Finance and Mathematics with a Year in Business

Core Modules

Year 1

You will take the following modules in Finance:

• Principles of Economics

  Principles of Economics is a first-year undergraduate module in how the economy works. The module is suitable for students with or without A-Level economics or equivalent. We will cover the basic theories of macroeconomics (that of the economy as a whole) and microeconomics (the behaviour of individuals, firms and governments and the interactions between them).

  The module adopts the state-of-the-art CORE approach (Curriculum Open-access Resources in Economics) to teaching Principles of Economics. The approach has three pillars which we rely on throughout the module:

  • Formulate a problem that our society is facing now or has faced in the past;
  • Build a theory to explain and solve the problem;
  • Evaluate the usefulness of the theory by using data observations and more novel theories.
• Applied Economics and Policy

  On this module you’ll learn how to think like an economist, by learning to write essays, how to present data, how to collect economic data and how to do economic research. There are four sections each lasting half a term. The sections are economic institutions, economic history, applied economics and policy & experimental & behavioural economics. In each section, students get an overview/introduction to the topic based on economic theory which is followed by some applications/data.

• Academic Integrity

  This module will describe the key principles of academic integrity, focusing on university assignments. Plagiarism, collusion and commissioning will be described as activities that undermine academic integrity, and the possible consequences of engaging in such activities will be described. Activities, with feedback, will provide you with opportunities to reflect and develop your understanding of academic integrity principles.

   

• Statistical Methods 1

  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.

• Calculus I

  This module aims to develop the students’ confidence and skill in dealing with mathematical expressions, to extend their understanding of calculus, and to introduce some topics which they may not have met at A-level. It also aims to ease the transition to university work and to encourage the student to develop good study skills.

• Calculus II

  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.

• Introduction to Pure Mathematics

  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.

• Linear Algebra I

  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 three-dimensional 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.

• Employability 1

  This module will be composed of an introduction to Employability, library resources, team building, and CV making. Career services will provide a session on self-awareness and decision making and library services will present their relevant resources. Finally, the Economics department will organise some team building exercises.

Year 2

• Microeconomics

  In this module you will develop an understanding of the models of individual optimisation and their applications. You will look at the key determinants of an individual’s behaviour in a variety of circumstances and the behaviour of firms in different market environments, such as perfect competition, monopoly and oligopoly. You will consider how changing circumstances and new information influences the actions of the economic agents concerned, and examine the properties of competitive markets and the need for government intervention to correct market failures.

• Macroeconomics

  In this module you will develop an understanding of macroeconomics and macroeconomic policy-making. You will look at a variety of contemporary and historical macroeconomic events, and the differences between the short, medium and long run. You will consider why some countries are rich and some are poor, why different economies grow at different rates, and what determines economic growth and prosperity. You will examine the role of monetary and fiscal policy, its impact on the economy and its limitations. You will also analyse how taxation, budget deficits, and public debt affect the economy.

• Financial Markets and Institutions

  In this module you will develop an understanding of the economic principles underlying the working of national and international financial institutions. You will look at what a financial system is and does, and the distinct functions of each component. You will consider the key financial instruments and the relationship between assets, agents, and institutions, and learn to solve simple problems using quantitative and graphical tools. You will critically evaluate country differences and analyse the interdependencies and rapid change of the modern financial world.

• 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 non-parametric methods, such as the Wilxocon and Kolmogorov-Smirnov goodness-of-fit tests, and learn to use the R open source software package.

• Probability Theory

  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 formulate and explain fundamental limit theorems, such as the weak law of large numbers and the central limit theorem.

• Employability 2

  Career services will provide a session on how to be ready to apply for an internship at the end of the second year. Students will prepare for a psychometric test and will undertake a series of a mock interviews in order to improve their interview technique. Finally, students will attend at least one Econ@Work talk to be aware of professional life and challenges.

Year 3

• Year in Business

  This year will be spent on a work placement. You will be supported by the Placements Office and the Royal Holloway Careers and Employability Service to find a suitable placement. However, Royal Holloway cannot guarantee that all students who are accepted onto this degree programme will secure a placement, and the ultimate responsibility lies with yourself. This year forms an integral part of the degree programme and you will be asked to complete assessed work. The mark for this work will count towards your final degree classification.

Year 4

• Financial Economics 1

  The module aims to introduce the student to what factors affect corporate financial decisions. Particular emphasis is given to the concepts of Net Present Value and Risk. The learning outcomes include: Understand what the goals of a firm are; Understand how investments are valued (Internal rate of returns) in order to help with good financial planning); Understand the concepts of risk, agency costs and how they feed into financial decision making; Understand the process of price formation in financial markets; Understand venture capital and different types of debt finance and debt valuation, including leverage.

