Core modules
The first two years of the MORSE degrees follow a (mainly) fixed set of courses, laying the foundations of the four main subjects. For part of the first two years, and the whole of the third, students are free to choose from a wide range of topics. Final year students can elect to specialise in one or two of the main subject areas or can continue a balanced programme by selecting topics from all four departments.
Year One: The compulsory modules in year one concentrate on the underlying mathematical ideas. You also study basic material from economics and OR.
Year Two: In year two the statistics, economics and OR are developed further, and there is a wide range of optional modules.
At the end of year two, you finalise your choice between the three-year MORSE degree and the four-year MMORSE (the latter requiring you averaged of at least 60%).
Final years: The third year includes compulsory modules on advanced probability, statistical modelling, and financial mathematics. The fourth (final) year of MMORSE offers many modules in probability, statistics, economics, operational research and financial mathematics, and you also choose a masters-level dissertation topic from one of these areas.
Year One
Introduction to Quantitative Economics
The focus of this module is mainly on economic theory but "real world" applications of relevant theories will also be examined, subject to time limitations. The module covers aspects of microeconomics and macroeconomics. Microeconomics is concerned with the economic behaviour of individual consumers and producing firms, and their interaction in markets for goods, services and factors of production. Macroeconomics, on the other hand, is concerned with aggregate economic variables or the workings of the national economy as a whole such as Gross Domestic Product, unemployment, inflation and interest rates, and with government economic policies to influence these variables.
Mathematical Programming I
Operational Research is concerned with advanced analytical methods to support decision making, for example for resource allocation, routing or scheduling. A common problem in decision making is finding an optimal solution subject to certain constraints. Mathematical Programming I introduces you to theoretical and practical aspects of linear programming, a mathematical approach to such optimisation problems.
Vectors and Matrices
Many problems in maths and science are solved by reduction to a system of simultaneous linear equations in a number of variables. Even for problems which cannot be solved in this way, it is often possible to obtain an approximate solution by solving a system of simultaneous linear equations, giving the "best possible linear approximation''.
The branch of maths treating simultaneous linear equations is called linear algebra. The module contains a theoretical algebraic core, whose main idea is that of a vector space and of a linear map from one vector space to another. It discusses the concepts of a basis in a vector space, the dimension of a vector space, the image and kernel of a linear map, the rank and nullity of a linear map, and the representation of a linear map by means of a matrix.
These theoretical ideas have many applications, which will be discussed in the module. These applications include:
Solutions of simultaneous linear equations. Properties of vectors. Properties of matrices, such as rank, row reduction, eigenvalues and eigenvectors. Properties of determinants and ways of calculating them.
Calculus 1/2
Calculus is the mathematical study of continuous change. In this module there will be considerable emphasis throughout on the need to argue with much greater precision and care than you had to at school. With the support of your fellow students, lecturers and other helpers, you will be encouraged to move on from the situation where the teacher shows you how to solve each kind of problem, to the point where you can develop your own methods for solving problems. By the end of the year you will be able to answer interesting questions like, what do we mean by `infinity’?
Sets and Numbers
It is in its proofs that the strength and richness of mathematics is to be found. University mathematics introduces progressively more abstract ideas and structures, and demands more in the way of proof, until most of your time is occupied with understanding proofs and creating your own. Learning to deal with abstraction and with proofs takes time. This module will bridge the gap between school and university mathematics, taking you from concrete techniques where the emphasis is on calculation, and gradually moving towards abstraction and proof.
Introduction to Statistical Modelling
This module is an introduction to statistical thinking and inference. You’ll learn how the concepts you met from Probability can be used to construct a statistical model – a coherent explanation for data. You’ll be able to propose appropriate models for some simple datasets, and along the way you’ll discover how a function called the likelihood plays a key role in the foundations of statistical inference. You will also be introduced to the fundamental ideas of regression. Using the R software package you’ll become familiar with the statistical analysis pipeline: exploratory data analysis, formulating a model, assessing its fit, and visualising and communicating results. The module also prepares you for a more in-depth look at Mathematical Statistics in Year Two.
Probability 1
Probability is a foundational module that will introduce you both to the important concepts in probability but also the key notions of mathematical formalism and problem-solving. Want to think like a mathematician? This module is for you. You will learn how to to express mathematical concepts clearly and precisely and how to construct rigorous mathematical arguments through examples from probability, enhancing your mathematical and logical reasoning skills. You will also develop your ability to calculate using probabilities and expectations by experimenting with random outcomes through the notion of events and their probability. You’ll learn counting methods (inclusion–exclusion formula and binomial co-efficients), and study theoretical topics including conditional probability and Bayes’ Theorem.
