The course
“Without mathematics, there's nothing you can do. Everything around you is mathematics. Everything around you is numbers.”  Shakuntala Devi
Want to use mathematics at the highest levels in the world of work or research? Then our fouryear MSci course is designed for you. We will challenge you to take your love of mathematics further and help you to develop an invaluable portfolio of skills that will set you apart in your future career. Guided by internationally recognised experts in the field, you will gain a thorough knowledge and understanding of all the key methods and concepts of pure and applied mathematics, probability, statistics, financial mathematics and the mathematics of information in your first year. You will then be free to tailor your studies to the areas that interest you the most, thanks to the courses modular structure, and in year 4 you will have the opportunity to complete a supervised project, alongside further optional modules.
This course is particularly aimed at those who want to explore the full breadth of mathematical fields, and go on to do postgraduate study or to pursue a scientific or technical career. Our graduates are in demand for their numeracy, analytical skills, data handling powers, logical thinking, advanced research skills and creative problem solving abilities. Our comprehensive curriculum is influenced by the department’s worldclass research activities and you will study topics ranging from pure mathematics to mathematical modelling, discrete mathematics, statistics, cryptography, quantum mechanics, informatics, and financial mathematics. Our staff are involved in pioneering research that is making an impressive impact on the global stage and we offer an extensive array of postgraduate opportunities. We are internationally renowned for our work in pure mathematics, information security, quantum dynamics, statistics and theoretical physics.
Join our friendly and inspiring department and you will benefit from a thoroughly supportive learning environment. We offer small group tutorials, problem solving sessions, practical workshops and IT classes, as well as generous staff office hours and a dedicated personal adviser to guide you through your studies. We also offer CV writing workshops and a competitive work placement scheme.
 Take your love of mathematics to a higher level and tailor your studies to your own interests, culminating in a supervised research project.
 Learn from our inspirational teachers.
 Feel at home in a lively, friendly department with a strong focus on small group teaching, where you will be known as an individual.
Course structure
Core Modules
Year 1
In this module you will develop an understanding of how mathematics has been used to describe space over the last 2,500 years. You will look at ruler and compass constructions from ancient Greece, the influence of algebra on geometry in the renaissance, and the intricate and beautiful fractal patterns developed by Benoît Mandelbrot in the 1970s. You will learn to sketch simple curves using polar coordinates, draw and classify conics, and use simple arguments to distinguish between countable and uncountable sets.

In this module, you will develop an understanding of how the techniques for solving differential equations can be applied to describe the real world. You will look at situations from balls flying through the air to planets orbiting the stars, including why the moon continues to orbit the Earth and not the Sun. You will consider the chatotic motion of a pendulum, and examine Einstein's theory of special relativity to describe the propagation of matter and light at high speeds.

In this module you will develop an understanding of the notion of probability and the basic theory and methods of statistics. You will look at random variables and their distributions, calculate probabilities of events that arise from standard distributions, estimate means and variances, and carry out t tests for means and differences of means. You will also consider the notions of types of error, power and significance levels, gaining experience in sorting a variety of data sets in a scientific way.

In this module, you will develop an understanding of the key concepts in Calculus, including differentiation and integration. You will learn how to factorise polynomials and separate rational functions into partial fractions, differentiate commonly occurring functions, and find definite and indefinite integrals of a variety of functions using substitution or integration by parts. You will also examine how to recognise the standard forms of firstorder differential equations, and reduce other equations to these forms and solve them.

In this module you will develop an understanding of the calculus functions of more than one variable and how it may be used in areas such as geometry and optimisation. You learn how to manipulate partial derivatives, construct and manipulate line integrals, represent curves and surfaces in higher dimensions, calculate areas under a curve and volumes between surfaces, and evaluate double integrals, including the use of change of order of integration and change of coordinates.

In this module you will develop an understanding of the fundamental algebraic structures, including familiar integers and polynomial rings. You will learn how to apply Euclid's algorithm to find the greatest common divisor of two integers, and use mathematical induction to prove simple results. You will examine the use of arithmetic operations on complex numbers, extract roots of complex numbers, prove De Morgan's laws, and determine whether a given mapping is bijective.