• Financial Economics 2

  In this module you will develop an understanding of the wide range of fixed income securities and derivatives available to investors in the financial markets. You will look at the basic institutional features of derivatives markets, as well as the pricing of bonds and of derivative instruments and using them for hedging purposes. You will consider investment and trading strategies that use bonds and derivatives, and evaluate the use of bonds in immunising portfolios based on the bond's duration. You will also explore the features and uses of the most popular types of derivatives available today, including options, futures, forwards, and swaps.

• Financial Mathematics I

  This module shows how mathematics is used by those who work in securities markets. The topics in the module are chosen to demonstrate important applications, and may include the basic asset pricing models, the general behaviour of markets, the theory of derivative securities and elements of portfolio theory.

   

• Financial Mathematics II

  This module continues the study of financial mathematics begun in Financial Mathematics I. In this module, students will develop a further understanding of the role of mathematics in securities markets. The topics in the module may include models of interest rates, elements of portfolio theory and models for financial returns.

Optional Modules

Below is a taster of some of the exciting optional modules that students on the course could choose from during this academic year. Please be aware these do change over time, and optional modules may be withdrawn or new ones added.

Year 1

• All modules are core

Year 2

• Applications of Vector Calculus

  In this module the vector calculus methods are applied to a variety of problems in the physical sciences, with a focus on electromagnetism and optics. On completion of the module, the student should be able to calculate relevant physical quantities such as field strengths, forces, and energy distributions in static as well as dynamical electromagnetic systems and be able to treat mathematically the interactions between moving electrical charges, magnets and optical fields.

• Probability Theory

  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 formulate and explain fundamental limit theorems, such as the weak law of large numbers and the central limit theorem.

• Ordinary Differential Equation

  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 delta-function and apply the Fourier transform. You will also examine how to solve differential equations where the coefficients are variable.

• Linear Algebra II

  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 so-called 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.

• 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.

• 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.

• 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 Cauchy-Riemann 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.

• 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.

Year 4

• Industrial Economics 1

  In this module you will develop an understanding of the methods and models applied by economists in the analysis of firms and industries. You will learn how to manipulate these models and analytically solve problems relating to industrial economics. You will consider the applications of the models to important policy areas, exploring topics such as collusion, mergers, product differentiation and asymmetric information. You will also also examine the limitations of the theory.

• Industrial Economics 2

  In this module you will develop an understanding of advanced topics in industrial organisation, with a special focus on the role that information plays in markets. You will explore topics such as collusion, mergers, product differentiation, and asymmetric information, and become familiar with a broad range of methods and models applied by economists in the analysis of firms and industries.

• Advanced Econometrics I

  In this module you will develop an understanding of the theoretical properties of different econometric estimation and testing procedures under various modelling assumptions. You will learn to formulate, estimate, test and interpret suitable models for the empirical study of economic phenomena. You will consider how to apply regression techniques and evaluate the appropriateness of each econometric estimation method under different data limitations.

• Advanced Econometrics II

  In this module you will develop an understanding of the theoretical properties of different econometric estimation and testing procedures under various modelling assumptions. You will look at regression techniquies and learn how to apply relevant econometric and statistical methods to your own research. You will also evaluate the appropriateness of each of the economic estimation methods and the impact of consider data limirations.

• Understanding Financial Crises

  The module's aim is to provide an honours level analysis of the fragility of the financial system and its relevance to the current financial crisis. The course will broadly follow the book "Understanding Financial Crises", Clarendon Lectures in Finance by Franklin Allen and Douglas Gale - both of whom are now Professors at Imperial College London - which is a book that has been used as the basis of an honours level course recently at UCL. The course will include analysis of bank runs and the theory of optimal financial regulation and analysis of the implications of asset price bubbles for financial stability.

• Environmental Economics

  In this module you will develop an understanding of how economic methods can be applied to environmental issues facing society. You will consider the difficulties arising in using economic analysis in environmental policy design and learn how to solve and manipulate a variety of diagrammatic and algebraic models in environmental economics. You will evaluate a number of real-world environmental policy problems and see how economic analysis has been applied in their solution.

• Behavioural Economics

  In this module you will develop an understanding of the use of experiments to test economic theories. You will look at how individuals make decisions in markets, how individuals decide to spend money today or save it for future spending, the assumption of self-regarding preferences commonly made in standard economic models, and the ability to act rationally in a strategic environment. You will consider the issues raised by experimental design and critically evaluate the advantages and disadvantages of experimental methods.

• Monetary Economics

  This is a half-year module in monetary economics. Its aim is to gain insight into more recent approaches to monetary policy, and to developments in understanding and applying such policy. The main objectives of the course will be to understand the role of money in the economy; identify the links between central bank decision making and commercial bank behaviour; know the basic theoretical underpinnings of monetary policy, monetary policy operating procedures and the central banking mechanisms. A significant part of the class discussions are focused on how the political forces affecting monetary policy making may affect inflation, which has important implications for contemporary policy making.

