The course
By studying Italian alongside your mathematics, you will have the chance to explore Italy’s role as the cradle of European culture and its influential contributions to modern literature, film, the visual arts, fashion and design. There has always been a logical connection between the study of mathematics and languages and this programme allows you to immerse yourself in both and leave the doors open as to your future career options.
Galileo famously described the universe as a book written in a mathematical language, and this concept of mathematics as a universal language has never been more relevant in our technologically advanced and globalised world. On this programme you will gain a thorough grounding in the key concepts and methods of mathematics, which comprises 75% of the programme, whilst honing your Italian language skills and gaining valuable crosscultural perspectives. Our Italian classes are taught by language specialists with research interests in Italian film, literature and the visual arts. Most of them are native speakers and our oral classes are all taught in Italian. We teach modern languages at beginner or advanced levels, depending on your previous experience.
All three years of this joint honours programme are completed in the UK. Its modular structure allows you to tailor your studies to your own interests. Our Department of Mathematics is internationally renowned for its work in pure mathematics, information security, statistics and theoretical physics, and our broad curriculum spans pure and applied mathematics, statistics and probability, and the mathematics of information and of financial markets. You will not only improve your numerical skills and your proficiency at reading, writing, listening and speaking in Italian, but you will also develop transferrable skills such as critical thinking, analysis, research, data handling and creative problem solving. Both departments offer friendly and motivating learning environments and a strong focus on small group teaching and academic support, with a personal adviser to guide you throughout your studies. We also offer a competitive work placement scheme.
 Specialise in mathematics but hone your language skills by learning Italian to an advanced level.
 Work across two friendly departments, where the focus is on small group teaching and you will be known as an individual.
 Tailor your degree to your own interests, with our wide range of optional modules.
Course structure
Core Modules
Year 1
Calculus
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.

Number Systems
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.

Numbers and Functions
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.

Functions of Several Variables
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.

Matrix Algebra
In this module you will develop an understanding of basic linear algebra, in particular the use of matrices and vectors. You will look at the basic theoretical and computational techniques of matrix theory, examining the power of vector methods and how they may be used to describe threedimensional space. You will consider the notions of field, vector space and subspace, and learn how to calculate the determinant of an n x n matrix.
 Intensive Italian for Beginners
 Advanced Italian I
 Italian Language: Culture and Translation

Linear Algebra and Project
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. Working in small groups, you will put together different aspects of mathematics in a project on a topic of your choosing, disseminating your findings in writing and giving an oral presentation to your peers.

Complex Variable
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.
 Advanced Italian II for Post Beginners
 Advanced Italian II
 Advanced Italian Translation: Skills and Practice
 Advanced Italian III
Optional Modules
Year 1
From Euclid to Mandelbrot
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.

Introduction to Applied Mathematics
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.

Principles of Statistics
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.

Vector Analysis and Fluids
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.

Statistical Methods
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 Minitab statistical software package.

Rings and Factorisation
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.

Real Analysis
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.

Probability
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.

Graphs and Optimisation
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.

Ordinary Differential Equations and Fourier Analysis
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.

Groups and Group Actions
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.

Further Linear Algebra and Modules
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.

Mathematics Project
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.

Number Theory
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.

Complexity Theory
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.

Quantum Theory 1
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.

NonLinear Dynamic Systems
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.

Inference
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.

Time Series Analysis
In this module you will develop an understanding of some of the descriptive methods and theoretical techniques that are used to analyse time series. You will look at the standard theory around several prototype classes of time series models and learn how to apply appropriate methods of times series analysis and forecasting to a given set of data using Minitab, a statistical computing package. You will examine inferential and associated algorithmic aspects of timeseries modelling and simulate time series based on several prototype classes.

Channels
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.

Mathematics of Financial Markets
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.

Combinatorics
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.

Cipher Systems
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.

Electromagnetism
In this module you will develop an understanding of the elementary ideas of electromagnetism. You will learn how to calculate electric fields and electric potentials from given fixed charge distributions and how to calculate magnetic fields and vector potentials from given steady current distributions. You will examine the magnetic effects of currents, including electromagnetic induction and displacement currents, and analyse the BiotSavart law and Ampere's law. You will examine Maxwell's equations, and the properties of electromagnetic waves in free space, as well as electric and magnetic dipoles and the electromagnetism of matter.

Applications of Field Theory
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.

Mathematics in the Classroom
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.

Computational Number 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.

Principles of Algorithm Design
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.

Dynamics of Real Fluids
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.

Quantum Theory 2
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.

Applied Probability
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.

