MathematicsBSc Hons

What you'll do

You'll explore ways in which calculus underpins much of modern science.

What you'll learn

When you complete this module successfully, you'll be able to:

• Compute limits, derivatives, and integrals of functions
• Use differential calculus to study the behaviour of functions
• Use integral calculus to compute areas
• Determine convergence of infinite series
• Use infinite series in differentiation and integration

What you'll do

You'll be introduced to fundamental concepts of Python and MATLAB programming languages. We'll demonstrate to you how to break a problem down into a series of steps. You'll then convert these steps to computer code. This will enable you to present model interfaces, reports and data visualisations.nd intuitive manner
Use computational methods for data analysis and visualisation 

What you'll learn

When you complete this module successfully, you'll be able to:

• Formulate and analyse physical problems and develop coherent mathematical models of physical systems expressed as algorithms
• Write computer programs that implement simple models using appropriate numerical methods using a high-level language
• Present model interfaces, documentation and reports in a clear a

What you'll do

You'll cover topics including Gaussian elimination, matrices, vector spaces and eigentheory. You'll also meet applications such as differential equations, conic sections, graphs and networks.

What you'll learn

When you complete this module successfully, you'll be able to:

• Solve a linear system of equations, including the case of infinitely many solutions
• Describe the solution of linear systems in geometrical terms such as lines and planes
• Develop the algebra of matrices and vectors including determinants and dot products
• Find the eigenvalues and eigenvectors of a matrix
• Use matrices to represent and interpret linear transformations
• Analyse the vector space properties (such as basis and dimension) of certain sets

What you'll do

You'll study different types of proof, learn to identify the circumstances in which they are useful, and practise the skills needed for finding proofs. You'll also learn about logic, set theory, complex numbers and elementary properties of integers. 

What you'll learn

When you complete this module successfully, you'll be able to:

• Construct clear, logical arguments demonstrating the difference between experimental evidence and proof
• Manipulate elementary mathematical constructs and complex numbers
• Demonstrate an understanding of cryptographic systems and techniques by enciphering and deciphering messages

What you'll do

You'll cover models such as linear difference equations, fractals and multivariable problems, and reflect on their applications to subjects including biology, ecology, epidemiology and physics/astronomy.

What you'll learn

When you complete this module successfully, you'll be able to:

• Formulate mathematical models using algebraic equations/inequalities, difference equations and ordinary differential equations (ODEs)
• Formulate and solve linear programming models using graphical methods
• Solve and interpret the solutions to linear 1st and 2nd order difference equations, e.g. in biology and economics
• Solve and interpret the solutions to 1st and 2nd order ODEs with application to real world problems, e.g. in mechanics
• Demonstrate familiarity with numerical techniques and their applications to mathematical modelling

What you'll do

You'll apply your understanding of statistical theory to techniques used widely in business and social science.

What you'll learn

When you complete this module successfully, you'll be able to:

• Interpret, summarise and present data clearly using a variety of methods
• Identify and apply standard probability models
• Test hypotheses about population parameters (means, proportions, correlations) using inferential methods
• Fit and test simple linear regression models to data
• Use appropriate software to analyse data

What you'll do

You'll work in a group to select a problem, then produce and present a report on your solutions.

What you'll learn

When you complete this module successfully, you'll be able to:

• Identify and apply mathematical techniques to solve a problem
• Use high-level software to model and analyse problems
• Work in and lead a team
• Communicate mathematical findings in written and oral form
• Identify and demonstrate your skills, priorities and constraints in the context of career decision-making

What you'll do

You'll begin exploring Fourier series.

To choose this module, you should have taken Mathematical Foundations and Calculus I core modules, and be familiar with the basics of calculus.

What you'll learn

When you complete this module successfully, you'll be able to:

• Determine the gradient, divergence, curl, and expressions involving these operators
• Evaluate line, surface and volume integrals
• Apply the integral theorems of vector calculus
• Find the Fourier series of a function
• Solve linear ordinary differential equations using standard analytical techniques including complementary function plus particular integral, series solutions and Laplace transform
• Interpret the solution of linear ordinary differential equations

What you'll do

To choose this module, you should have taken the Mathematical Foundations and Calculus I core modules in year 1.

What you'll learn

When you complete this module successfully, you'll be able to:

• Construct simple proofs and counter examples
• Give clear definitions and state basic theorems of analysis
• Illustrate simple complex mappings in the complex plane
• Demonstrate your understanding of the properties of standard functions of complex variables (including continuity, differentiability, series representation and integration)
• Apply the techniques of complex analysis to evaluate derivatives and integrals, and to solve appropriate problems

What you'll do

To choose this module, you need to take the Mathematical Foundations and Linear Algebra modules in year 1, to gain knowledge of complex numbers and linear algebra (particularly vector spaces, bases, dimension and matrices) and experience in mathematical proofs.

