“There is no branch of mathematics, however abstract, which may not someday be applied to phenomena of the real world.” – Nikolai Ivanovich Lobachevsky
If you're looking to take your undergraduate mathematics experience to new levels and develop advanced research skills, this intensive programme covers the wide spectrum of discrete mathematics, applied mathematics and statistics, and addresses some of the key quantifiable challenges and opportunities in the world around us. An interdisciplinary subject by nature, we will help you to apply mathematical concepts and methods to the ever-changing worlds of science, engineering, business, digital technology and industry, and particularly to communication theory, mathematical physics and financial mathematics, where some of our key research interests lie.
The skills you gain will open up a range of career options and provide a solid foundation if you wish to progress to a PhD. You will be guided by renowned specialists in the field who publish in internationally competitive journals and work closely with partners in industry.
Join our friendly and inspiring department and you will benefit from a thoroughly supportive learning environment, with generous staff office hours and a dedicated personal advisor to help you with any queries and guide you through your degree. Our graduates are in demand for their skills in research, numeracy, data handling and analysis, logical thinking and creative problem solving.
- Apply your mathematics to real-world situations and gain the skills to work at a high level in industry, business or research.
- Learn from internationally renowned mathematicians. We rank second in the UK for our research impact and fourth for world leading or internationally excellent research output (Research Excellence Framework 2014).
- Feel at home in a friendly department where you will be known as an individual.
You will carry out a detailed study into a topic of your choosing in mathematics, analysing information from a range of sources. You will submit a written report of between 8,000 and 16,000 words in length.
In addition to these mandatory course units there are a number of optional course units available during your degree studies. The following is a selection of optional course units that are likely to be available. Please note that although the College will keep changes to a minimum, new units may be offered or existing units may be withdrawn, for example, in response to a change in staff. Applicants will be informed if any significant changes need to be made.
Theory of Error-Correcting Codes
In this module you will develop an understanding of the theory of error-correcting codes, employing the methods of elementary enumeration, linear algebra and finite fields. You will learn how to calculate the probability of error or the necessity of retransmission for a binary symmetric channel with given cross-over probability, with and without coding. You will prove and apply various bounds on the number of possible code words in a code of given length and minimal distance, and use Hadamard matrices to construct medium-sized linear codes of certain parameters. You will also consider how to reduce a linear code to standard form, find a parity check matrix, and build standard array and syndrome decoding tables.
In this module you will develop an understanding of the problems of data compression and information transmission in both noiseless and noisy environments. You will look at a range of information-theoretic equalities and inequalities, and data-compression techniques for ergodic as well as memoryless sources. You will consider the proof of the noiseless coding theorem, and define and use the concept of channel capacity of a noisy channel. You will also examine a range of further applications of the theory, such as hash codes or the information-theoretic approach to cryptography and authentication.
Advanced Cipher Systems
In this module you will develop an understanding of the mathematical and security properties of both symmetric key cipher systems and public key cryptography. You will look at the concepts of secure communications and cipher systems, and learn how to use use statistical information and the concept of entropy. You will consider the main properties of Boolean functions, their applications and use in cryptographic algorithms, and the structure of stream ciphers and block ciphers. You will examine how to construct keystream generators, and how to manipulate the concept of perfect secrecy. You will also analyse the concept of public key cryptography, including the details of the RSA and ElGamal cryptosystems.
Public Key Cryptography
In this module you will develop an understanding of the mathematical ideas that underpin public key cryptography, such as discrete logarithms, lattices and ellipticcurves. You will look at the RSA and Rabin cryptosystems, the hard problems on which their security relies, and attacks on them. You will consider finite fields, elliptic curves, and the discrete logarithm problem. You will examine security notions and attack models relevant for modern theoretical cryptography, such as indistinguishability and adaptive chosen ciphertextattack. You will also gain experience in implementing cryptosystems and cryptanalytic methods using software such as Mathematica.
Applications of Field Theory
In this module you will develop an understanding of the basic theory of field extensions. You will learn how to classify finite fields and determine the number of irreducible polynomials over a finite field. You will consider the fundamental thorem of Galois theory and how to compute in a finite field. You will also examine the applications of fields.
Quantum Information and Coding
In this module you will develop an understanding of the principles of quantum superposition and quantum measurement. You will look at the many applications of quantum information theory, and learn how to manipulate tensor-product states and use the concept of entanglement. You will consider a range of problems involving one or two quantum bits and how to apply Grover’s search algorithm. You will also examine applications of entanglement such as quantum teleportation or quantum secret key distribution, and analyse Deutsch's algorithm and its implications for the power of a quantum computer.
Principles of Algorithm Design
In this module you will develop an understanding of the fundamental principles of alogrithm design, including basic data-structures and asymptotic notation. You will look at how algorithms are designed to meet to desired specifications, and consider the importance of algorithmic efficiency. You will also examine fundamental problems such as sorting numbers and multiplying matrices.
Advanced Financial Mathematics
In this module you will develop an understanding of the autoregressive conditionally heteroscedastic family of models in time series and the ideas behind the use of the BDS test and the bispectral test for time series. You will consider the partial differential equation for interest rates and its assumptions, and model forward and spot rates. You will consider the validity of various linear and non-linear time series occurring in finance, and apply stochastic calculus to interest rate movements and credit rating. You will also examine how to model the prices for Asian and barrier options.