Quantum Information and Coding
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.

Advanced Financial Mathematics
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.

Error Correcting Codes
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.

Public Key Cryptography
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.

Groups and Group Actions
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.

Further Linear Algebra and Modules
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.

Topology
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 programme has a flexible, modular structure and you will take a total of 12 course units at a rate 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. Some contribute 15 credits to your overall award while others contribute the full 30.
We use a variety of teaching methods and there is a strong focus on small group teaching throughout the programme. You will attend 12 to 15 hours of formal teaching in a typical week, including lectures, seminars, group tutorials, statistics and IT classes, problem solving workshops in mathematics, and role play and conversational classes in Italian. You will also be expected to work on worksheets, revision and project work outside of class time, and you will have access to a host of online resources on Moodle, the University's elearning facility. In year 2, much of our mathematics teaching will be delivered through lectures, workshops and practical classes, and in year 3, mostly through relatively small group lectures and supervised project work. Our language teaching is mainly through seminars and small group work, with some lectures.
Assessment is through a mixture of coursework and endofyear examinations, depending on the course units you choose to take. Statistics and computational courses may include project work and tests, and Italian coursework will include essays, language and translation exercises and written reports. Some Italian modules include oral presentations and computerbased tests to help assess grammar and comprehension skills. All students work in small groups to prepare a report and an oral presentation on a mathematical topic of their choice, which contributes towards one of the core subject marks in year 2, and two of the optional mathematics units in year 3 are examined solely by a project and presentation.
You will be required to take a study skills course during year 1, to equip you with and enhance the writing skills you will need to be successful in your degree. This does not count towards your final degree but you are required to pass it to progress to your second year.
Entry requirements
A Levels: AABABB
Required subject: Grade A in Mathematics, plus grade B in Italian for the advanced level language pathway, or, for the beginners language pathway, grade B in an essay based subject. At least five GCSEs at grade A*C 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 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.
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, students can progress on to selected undergraduate degree programmes at Royal Holloway, University of London.
Your future career
By combining your mathematics with Italian you will find yourself in demand for your valuable combination of numeracy and language skills, and you will be well placed to take advantage of the globalised jobs market. Employers will value your crosscultural awareness and understanding and your ability to communicate clearly, research effectively, analyse and handle complex data, approach problems with creativity and employ logical thinking. We have a strong track record of preparing our students for the world of work and research.
Graduates from the two departments have gone on to enjoy successful careers in international management and consultancy, computer analysis and programming, teaching, sales and marketing, media and publishing, banking, accountancy, law, the arts, the civil service, politics, travel and tourism, translating and interpreting, finance, risk analysis, research and engineering. They work for employers as diverse as KPMG, Ernst & Young, the Ministry of Defence, Barclays Bank, Lloyds Banking Group, the Department of Health, Logica, McLaren, TowersWatson FleishmanHillard, the Canadian High Commission in London, UBS Investment Bank, BBC, Pearson Education Limited, London Chamber of Commerce, the Foreign and Commonwealth Office, Thomson Reuters, Fremantle Media, Citigroup, Crédit Suisse, JP Morgan Chase, Mills & Reeve Solicitors, Deloitte & Touche LLP, Burberry, the Government Economic Service, Little Brown Book Group, Estée Lauder Companies, Systema Human Information Services, Bloomberg Tradebrook Europe, Pineapple, and Amazon UK.
Our Mathematics department is part of the School of Mathematics and Information Security and we enjoy strong ties with the information security sector as well as with industry at large. 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 offers tailored sessions for mathematics and modern languages students, on finding summer internships or holiday jobs and securing employment after graduation.
 Develop advanced numerical and language skills and a crosscultural perspective that will give you a competitive edge in a globalised world.
 Keep your options open by equipping yourself with language skills that could help you to live and work in different countries.
 Take advantage of our summer work placement scheme and finetune your CV before you enter your final year.
 Benefit from a personal adviser to guide you through all your options.
Fees & funding
Home and EU students tuition fee per year*: £9250
International students tuition fee per year**: £16500
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, scholarships and bursaries.
^{*}The tuition fee for UK and EU undergraduates is controlled by Government regulations, and for students starting a degree in the academic year 2018/19 will be £9,250 for that year. The UK Government has confirmed that EU students starting an undergraduate degree in 2018/19 will pay the same level of fee as a UK student for the duration of their degree.
^{**}Fees for international students may increase yearonyear in line with the rate of inflation. The policy at Royal Holloway is that any increases in fees will not exceed 5% for continuing students. For further information see fees and funding 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 etc., have not been included.