“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.
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.
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 System
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 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 elliptic curves. 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 ciphertext-attack. 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 algorithm design, including basic data-structures and asymptotic notation. You will look at how algorithms are designed to meet 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 axioms 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.
Teaching & assessment
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%.
A UK Honours degree 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.
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: 6.5 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.
For more information about country-specific entry requirements for your country please see here. For international students who do not meet the direct entry requirements, we offer a Pre-Master’s Diploma for International Students (PDIS), a one-year full-time programme that will prepare you for postgraduate study in the UK. For more information please see here.
Your future career
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.
Fees & funding
Home and EU students tuition fee per year*: £7200
International students tuition fee per year**: £14900
Other essential costs***: There are no single associated costs greater than £50 per item on this course.
* and ** 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.