In this module you will develop an understanding of key mathematical concepts such as the construction of real numbers, limits and convergence of sequences, and continuity of functions. You will look at the infinite processes that are essential for the development of areas such as calculus, determining whether a given sequence tends to a limit, and finding the limits of sequences defined recursively.

In this module you will develop an understanding of vectors and matrices within the context of vector spaces, with a focus on deriving and using various decompositions of matrices, including eigenvalue decompositions and the socalled normal forms. You will learn how these abstract notions can be used to solve problems encountered in other fields of science and mathematics, such as optimisation theory.

In this module you will develop an understanding of the basic complex variable theory. You will look at the definitions of continuity and differentiability of a complex valued function at a point, and how CauchyRiemann equations can be applied. You will examine how to use a power series to define the complex expontential function, and how to obtain Taylor series of rational and other functions of standard type, determining zeros and poles of given functions. You will also consider how to use Cauchy's Residue Theorem to evaulate real integrals.

In this module you will develop an understanding of the convergence of series. You will look at the Weierstrass definition of a limit and use standard tests to investigate the convergence of commonly occuring series. You will consider the power series of standard functions, and analyse the Intermediate Value and Mean Value Theorems. You will also examine the properties of the Riemann integral.

This course introduces you to the basics of Python programming for building solutions to mathematicsbased tasks. This will encourage a deeper understanding of the mathematics that you will learn across your degree, developing general mathematical skills and group working. Additionally, the module will provide guidance through the process of applying for a summer internship or job, as well as reviewing the range of career options available to Mathematicians upon graduation.
 All modules are optional

You will carry out a detailed investigation on a topic of your choosing, guided by an academic supervisor. You will prepare a written report around 10,000 words in length, and give a fifteenminute presentation outlining your findings.
Optional Modules
There are a number of optional course modules available during your degree studies. The following is a selection of optional course modules that are likely to be available. Please note that although the College will keep changes to a minimum, new modules may be offered or existing modules may be withdrawn, for example, in response to a change in staff. Applicants will be informed if any significant changes need to be made.
Year 1 All modules are core

In this module you will develop an understanding of the concepts of scalar and vector fields. You examine how vector calculus is used to define general coordinate systems and in differential geometry. You will learn how to solve simple partial differential equations by separating variables, and become familiar with how these concepts can be appield in the field of dynamics of inviscid fluids.

In this module you will develop an understanding of statistical modelling, becoming familiar with the theory and the application of linear models. You will learn how to use the classic simple linear regression model and its generalisations for modelling dependence between variables. You will examine how to apply nonparametric methods, such as the Wilxocon and KolmogorovSmirnov goodnessoffit tests, and learn to use the R open source software package.

In this module you will develop an understanding of the basic principles of the mathematical theory of probability. You will use the fundamental laws of probability to solve a range of problems, and prove simple theorems involving discrete and continuous random variables. You will learn how to forumulate an explain fundamental limit theorems, such as the weak law of large numbers and the central limit theorem.

In this module you will develop an understanding of the basic concepts of graph theory and linear programming. You will consider how railroad networks, electrical networks, social networks, and the web can be modelled by graphs, and look at basic examples of graph classes such as paths, cycles and trees. You will examine the flows in networks and how these are related to linear programming, solving problems using the simplex algorithm and the strong duality theorem.

In this module you will develop an understanding of the concepts arising when the boundary conditions of a differential equation involves two points. You will look at eigenvalues and eigenfunctions in trigonometric differential equations, and determine the Fourier series for a periodic function. You will learn how to manipulate the Dirac deltafunction and apply the Fourier transform. You will also examine how to solve differential equations where the coefficients are variable.

In this module you will develop an understanding of ring theory and how this area of algebra can be used to address the problem of factorising integers into primes. You will look at how these ideas can be extended to develop notions of 'prime factorisation' for other mathematical objects, such as polynomials. You will investigate the structure of explicit rings and learn how to recognise and construct ring homomorphisms and quotients. You will examine the Gaussian integers as an example of a Euclidean ring, Kronecker's theorem on field extensions, and the Chinese Remainder Theorem.

In this module you will develop an understanding of the algebraic structures known as groups. You will look at how groups represent symmetries in the world around us, examining examples that arise from the theory of matrices and permutations. You will see how groups are ubiquitous and used in many different fields of human study, including mathematics, physics, the study of crystals and atoms, public key cryptography, and music theory. You also will also consider how various counting problems concerning discrete patterns can be solved by means of group actions.