In this module you will develop an understanding of the standard techniques and concepts of combinatorics, including methods of counting and the principle of inclusion and exclusion. You will perform simple calculations with generation functions, and look at Ramsey numbers, calculating upper and lower bounds for these. You will consider how to calculate sets by inclusion and exclusion, and examine how to use simple probabilistic tools for solving combinatorial problems.
Computational Number Theory
In this module you will develop an understanding of the major methods used for testing and proving primality and for the factorisation of composite integers. You will look at the mathematical theory that underlies these methods, such as the theory of bionary quadratic forms, elliptic curves, and quadratic number fields. You willl also analayse the complexity of the fundamental number-theoretic algorithms.
In this module you will develop an understanding of the principal methods of the theory of stochastic processes, and probabilistic methods used to model systems that exhibit random behaviour. You will look at methods of conditioning, conditional expectation, and how to generate functions, and examine the structure and concepts of discrete and continuous time Markov chains with countable state space. You will also examine the structure of diffusion processes.
In this module you will develop an understanding of the mathematical theory underlying the main principles and methods of statistics, in particular, parametric estimation and hypotheses testing. You will learn how to formulate statistical problems in rigorous mathematical terms, and how to select and apply appropriate tools of mathematical statistics and advanced probability. You will construct mathematical proofs of some of the main theoretical results of mathematical statistics and consider the asymptotic theory of estimation.
In this module you will develop an understanding of what it means for knots and links to be equivalent. You will look at the properties of a metric space, and learn how to determine whether a given function defines a metric. You will consider how topological spaces are defined and how to verify the exioms for given examples. You will examine the concepts of subspace, product spaces, quotient spaces, Hausdorff space, homeomorphism, connectedness and compactness, and the notions of Euler characteristic, orientability and how to apply these to the classification of closed surfaces.
You will initially choose eight modules from the list of available options, of which you specify modules during the second term that will count towards your final award. You will also complete a core research project under the supervision of one of our academic staff. There is a strong focus on small group teaching throughout the programme.
Assessment is carried out through a variety of methods, including coursework, examinations and the main project. End-of-year examinations in May or June will count for 66.7% of your final award, while the dissertation will make up the remaining 33.3%.
UK 2:1 (Honours) or equivalent with Mathematics as a main field of study and good marks in relevant courses. Exceptionally, at the discretion of the course director, qualifications in other subjects (for example, physics or computer science) or degrees of lower classification may be considered.
English language requirements:
IELTS 6.5 overall with a minimum of 5.5 in all other subscores. For equivalencies, please see here
If you require Royal Holloway to sponser your study in the UK, your IELTS must be a UK government-approved Secure English Language Test (SELT).
International and EU entry requirements
Please select your country from the drop-down list below
Students from overseas should visit the International pages for information on the entry requirements from their country and further information on English language requirements. Royal Holloway offers a Pre-Master’s Diploma for International Students and English language pre-sessional courses, allowing students the opportunity to develop their study skills and English language before starting their postgraduate degree.
By the end of this programme you will have completed a major research project and acquired an advanced knowledge and understanding of: the role and limitations of mathematics in solving problems that arise in real-world scenarios. You will also have impressive skills in selected areas of mathematics and their applications, and the ability to synthesise and interpret information from multiple sources with insight and critical awareness. We will teach you to formulate problems clearly and express your technical work and conclusions clearly in writing, and you will develop valuable transferable skills such as time management, adaptability and self-motivation.
Our graduates have gone on to carry out cutting-edge research in the fields of communication theory and cryptography, as well as successful careers in industries such as: information security, IT consultancy, banking and finance, higher education and telecommunication. They have taken up roles such as: Principal Information Security Consultant at Abbey National PLC; Senior Manager at Enterprise Risk Services, Deloitte & Touche; Global IT Security Director at Reuters; and Information Security Manager at London Underground.
You will have a dedicated personal adviser to guide you through your studies and advise you on postgraduate opportunities, and the campus Careers team will be on hand to offer advice and guidance on your chosen career. The University of London Careers Advisory Service offers regular, tailored sessions for Mathematics students, on finding summer internships or vacation employment and getting into employment.
- Open doors to a range of exciting opportunities in advanced research, science and industry.
- 90% of our graduates are in work or undertaking further study within six months of leaving (Unistats 2015).
- Our strong ties with industry mean we understand the needs of employers.
- Take advantage of our summer work placement scheme and fine-tune your CV before you enter your final year.
- Benefit from a personal advisor who will guide you through your studies and future options.
Home and EU students tuition fee per year 2018/19*: £7,200
International students tuition fee per year 2018/19*: £14,900
Other essential costs**: There are no single associated costs greater than £50 per item on this course
How do I pay for it? Find out more about funding options, including loans, grants, scholarships and bursaries.
* These tuition fees apply to students enrolled on a full-time basis. Students studying part-time are charged a pro-rata tuition fee, usually equivalent to approximately half the full-time fee. Please email firstname.lastname@example.org for further information on part-time fees. All postgraduate fees are subject to inflationary increases. Royal Holloway's policy is that any increases in fees will not exceed 5% for continuing students. For further information see tuition fees and our terms and conditions.
** These estimated costs relate to studying this particular degree programme at Royal Holloway. Costs, such as accommodation, food, books and other learning materials and printing, have not been included.