Course content What will you study on the BSc/MSci Mathematics with Computing? The programme begins where your previous learning ends, developing your knowledge of the core areas of mathematics fundamental to further study. You will develop your analytic and problem solving skills, learning how to communicate effectively complex ideas. You will learn to programme throughout your degree, starting in the first year. Additionally you will learn how to develop and study algorithms and determine their efficiency, and you will find out how computers can learn beyond what is programmed. Options in your third year allow you to specialise in computer graphics, artificial intelligence or in other areas, tailoring your development towards your own career choice. The four-year MSci Mathematics with Computing allows you to specialise in your final year to study a huge variety of subjects, thereby letting you develop your own work in a number of cutting-edge areas. BSc Modules Year 1 Vectors and Matrices (30 credits) – Compulsory This module aims to provide an introduction to vector spaces and linear maps. The foundations are laid by studying basic manipulations of complex numbers, vectors and matrices. The underlying geometric meanings of these manipulations are emphasised and concrete examples are explored both by hand and with the help of computer software. Once the foundations have been developed, more advanced and abstract notions are studied for a deeper understanding. Calculus and Differential Equations (30 credits) – Compulsory Integration and differentiation are used to model situations in physics and engineering, as well as in other applications. In this module, we’ll look at how to describe these kinds of equations and you’ll be introduced to the importance of rigour in maths. Logic and Structures (30 credits) – Compulsory One of the fundamental concepts in maths is how unknown ideas are deduced from things that are known. This module takes a closer look at the logic behind argument and develops a keener understanding about the structures that underlie this. The module will develop your appreciation of the way mathematicians think about topics and how we critically analyse arguments. Data and Information (30 credits) – Compulsory We’ll begin to look at how maths is used to analyse information in this module. You will be introduced to some of the ideas behind how patterns and shapes can be deduced from given data and how we can use this information to model and estimate future trends. Year 2 Algorithmic Complexity and Machine Learning (30 credits) – Compulsory This module has two components. First, the Algorithmic Complexity component introduces students to the theory of algorithms and data structures. Algorithms are at the core of every non-trivial computer program and application. Students will learn how to measure the efficiency of an algorithm in terms of its time and space requirements and distinguish between efficient and inefficient algorithmic solutions. General algorithmic design techniques as well as data structures for efficient data manipulation are taught. For this, we study fundamental problems such as sorting, searching, and discrete optimisation problems on graphs, strings and geometry. Second, the Machine Learning component introduces students to algorithmic approaches to learning from exemplar data. Students learn the process of representing training data within appropriate feature spaces for the purposes of classification. The major classifier types are taught before introducing students to specific instances of classifiers along with appropriate training protocols. Where classifiers have a relationship to statistical theory this is fully explored. Notions of structural risk with respect to model fitting are developed such that students are equipped with techniques for managing this in practical contexts. Groups and Rings (30 credits) – Compulsory Groups and Rings are structures used throughout maths to emulate objects like whole numbers or matrices. In this module you’ll be introduced to these structures and you’ll study their properties and what they look like. You’ll find that from very humble beginnings groups and rings lead to a deep and rich theory. Mathematical Analysis (30 credits) – Compulsory This module will begin by looking at what we really mean when we look at limits in mathematics and build on this to give you a greater understanding of series and calculus. The module builds on the ideas introduced in the level 4 modules about the need for rigour in maths. You will develop your ability to question mathematical arguments and to think logically about what definitions mean. Problem Solving Methods (30 credits) – Compulsory The HE Maths Curriculum Summit (2011) recently concluded that “problem-solving is the most useful skill a student can take with them when they leave university”. This module fosters this skill in you by building on the approaches developed throughout the programme to enhance your ability to approach problems in diverse areas of maths and solve them. You will learn how to approach a problem, analyse its properties and develop a strategy to solve it. Rowlett, P. (Ed.) (2011, January), HE Mathematics Curriculum Summit Year 3 Advanced Algebra (30 credits) – Compulsory This module builds on the topics covered in MSO2110 Groups and Rings. The module begins with a review of the material on rings encountered in the prerequisite and proceeds to build towards a study of fields, culminating in the development of Galois Theory. Students will develop their appreciation of the effect additional axioms have on the structure of rings by learning about commutative rings, Euclidean rings, integral domains, fields and other algebraic objects. In the second half of the module students will extend the work on fields and field extensions to develop Galois Theory. Real and Complex Analysis (30 credits) – Compulsory Following from the previous module on mathematical analysis, this module will continue to develop your understanding of infinite and infinitesimal processes. You will also learn how extending the ideas developed here and in your previous module to complex numbers leads to a very different theory. Communicating Mathematics (30 credits) – Compulsory Maths is often called the universal language of science but communicating it can be difficult. With the continuing move to more diverse platforms such as social media this leads to even more challenges in communicating maths. In this module you will look at how maths is communicated, be it to specialists, non-specialists, school