In this module you will develop an understanding of the language and concepts of linear algebra that are used within Mathematics. You will look at topics in linear algebra and the theory of modules, which can be seen as generalisations of vector spaces. You will learn how to use alternative matrix representations, such as the Jordan canonical or the rational canonical form, and see why they are important in mathematics.

You will carry out a detailed investigation on a topic of your choosing, guided by an academic supervisor. You will prepare a written report around 7,000 words in length, and give a tenminute presentation outlining your findings.

In this module you will develop an understanding of a range of methods for teaching children up to Alevel standard. You will act act as a role model for pupils, devising appropriate ways to convey the principles and concepts of mathematics. You will spend one session a week in a local school, taking responsibility for preparing lesson plans, putting together relevant learning aids, and delivering some of the classes. You will work with a specific teacher, who will act as a trainer and mentor, gaining valuable transferable skills.

In this module you will develop an understanding of how prime numbers are the building blocks of the integers 0, ±1, ±2, … You will look at how simple equations using integers can be solved, and examine whether a number like 2017 should be written as a sum of two integer squares. You will also see how Number Theory can be used in other areas such as Cryptography, Computer Science and Field Theory.

In this module you will develop an understanding of a range methods used for testing and proving primality, and for the factorisation of composite integers. You will look at the theory of binary quadratic forms, elliptic curves, and quadratic number fields, considering the principles behind stateofthe art factorisation methods. You will also look at how to analyse the complexity of fundamental numbertheoretic algorithms.

In this module you will develop an understanding the different classes of computational complexity. You will look at computational hardness, learning how to deduce cryptographic properties of related algorithms and protocols. You will examine the concept of a Turing machine, and consider the millennium problems, including P vs NP, with a $1,000,000 prize on offer from the Clay Mathematics Institute if a correct solution can be found.

In this module you will develop an understanding of efficient algorithm design and its importance for handling large inputs. You will look at how computers have changed the world in the last few decades, and examine the mathematical concepts that have driven these changes. You will consider the theory of algorithm design, including dynamic programming, handling recurrences, worstcase analysis, and basic data structures such as arrays, stacks, balanced search trees, and hashing.

In this module you will develop an understanding of quantum theory, and the development of the field to explain the behaviour of particles at the atomic level. You will look at the mathematical foundations of the theory, including the Schrodinger equation. You will examine how the theory is applied to one and three dimensional systems, including the hydrogen atom, and see how a probabilistic theory is required to interpret what is measured.

In this module you will develop an understanding of how the RayleighRitz variational principle and perturbation theory can be used to obtain approximate solutions of the Schrödinger equation. You will look at the mathematical basis of the Period Table of Elements, considering spin and the Pauli exclusion principle. You will also examine the quantum theory of the interaction of electromagnetic radiation with matter.

In this module you will develop an understanding of how the theory of ideal fluids can be used to explain everyday phenomena in the world around us, such as how sound travels, how waves travel over the surface of a lake, and why golden syrup (or volcanic lava) flows differently from water. You will look at the essential features of compressible flow and consider basic vector analysis techniques.

In this module, you will develop an understanding of nonlinear dynamical systems. You will investigate whether the behaviour of a nonlinear system can be predicted from the corresponding linear system, and see how dynamical systems can be used to analyse mechanisms such as the spread of disease, the stability of the universe, and the evolution of economic systems. You will gain an insight into the 'secrets' of the nonlinear world and the appearance of chaos, examining the significant developments achieved in this field during the final quarter of the 20th Century.

In this module you will develop an understanding of the mathematics of communication, focusing on digital communication as used across the internet and by mobile telephones. You looking at compression, considering how small a file, such as a photo or video, can be made, and therefore how the use of data can be minimised. You will examine error correction, seeing how communications may be correctly received even if something goes wrong during the transmission, such as intermittent wifi signal. You will also analyse the noiseless coding theorem, defining and using the concept of channel capacity.

In this module you will develop an understanding of how the behaviour of quantum systems can be harnessed to perform information processing tasks that are otherwise difficult, or impossible, to carry out. You will look at basic phenomena such as quantum entanglement and the nocloning principle, seeing how these can be used to perform, for example, quantum key distribution. You will also examine a number of basic quantum computing algorithms, observing how they outperform their classical counterparts when run on a quantum computer.