What you'll learn

When you complete this module successfully, you'll be able to:

• Apply appropriate terminology and techniques involving different algebraic structures to their solutions
• Construct simple proofs and counter examples for a wide set of given mathematical propositions
• Give clear definitions and statements of basic results in Abstract Algebra
• Conceptualise the notion of a group by using the group axioms to construct relevant proofs and by describing structural properties
• Conceptualise the notion of a group and its categorical properties by constructing proofs of structural properties
• Display your familiarity with sets, general abstract objects, and their correspondences

What you'll learn

The learning objectives of this module are to be confirmed.

What you'll do

You'll develop practical programming skills (Python) and apply them creatively, using industry standard computer tools. 

To choose this module, you need to show knowledge of calculus and stochastic methods.

What you'll learn

When you complete this module successfully, you'll be able to:

• Model and solve financial problems mathematically
• Simulate and analyse financial scenarios using computer tools
• Apply advanced analysis techniques to model financial market

What you'll do

You'll then study linear and nonlinear differential and difference equations in the context of methods and techniques of dynamical systems.

What you'll learn

When you complete this module successfully, you'll be able to:

• Model simple mechanical systems (for example a double pendulum) in terms of coordinates and velocities
• Derive a special function known as a ""Langrarian"", from which the equations of motion (a set of differential equations) are derived
• Simulate/solve these equations, taking advantage of any symmetry in the model, to predict the behaviour of the system
• Broaden the context of the study of differential/difference equations in general, both linear and nonlinear
• Use special techniques to describe the qualitative behaviour of the solutions to these equations

What you'll learn

When you complete this module successfully, you'll be able to:

• Understand the numerical solution of systems of linear and nonlinear equations
• Understand the numerical solution of differential equations
• Fit mathematical models, including differential equations, to data
• Estimate model parameters, and select models, using Bayesian methods
• Implement methods and visualize data using R or Python

What you'll do

The knowledge you'll develop will help in future operational research topics such as simulation, planning, scheduling, forecasting, supply chain management and advanced modelling.

What you'll learn

When you complete this module successfully, you'll be able to:

• Model operational research problems
• Formulate and solve linear, nonlinear and dynamic programming models
• Formulate and solve game theory models

What you'll do

You'll also apply your understanding of statistical methods to quality control systems. To choose this module, you need to take the Statistical Theory and Methods I module in year 1, or equivalent.

What you'll learn

When you complete this module successfully, you'll be able to:

• Formulate problems in statistical terms, apply appropriate analyses and interpret the results in terms of the problem
• Recognise, analyse and interpret results from simple designed experiments using ANOVA
• Fit, test and interpret multiple linear regression models
• Estimate and make inferences about parameters of statistical models using a variety of approaches
• Analyse bi-variate probability distributions
• Perform transformations of uni-variate and bi-variate random variables

What you'll do

You'll develop your understanding in celestial coordinate systems, telescope design, comparative planetology, stellar evolution, the formation and evolution of galaxies, and the dynamics and matter content Universe.

What you'll learn

When you complete this module successfully, you'll be able to:

• Derive and apply mathematical equations to solve astronomical problems
• Identify and apply physical principles underlying the properties and behaviour of planets, stars and galaxies
• Make astronomical observations and analyse the results with appropriate software

What you'll do

You'll study solution techniques such as the heat conduction (or diffusion) equation, Laplace's equation, wave equations, and classifications of elliptic, parabolic and hyperbolic equations.

What you'll learn

When you complete this module successfully, you'll be able to:

• Classify partial differential equations and boundary conditions
• Solve partial differential equations using the method of characteristics, transform methods, numerical methods, separation of variables and Fourier series

What you'll do

You'll learn about data envelopment analysis and decision analysis, explore realistic case studies, and work toward optimal solutions for computationally difficult problems.

To choose this module, you need to take the Statistical Theory and Methods I module in year 1.

What you'll learn

When you complete this module successfully, you'll be able to:

• Use computer software to formulate and solve linear, integer programming models
• Formulate data envelopment analysis models
• Formulate and solve multiple decision analysis models
• Understand and use the discrete-event simulation process
• Interpret and report on the results of solutions of your models and processes

What you'll do

You'll focus on equity options and portfolio construction, exploring no-arbitrage pricing, the Black-Scholes partial differential equation, and hedging. To choose this module, you need to take the Calculus II and Mathematics for Finance modules in year 2 or an equivalent module covering basic interest rate structures and elementary stochastic processes.

What you'll learn

When you complete this module successfully, you'll be able to:

• Define and distinguish between various types of derivative contracts, deduce their future payoffs and analyse investment portfolios created from them
• Prove standard relations (parity) between the prices of different contracts under the assumption of fairness (no-arbitrage)
• Derive the famous Black-Scholes partial differential equation and learn how to solve it for various asset types
• Use stochastic processes to price options using risk-neutral valuation via change of measure and expectation
• Perform partial differential equation changes of variables and other solution methods for certain exotic options
• Put theory into practice by implementing numerical schemes on the computer

What you'll do

To choose this module, Physics students need to take the Mathematical Physics (level 5) and Introduction to Modern Physics and Astrophysics (level 5) modules.

To take this module, Maths students need to take the Applied Mathematics (level 5) module.