• Topics in Game Theory

  Game Theory uses mathematical models to study and analyze strategic interactions between agents. This module is designed to provide an understanding of the fundamental concepts in game theory as applied to economics in general and microeconomics in particular. The module will cover both the theory and its applications.

• Advanced Economic Theory

  In this module you will develop an understanding of the mathematics of optimisation and of equilibrium models. You will look at the linkage between markets and Pareto optimality and consider the social outcomes that can be implemented in game-theoretic equilibrium. You will also examine the basic types of auctions and when and why they implement identical outcomes.

   

• Financial Econometrics with Data Science

  The course aims to familiarise students with the principal techniques used in Financial Econometrics. Some in depth discussion of the key technical concepts needed to understand the Financial Econometrics literature. Furthermore, the course makes a real effort to facilitate awareness in students of how these techniques can be used and applied on real data, and provides the necessary background to understand and critically assess empirical findings reported in the financial literature, as well as to carry out their own empirical research in the future.

• Labour Economics

  This course will analyse the economic issues of behaviour and outcomes in labour markets. It will focus on topics relating to labour supply and demand, wage formation and earnings inequalities:

  • Labour Demand
  • Labour Supply
  • Human Capital and Compensating Wage Differentials
  • Inequality in Earnings
  • Labor Mobility
  • Discrimination
  • Unemployment
• Philosophy of Economics

  In this module you will develop an understanding of the different approaches to national economic policy. You will consider the economic advantages and disadvantages of globalisation and look at the effects of tax-cutting, deregulation, privatisation, mixed economy, efficiency and income distribution.

• Public Economics

  Public Economics is concerned with the study of the effects of government policy upon the economy and the design of optimal policies. The module begins with a review of empirical methods for policy evaluation. A number of recent research areas in public economics are then discussed including income taxation, welfare support, behavioural responses, and social security. Throughout the module, the emphasis is placed upon analytical techniques, policy applications, and empirical evidence.

• Topics in Economic History

  This is a survey course covering several important topics in economic history. It is designed to be within the mainstream economics tradition in that the focus is on topics and methodology rather than time periods or countries. The aim is to teach students how economists have analysed important and ongoing historical events and trends.

• Topics in Development Economics

  This module provides an overview of the current literature in development microeconomics, with strong policy and empirical components. The topics covered offer a political economy perspective over development issues. Students will be introduced to theoretical and empirical approaches to the understanding of binding constraints to development such as the quality of political institutions, democracy and elections and ethnical diversity. We will also cover the motivations and solutions to corruption and conflict as large deterrents to development. And how grass-root movements and innovative behavioral mechanisms can improve people’s lives.
  
  Week 1: Development Economics: stylised facts, definitions, and measurement issues
  Week 2: Methodological Issues in Development
  Week 3: Macro perspective of poverty and Foreign Aid
  Week 4: History and Institutions
  Week 5: Ethnic Divisions
  Week 6: Democracy and Elections
  Week 7: Corruption
  Week 8: Conflict
  Week 9: Improving governance in the field
  Week 10: Behavioral insights in development

• Advanced Topics in Game Theory
  In this module you will develop an understanding of the mathematical models used to study and analyse strategic interactions between agents. You will look at the fundamental concepts in game theory as applied to economics in general and microeconomics in particular. You will become familiar with basic equilibrium concepts such as Nash equilibrium and subgame perfect equilibrium, and be able to find equilibrium outcomes of simple games including the use of backward induction.
• Economics of Inequality

  In this module you will develop an understanding of economic inequality. You will look at the factors that determine wage differentials among workers from a theoretical and empirical point of view. You will consider why similar workers are paid differently and examine how labour mobility can improve the allocation of workers to firms, enhance aggregate productivity, and reduce inequality.

• 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 ten-minute 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.

• Non-Linear Dynamic Systems

  In this module you will develop an understanding of non-linear dynamical systems. You will investigate whether the behaviour of a non-linear 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 non-linear world and the appearance of chaos.

• Statistical Inference

  In this module you will develop an understanding of the main principles and methods of statistics, 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 Neyman-Pearson 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 time-series 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 Black-Scholes formula for the pricing of options.

• Combinatorics

  In this module you will develop an understanding of some of the 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 Biot-Savart 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 X²⁰¹⁷=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 A-level standard. You will 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 state-of-the art factorisation methods. You will also look at how to analyse the complexity of fundamental number-theoretic 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, worst-case 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 Rayleigh-Ritz 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.

• Markov Chains and Applications

  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 Theory

  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 no-cloning 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.

• Financial Mathematics II

  This module continues the study of financial mathematics begun in Financial Mathematics I. In this module, students will develop a further understanding of the role of mathematics in securities markets. The topics in the module may include models of interest rates, elements of portfolio theory and models for financial returns.

• 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.