In this module you will develop an understanding of how financial markets operate, with a focus on the ideas of risk and return and how they can be measured. You will look at the random behaviour of the stock market, Markowitz portfolio optimisation theory, the Capital Asset Pricing Model, the Binomial model, and the BlackScholes formula for the pricing of options.

In this module you will develop an understanding of the role of mathematics and statistics in securities markets. You will investigate the validity of various linear and nonlinear time series occurring in finance, and apply stochastic calculus, including partial differential equations, for interest rate and credit analysis. You will also consider how spot rates and prices for Asian and barrier exotic options are modelled.

In this module you will develop an understanding of some of the standard techniques and concepts of combinatorics, including methods of counting, generating functions, probabilistic methods, permutations, and Ramsey theory. You will see how algebra and probability can be used to count abstract mathematical objects, and how to calculate sets by inclusion and exclusion. You will examine the applications of number theory and consider the use of simple probabilistic tools for solving combinatorial problems.

In this module you will develop an understanding of how error correcting codes are used to store and transmit information in technologies such as DVDs, telecommunication networks and digital television. You will look at the methods of elementary enumeration, linear algebra and finite fields, and consider the main coding theory problem. You will see how error correcting codes can be used to reconstruct the original information even if it has been altered or degraded.

In this module you will develop an understanding of how error correcting codes are used to store and transmit information in technologies such as DVDs, telecommunication networks and digital television. You will look at the methods of elementary enumeration, linear algebra and finite fields, and consider the main coding theory problem. You will see how error correcting codes can be used to reconstruct the original information even if it has been altered or degraded.

In this module you will develop an understanding of public key cryptography and the mathematical ideas that underpin it, including discrete logarithms, lattices and elliptic curves. You will look at several important public key cryptosystems, including RSA, Rabin, ElGamal encryption and Schnorr signatures. You will consider notions of security and attack models relevant for modern theoretical cryptography, such as indistinguishability and adaptive chosen ciphertext attack.

In this module you will develop an understanding of Field Theory. You will learn how to express equations such as X^{2017}=1 in a formal algebraic setting, how to classify finite fields, and how to determine the number of irreducible polynomials over a finitie field. You will also consider some the applications of fields, including ruler and compass constructions and why it is impossible to generically trisect an angle using them.

In this module you will develop an understanding of the algebraic structures known as groups. You will look at how groups represent symmetries in the world around us, examining examples that arise from the theory of matrices and permutations. You will see how groups are ubiquitous and used in many different fields of human study, including mathematics, physics, the study of crystals and atoms, public key cryptography, and music theory. You also will also consider how various counting problems concerning discrete patterns can be solved by means of group actions.

In this module you will develop an understanding of the language and concepts of linear algebra that are used within Mathematics. You will look at topics in linear algebra and the theory of modules, which can be seen as generalisations of vector spaces. You will learn how to use alternative matrix representations, such as the Jordan canonical or the rational canonical form, and see why they are important in mathematics.

In this module you will develop an understanding of geometric objects and their properties. You will look at objects that are preserved under continuous deformation, such as through stretching or twisting, and will examine knots and surfaces. You will see how colouring a knot can be used to determine whether or not it can be transformed into the unknot without any threading. You will also consider why topologists do not distinguish between a cup and a donut.

In this module you will develop an understanding of a range of methods used for testing and proving primality, and for the factorisation of composite integers. You will look at the theory of binary quadratic forms, elliptic curves, and quadratic number fields, considering the principles behind stateoftheart factorisation methods. You will also look at how to analyse the complexity of fundamental numbertheoretic algorithms.

In this module you will develop an understanding the different classes of computational complexity. You will look at computational hardness, learning how to deduce cryptographic properties of related algorithms and protocols. You will examine the concept of a Turing machine, and consider the millennium problems, including P vs NP, with a $1,000,000 prize on offer from the Clay Mathematics Institute if a correct solution can be found.

In this module you will develop an understanding of efficient algorithm design and its importance for handling large inputs. You will look at how computers have changed the world in the last few decades, and examine the mathematical concepts that have driven these changes. You will consider the theory of algorithm design, including dynamic programming, handling recurrences, worstcase analysis, and basic data structures such as arrays, stacks, balanced search trees, and hashing.

