail. 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 15 First Semester 100:0 1. To introduce the physical principles behind Special and General Relativity and their main consequences.2. To develop the competence in the mathematical framework of the subjects: Lorentz transformations and Minkowski space-time, semi-Riemannian geometry and curved space-time, symmetries and conservation laws, variational principles.3. To develop the understanding of the dynamcis of particles and of the Maxwell field in Minkowski space-time, and of particles in a curved space-time.4. To develop the knowledge of tests of General Relativity, including the classical tests (perihelion shift, gravitational deflection of light).5. To understand the basic concepts of black holes, and, time permitting, of relativistic cosmology and gravitational waves.​ ​ ​To be be proficient in calculations involving Lorentz transformations, the kinematical and dynamical quantities associated to particles in Minkowski space-times, and the application of the conservation law for the four-momentum to scattering processes.​To know the relativistically covariant form of the Maxwell equations​.​To know the action principles for relativistic particles, the Maxwell field and the gravitational field​.​ ​To be be proficient at calculations in semi-Riemannian geometry as far as needed for General Relativity, including calculations involving general coordinate transformations, tensor fields, covariant derivatives, parallel transport, geodesics and curvature​​ ​To understand the arguments leading to Einstein''s field equations and how Newton''s law of gravity arises as a limit​.To ​be able to calculate the trajectories of bodies in a Schwarzschild space-time​.
• ##### Mathematical Economics (MATH331)
Level 3 15 Second Semester 100:0 ·      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. 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 15 Second Semester 100:0 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. 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 15 Second Semester 100:0 To give an account of elementary number theory with use of certain algebraic methods and to apply the concepts to problem solving. To understand and solve a wide range of problems about integers numbers. ​​​​ To ​have a better understanding of the properties of prime numbers.
• ##### Differential Geometry (MATH349)
Level 3 15 Second Semester 85:15 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.​ ​​​1a. Knowledge and understanding: Students will have a reasonable understanding of invariants used to describe the shape of explicitly given curves and surfaces. ​1b. Knowledge and understanding: Students will have a reasonable understanding of special curves on surfaces.​1c. Knowledge and understanding: Students will have a reasonable understanding of the difference between extrinsically defined properties and those which depend only on the surface metric.​1d. Knowledge and understanding: Students will have a reasonable understanding of the passage from local to global properties exemplified by the Gauss-Bonnet Theorem.​2a. Intellectual abilities: Students will be able to use differential calculus to discover geometric properties of explicitly given curves and surfaces.​2b. Intellectual abilities: Students will be able to understand the role played by special curves on surfaces.​3a. Subject-based practical skills: Students will learn to compute invariants of curves and surfaces.​3b. Subject-based practical skills: Students will learn to interpret the invariants of curves and surfaces as indicators of their geometrical properties.​​4a. General transferable skills: Students will improve their ability to think logically about abstract concepts,​4b. General transferable skills: Students will improve their ability to combine theory with examples in a meaningful way.​​

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.