pupils or CEOs and how to motivate it for these diverse audiences. Project (30 credits) – Compulsory This is your chance to study an area of maths that you’re interested in and write your own report on it. You can choose your topic yourself or from a list given by staff. If you choose this module then you will have the opportunity to publish your work as an article in a journal. Computer Graphics (30 credits) – Compulsory The aim of this module is to examine in depth the concepts and techniques needed in the construction of interactive graphics and visualisation systems covering advanced graphics programming techniques. It will cover theory and mathematics as required and It aims to provide students with practical experience via significant individual project work developing 2D and 3D programs using an industry standard environment. Artificial Intelligence (30 credits) – Compulsory The aim of the module is to introduce students to a range of AI theories and techniques, including the most commonly used. This will extend to the ability to implement these techniques, and the students will extend their own development skills. Combinatorics (30 credits) – Compulsory This module builds on the discrete mathematics introduced in the second year to provide students with a range of methods for analysing and solving combinatorial problems, with applications across mathematics and to computer science, physics and beyond. It explores ideas of graph theory and design theory, and a range of techniques for solving counting problems both by hand and using computing. It also aims to develop students’ analytical and reasoning skills, and their ability to apply familiar techniques in unfamiliar settings. Multivariate Statistics (30 credits) – Compulsory Understanding and recognising patterns in data can be difficult when it comes from many different sources. In this module, you will begin to develop the techniques and tools that will enable you to study these kinds of relationships. You will develop a critical view of the pros and cons of the methods and the assumptions being made. Simulation and Decision Making (30 credits) – Compulsory Simulating systems such as queuing times at hospitals, or traffic congestion in towns and cities is one of the most important tools used to inform management decisions. In this module, you will be introduced to mathematical simulation and you will learn how to develop models to study systems and improve efficiency. Functional Analysis (30 credits) – Compulsory This module introduces students to the main principles and ideas of functional analysis – a modern branch of mathematical analysis, largely influenced by progress in physics during the 20th century, such as quantum mechanics.  The main object of study is a vector space, the elements of which are functions, so that the space is usually infinite-dimensional.  Starting from simple geometric objects in vector spaces, the module will gradually introduce ideas from other areas of mathematics, such as topology, differentiation, integration, measure, optimisation, differential and integral equations.  Functional analysis will help students to build a unified and beautiful picture about these topics, and achieve deeper understanding of various branches of mathematics and their applications. Differential Equations (30 credits) – Compulsory Differential equations (DEs) are essential mathematical tools used to describe fundamental laws of nature. This module will equip you with general knowledge of the main types of these equations (ordinary and partial, linear and non-linear), which will be illustrated by examples from the natural sciences. Several techniques for solving differential equations will be presented, alongside a broad geometric understanding of possible solution behaviours. MSci Modules Year 1 Vectors and Matrices (30 credits) – Compulsory Vectors and matrices are the mathematical building blocks used in areas ranging from theoretical physics to computer graphics, as well as providing the basis for an understanding of how structures in maths interact. This module will introduce you to the methods and techniques used to analyse vectors and matrices. Calculus and Differential Equations (30 credits) – Compulsory Integration and differentiation are used to model situations in physics and engineering, as well as in other applications. In this module, we’ll look at how to describe these kinds of equations and you’ll be introduced to the importance of rigour in maths. Logic and Structures (30 credits) – Compulsory One of the fundamental concepts in maths is how unknown ideas are deduced from things that are known. This module takes a closer look at the logic behind argument and develops a keener understanding about the structures that underlie this. The module will develop your appreciation of the way mathematicians think about topics and how we critically analyse arguments. Data and Information (30 credits) – Compulsory We’ll begin to look at how maths is used to analyse information in this module. You will be introduced to some of the ideas behind how patterns and shapes can be deduced from given data and how we can use this information to model and estimate future trends. Year 2 Algorithmic Complexity and Machine Learning (30 credits) – Compulsory This module has two components. First, the Algorithmic Complexity component introduces students to the theory of algorithms and data structures. Algorithms are at the core of every non-trivial computer program and application. Students will learn how to measure the efficiency of an algorithm in terms of its time and space requirements and distinguish between efficient and inefficient algorithmic solutions. General algorithmic design techniques as well as data structures for efficient data manipulation are taught. For this, we study fundamental problems such as sorting, searching, and discrete optimisation problems on graphs, strings and geometry. Second, the Machine Learning component introduces students to algorithmic approaches to learning from exemplar data. Students learn the process of representing training data within appropriate feature spaces for the purposes of classification. The major classifier types are taught before introducing students to specific instances of classifiers along with appropriate training protocols. Where classifiers have a relationship to statistical theory this is fully explored. Notions of structural risk with respect to model fitting are developed such that students are equipped with techniques for managing this in practical contexts. Groups