In this module you will develop an understanding of how the RayleighRitz variational principle and perturbation theory can be used to obtain approximate solutions of the Schrödinger equation. You will look at the mathematical basis of the Period Table of Elements, considering spin and the Pauli exclusion principle. You will also examine the quantum theory of the interaction of electromagnetic radiation with matter.

In this module you will develop an understanding of the main priciples and methods of statstics, in particular the theory of parametric estimation and hypotheses testing.You will learn how to formulate statistical problems in mathematical terms, looking at concepts such as Bayes estimators, the NeymanPearson framework, likelihood ratio tests, and decision theory.

In this module you will develop an understanding of the probabilistic methods used to model systems with uncertain behaviour. You will look at the structure and concepts of discrete and continuous time Markov chains with countable stable space, and consider the methods of conditional expectation. You will learn how to generate functions, and construct a probability model for a variety of problems.

In this module you will develop an understanding of the mathematics of communication, focusing on digital communication as used across the internet and by mobile telephones. You looking at compression, considering how small a file, such as a photo or video, can be made, and therefore how the use of data can be minimised. You will examine error correction, seeing how communications may be correctly received even if something goes wrong during the transmission, such as intermittent wifi signal. You will also analyse the noiseless coding theorem, defining and using the concept of channel capacity.

In this module you will develop an understanding of how the behaviour of quantum systems can be harnessed to perform information processing tasks that are otherwise difficult, or impossible, to carry out. You will look at basic phenomena such as quantum entanglement and the nocloning principle, seeing how these can be used to perform, for example, quantum key distribution. You will also examine a number of basic quantum computing algorithms, observing how they outperform their classical counterparts when run on a quantum computer.

In this module you will develop an understanding of the role of mathematics and statistics in securities markets. You will investigate the validity of various linear and nonlinear time series occurring in finance, and apply stochastic calculus, including partial differential equations, for interest rate and credit analysis. You will also consider how spot rates and prices for Asian and barrier exotic options are modelled.

In this module you will develop an understanding of some of the standard techniques and concepts of combinatorics, including methods of counting, generating functions, probabilistic methods, permutations, and Ramsey theory. You will see how algebra and probability can be used to count abstract mathematical objects, and how to calculate sets by includion an exclusion. You will examine the applications of number theory and consider the use of simple probabilistic tools for solving combinatorial problems.

In this module you will develop an understanding of how error correcting codes are used to store and transmit information in technologies such as DVDs, telecommunication networks and digital television. You will look at the methods of elementary enumeration, linear algebra and finite fields, and consider the main coding theory problem. You will see how error correcting codes can be used to reconstruct the original information even if it has been altered or degraded.

In this module you will develop an understanding of secure communication and how cryptography is used to achieve this. You will look at some of the historical cipher systems, considering what security means and the kinds of attacks an adversary might launch. You will examine the structure of stream ciphers and block ciphers, and the concept of public key cryptography, including details of the RSA and ElGamal cryptosystems. You will see how these techniques are used to achieve privacy and authentication, and assess the problems of key management and distribution.

In this module you will develop an understanding of public key cryptography and the mathematical ideas that underpin it, including discrete logarithms, lattices and elliptic curves. You will look at several important public key cyptosystems, including RSA, Rabin, ElGamal encryption and Schnorr signatures. You will consider notions of security and attack models relevant for modern theoretical cryptography, such as indistinguishability and adaptive chosen ciphertext attack.

In this module you will develop an understanding of Field Theory. You will learn how to express equations such as X^{2017}=1 in a formal algebraic setting, how to classify finite fields, and how to determine the number of irreducible polynomials over a finitie field. You will also consider some the applications of fields, including ruler and compass constructions and why it is impossible to generically trisect an angle using them.