Probability 2
This module continues from Probability 1, which prepares you to investigate probability theory in further detail here. Now you will look at examples of both discrete and continuous probability spaces. You’ll scrutinise important families of distributions and the distribution of random variables, and the light this shines on the properties of expectation. You’ll examine mean, variance and co-variance of distribution, through Chebyshev's and Cauchy-Schwarz inequalities, as well as the concept of conditional expectation. The module provides important grounding for later study in advanced probability, statistical modelling, and other areas of potential specialisation such as mathematical finance.
Year Two
Stochastic Processes
The concept of a stochastic (developing randomly over time) process is a useful and surprisingly beautiful mathematical tool in economics, biology, psychology and operations research. In studying the ideas governing stochastic processes, you’ll learn in detail about random walks – the building blocks for constructing other processes as well as being important in their own right, and a special kind of ‘memoryless’ stochastic process known as a Markov chain, which has an enormous range of application and a large and beautiful underlying theory. Your understanding will extend to notions of behaviour, including transience, recurrence and equilibrium, and you will apply these ideas to problems in probability theory.
Mathematical Methods for Statistics and Probability
Following the mathematical modules in Year One, you’ll gain expertise in the application of mathematical techniques to probability and statistics. For example, you’ll be able to adapt the techniques of calculus to compute expectations and conditional distributions relating to a random vector, and you’ll encounter the matrix theory needed to understand covariance structure. You’ll also gain a grounding in the linear algebra underlying regression (such as inner product spaces and orthogonalization). By the end of your course, expect to apply multivariate calculus (integration, calculation of under-surface volumes, variable formulae and Fubini’s Theorem), to use partial derivatives, to derive critical points and extrema, and to understand constrained optimisation. You’ll also work on eigenvalues and eigenvectors, diagonalisation, orthogonal bases and orthonormalisation.
Probability for Mathematical Statistics
If you have already completed Probability in Year One, on this module you’ll have the opportunity to acquire the knowledge you need to study more advanced topics in probability and to understand the bridge between probability and statistics. You’ll study discrete, continuous and multivariate distributions in greater depth, and also learn about Jacobian transformation formula, conditional and multivariate Gaussian distributions, and the related distributions Chi-squared, Student’s and Fisher. You will also cover more advanced topics including moment-generating functions for random variables, notions of convergence, and the Law of Large Numbers and the Central Limit Theorem.
Mathematical Statistics
If you’ve completed “Probability for Mathematical Statistics”, this second-term module is your next step, where you’ll study in detail the major ideas behind statistical inference, with an emphasis on statistical modelling and likelihoods. You’ll learn how to estimate the parameters of a statistical model through the theory of estimators, and how to choose between competing explanations of your data through model selection. This leads you on to important concepts including hypothesis testing, p-values, and confidence intervals, ideas widely used across numerous scientific disciplines. You’ll also discover the ideas underlying Bayesian statistics, a flexible and intuitive approach to inference which is especially amenable to modern computational techniques. Overall this module will provide you a very firm foundation for your future engagement in advanced statistics – in your final years and beyond.
At least one of Mathematical Economics 1A, Economics 2: Microeconomics, or Economics 2: Macroeconomics
You will choose at least one of three key modules in economics. The choice will provide you with a sense of the importance of strategic considerations in economic problem solving. You will see that simple, intuitive principles, formulated precisely, can go a long way in understanding the fundamental aspects of many economic problems. You will also have the flexibility to tailor the specific area of economics to your own interests: Mathematical Economics 1A focuses on game theory, Economics 2: Microeconomics focuses on microeconomics from the points of view of consumers, producers, and competing firms, and Economics 2: Macroeconomics covers a collection of macroeconomic topics such as labour markets, exchange rates, fiscal and monetary policy, and the relationship between unemployment and inflation.
Mathematical Programming II
This module builds on the first year module Mathematical Programming 1. You will learn how to identify the business problems that can be modelled using optimisation techniques and formulate them in a suitable mathematical form. You will then apply optimisation techniques to the solution of the problems using spreadsheets and other appropriate software and learn how to report on the meaning of the optimal solution in a manner suited to a business context.
Year Three
The third year of MMORSE includes compulsory modules which may be on advanced probability, statistical modelling or financial mathematics, depending on the stream chosen.
Year Four
The fourth (final) year of MMORSE offers many modules in probability, statistics, economics, operational research and financial mathematics, and you also choose a Master's level dissertation topic from one of these areas.
Optional modules
Optional modules can vary from year to year. Example optional modules may include:
- Groups and Rings
- Introduction to Mathematical Biology
- Games and Decisions
- Visualisation and Communication of Data
- Simulation
- Introduction to Mathematical Finance
- Programming for Data Science
- Bayesian Forecasting and Intervention
- Mathematics of Machine Learning
- Principles of Entrepreneurship
- Statistical learning and Big Data (MMORSE)
- Advanced Trading Strategies (MMORSE)