What you'll learn

When you complete this module successfully, you'll be able to:

• Analyse the 4-dimensional spacetime formulation of Special Relativity
• Carry out basic calculations in tensor algebra and calculus, and apply these to physical problems
• Apply Einstein field equations to the calculation of the simplest exact and approximate solutions for relativistic stars and black holes and in cosmology, as well as in the weak field regime and for gravitational waves
• Analyse a problem and associate it with the physical and mathematical principle of General Relativity
• Apply the specific mathematical techniques of General Relativity to solve exercises and problems, conceptualising and generalising from previously seen problems
• Discuss the use of physical and mathematical principles and hypotheses in the solution of exercises and problems

What you'll do

You'll consider  the physics of stars, black holes and galaxies, and their formation mechanisms. To choose this module, you need to take the Mathematical Physics, and Introduction to Modern Physics and Astrophysics module.

What you'll learn

When you complete this module successfully, you'll be able to:

• Analyse fundamental physical processes in astrophysics, and apply them to the physics of stars, black holes and galaxies in multiple contexts
• Apply the physics of gravitational collapse to solve problems related to the formation of stars and galaxies, and compact objects
• Demonstrate your understanding of fundamental nuclear reactions and energetic balance, and evaluate the energetics of stars and galaxies
• Demonstrate your understanding of the quest for dark matter in galaxy formation and evolution and evaluate the observational evidence

What you'll do

You'll study perturbation methods and Turing instability, and the software packages to apply them to contexts including biology, chemistry, physics and mathematics.

To choose this module, you need to take the Mechanics and Dynamics module in year two.

What you'll learn

When you complete this module successfully, you'll be able to:

• Recognise the solution behaviour of 1 to 3-dimensional nonlinear systems (fixed points, stability properties and bifurcation scenarios)
• Classify regular and chaotic regimes by applying analytical methods based on perturbation theory
• Perform numerical studies related to the long-­term behaviour of realistic problems

What you'll do

You'll manage your own plan, from outline to dissertation, with supervision from relevant lecturers.

What you'll learn

When you complete this module successfully, you'll be able to:

• Locate and critically review relevant literature for your topic
• Synthesise information and ideas from multiple sources
• Write a formal, well-structured dissertation, including an abstract and appropriate conclusions
• Manage a project
• Evaluate and discuss your work

What you'll do

You'll learn about game theory and its application in logistics, network flow, revenue and inventory management, and scheduling. To choose this module you need to show a general knowledge of Operational Research techniques.

What you'll learn

When you complete this module successfully, you'll be able to:

• Design and solve game theory models and study their efficiency in logistics
• Formulate and solve network flow and revenue management problems
• Formulate and solve inventory management, planning and scheduling models

What you'll do

You'll explore advanced regression modelling, modern statistical learning methods, the use of open source statistical tools, and forecasting methodologies. To choose this module, you need basic knowledge of probability, calculus and linear algebra.

What you'll learn

When you complete this module successfully, you'll be able to:

• Apply statistical learning techniques to business problems, and interpret your results
• Use Python and/or R language to apply statistical learning techniques
• Demonstrate understanding of the bias variance trade-off and cross validation
• Fit and test general linear models to numerical and categorical data
• Fit a variety of predictive models to real world data
• Demonstrate understanding of advanced techniques such as regularisation, nonlinear models and clustering

What you'll do

You'll explore multiple procedures for data analysis, use the statistical language and software environment R to model data, and examine the strengths and weaknesses of various study designs. To choose this module, you need to take the Statistical Theory and Methods I module in year 1.

What you'll learn

When you complete this module successfully, you'll be able to:

• Formulate and employ statistical methods commonly used in the study of epidemiology
• Construct lifetables and compare survival patterns in population subgroups
• Conceptualize the basic principles underpinning the design and analysis of clinical trials
• Employ a variety of multivariate techniques.
• Formulate the principles relating to questionnaire design and validation.
• Use R language to apply the learnt statistical techniques

What you'll do

You'll apply these to subjects such as earthquakes, disease spread, financial markets or population dynamics, and demonstrate your understanding of the wider applications of mathematics.

What you'll learn

When you complete this module successfully, you'll be able to:

• Apply distributions of random discrete and continuous random variables
• Determine the probability of the extinction of branching processes
• Solve the gambler's ruin problem
• Solve partial differential equations that arise in birth-death processes
• Apply appropriate techniques to analyse continuous time stochastic processes

What you'll do

Over ten half-days, you'll be mentored by a maths teacher as you gain experience of teaching mathematics, leading special projects and offering classroom support. To choose this module, you need to show you know the fundamental concepts of basic probability, calculus and linear algebra.

What you'll learn

When you complete this module successfully, you'll be able to:

• Demonstrate your understanding of teaching mathematics, and of educational theories and debates
• Work in a challenging and unpredictable working environment
• Communicate difficult principles or concepts, whether you're speaking one-to-one or to an audience
• Reflect on stereotypes of mathematics and mathematicians, and how to combat them