In this module you will develop an understanding of geometric objects and their properties. You will look at objects that are preserved under continuous deformation, such as through stretching or twisting, and will examine knots and surfaces. You will see how colouring a knot can be used to determine whether or not it can be transformed into the unknot without any threading. You will also consider why topologists do not distinguish between a cup and a donut.
Teaching & assessment
The course has a flexible, modular structure and you will study the equivalent of four, 30credit modules per year. In addition to our compulsory core modules you will be free to choose between a number of optional courses in years 2 to 4. Most contribute 15 credits to your overall award, while the supervised project in year 4 is worth 30 credits. You will have the option of transferring onto our threeyear Mathematics BSc course during your studies, if you wish to.
We use a variety of teaching methods and there is a strong focus on small group teaching in the department. You will attend 12 to 15 hours of formal teaching in a typical week, including lectures, tutorials, problem solving workshops and practical sessions in statistics and computational mathematics. You will also be expected to work on worksheets, revision and project work outside of these times. In year 2, teaching will mainly be delivered through lectures and workshops and in years 3 and 4, mostly through relatively small group lectures and tutorials.
Our courses are mostly examined by written exams taken in the summer term, but some of our statistics and computational courses also have project components and between 10% and 30% of your final mark for each core module in year 1 will come from coursework. Some of the first year modules also include tests that contribute 10% of the final mark. In the first term of year 2 you will work in small groups to prepare a report and an oral presentation on a mathematical topic of your choice, which will contribute to one of your core module grades. CV writing skills are also embedded into that course. In year 3 there are two optional courses which are examined solely by a project and presentation, and in year 4 you will complete a supervised project.
Entry requirements
A Levels: AAAAAB
Required subjects:
 Alevel in Mathematics at grade A
 At least five GCSEs at grade A*C or 94 including English and Mathematics.
Where an applicant is taking the EPQ alongside A  levels, the EPQ will be taken into consideration and result in lower Alevel grades being required. Socio  economic factors which may have impacted an applicant's education will be taken into consideration and alternative offers may be made to these applicants.
Other UK and Ireland Qualifications
International & EU requirements
English language requirements
All teaching at Royal Holloway (apart from some language courses) is in English. You will therefore need to have good enough written and spoken English to cope with your studies right from the start.
The scores we require
 IELTS: 6.0 overall. No subscore lower than 5.5.
 Pearson Test of English: 61 overall. No subscore lower than 51.
 Trinity College London Integrated Skills in English (ISE): ISE III.
 Cambridge English: Advanced (CAE) grade C.
Countryspecific requirements
For more information about countryspecific entry requirements for your country please visit here.
For international students who do not meet the direct entry requirements, we offer an International Foundation Year, run by Study Group at the Royal Holloway International Study Centre. Upon successful completion, you may progress on to selected undergraduate degree programmes at Royal Holloway, University of London.
Your future career
Graduates from this course are in high demand for their broad and deep understanding of mathematical methods and concepts, their ability to handle complex data sets, approach problems creatively and logically and undertake specialist research. We have a strong track record of producing high achievers who go on to enjoy rewarding and high profile careers.
Our department is part of the School of Mathematics and Information Security and we enjoy strong ties with both the information security sector and industry at large. Graduates from our department have successfully secured positions in business management, IT consultancy, computer analysis and programming, accountancy, the civil service, teaching, actuarial science, finance, risk analysis, research and engineering. We have graduates working for organisations such as KPMG, Ernst & Young, the Ministry of Defence, Barclays Bank, Lloyds Banking Group, the Department of Health, Logica, McLaren and TowersWatson, and in research teams tackling problems as diverse as aircraft design, operational research and cryptography. We also offer an extensive array of further postgraduate opportunities and depending on your choice of courses, you could be eligible for certain membership exemptions from the Institute of Actuaries and other professional bodies.
We offer a competitive work experience scheme at the end of year 2, with shortterm placements available during the summer holidays. You will also attend a CV writing workshop as part of your core modules in year 2, and your personal adviser and the campus Careers team will be on hand to offer advice and guidance on your chosen career. The Royal Holloway Careers and Employability Service also offers tailored sessions for Mathematics students, on finding summer internships or holiday jobs and securing employment after graduation.
 Equip yourself with the knowledge and skills to take mathematics to the highest levels, in research, science or industry.
 Our strong ties with industry mean we understand the needs of employers.
 Take advantage of our summer work placement scheme and learn to finetune your CV.
 Benefit from a personal adviser who will guide you through your studies and career options.
Fees & funding
Home and EU students tuition fee per year*: £9,250
International students tuition fee per year**: £18,300
Other essential costs***: There are no single associated costs greater than £50 per item on this course