and Rings (30 credits) – Compulsory Groups and Rings are structures used throughout maths to emulate objects like whole numbers or matrices. In this module you’ll be introduced to these structures and you’ll study their properties and what they look like. You’ll find that from very humble beginnings groups and rings lead to a deep and rich theory. Mathematical Analysis (30 credits) – Compulsory This module will begin by looking at what we really mean when we look at limits in mathematics and build on this to give you a greater understanding of series and calculus. The module builds on the ideas introduced in the level 4 modules about the need for rigour in maths. You will develop your ability to question mathematical arguments and to think logically about what definitions mean. Problem Solving Methods (30 credits) – Compulsory The HE Maths Curriculum Summit (2011) recently concluded that “problem-solving is the most useful skill a student can take with them when they leave university”. This module fosters this skill in you by building on the approaches developed throughout the programme to enhance your ability to approach problems in diverse areas of maths and solve them. You will learn how to approach a problem, analyse its properties and develop a strategy to solve it. Rowlett, P. (Ed.) (2011, January), HE Mathematics Curriculum Summit Year 3 Advanced Algebra (30 credits) – Compulsory This module will look at more advanced subjects in algebra. The module will further enhance your understanding of the often abstract nature of structures used in maths and how seemingly simple definitions lead to a great deal of rich and interesting ideas. The main thrust of this module will be Galois Theory, an important area that develops a deep relationship between groups and field. Real and Complex Analysis (30 credits) – Compulsory Following from the previous module on mathematical analysis, this module will continue to develop your understanding of infinite and infinitesimal processes. You will also learn how extending the ideas developed here and in your previous module to complex numbers leads to a very different theory. Communicating Mathematics (30 credits) – Compulsory Maths is often called the universal language of science but communicating it can be difficult. With the continuing move to more diverse platforms such as social media this leads to even more challenges in communicating maths. In this module you will look at how maths is communicated, be it to specialists, non-specialists, school pupils or CEOs and how to motivate it for these diverse audiences. Project (30 credits) – Compulsory This is your chance to study an area of maths that you’re interested in and write your own report on it. You can choose your topic yourself or from a list given by staff. If you choose this module then you will have the opportunity to publish your work as an article in a journal. Computer Graphics (30 credits) – Optional The aim of this module is to examine in depth the concepts and techniques needed in the construction of interactive graphics and visualisation systems covering advanced graphics programming techniques. It will cover theory and mathematics as required and It aims to provide students with practical experience via significant individual project work developing 2D and 3D programs using an industry standard environment. Artificial Intelligence (30 credits) – Optional The aim of the module is to introduce students to a range of AI theories and techniques, including the most commonly used. This will extend to the ability to implement these techniques, and the students will extend their own development skills. Combinatorics (30 credits) – Optional Combinatorics can be seen as the study of interactions and connections between people or objects. Examples of its use range from developing computer architectures to logistic management. In this module, you will develop the topics introduced in your level 5 discrete maths module to better understand these connections and how they are studied. Multivariate Statistics (30 credits) – Compulsory Understanding and recognising patterns in data can be difficult when it comes from many different sources. In this module, you will begin to develop the techniques and tools that will enable you to study these kinds of relationships. You will develop a critical view of the pros and cons of the methods and the assumptions being made. Simulation and Decision Making (30 credits) – Optional Simulating systems such as queuing times at hospitals, or traffic congestion in towns and cities is one of the most important tools used to inform management decisions. In this module, you will be introduced to mathematical simulation and you will learn how to develop models to study systems and improve efficiency. Functional Analysis (30 credits) – Optional Functional analysis is the study of how structures made up of sets of functions behave; the subject is at the intersection of algebra, analysis and geometry. In this module, you will learn how infinite-dimensional algebraic structures act, what they look like geometrically and some of their properties. Differential Equations (30 credits) – Optional Physical systems like pollution in the atmosphere, molecules in a liquid or interactions between planetary systems are all modelled using differential equations. In this module, you will continue with the topics introduced in level 4 and developed in level 5 to study these equations and their solutions. Year 4 Reading Course (30 credits) – Optional This module allows you to work in small groups studying a substantial topic of advanced mathematics. You will be allowed to choose from a list of possible topics that are described in a body of literature, either in the form of a book or a number of research or other articles. As with the Project, this module will let you design your degree programme to suit your interests. Advanced Topics in Mathematics A and B (30 credits) – Optional These two modules allow you to study, in depth, an advanced topic in mathematics or an application of specialist mathematics. The subject of the module, either a contemporary or classical area of mathematics, will change periodically reflecting the interests of staff in the department, and the interests of the students studying it. As an MSci student these two modules give you the chance to experience the dynamic nature of modern mathematics and how it is applied, and serves to illustrate the ever changing character of the subject. This module allows you to encounter cutting edge areas of modern mathematics. Optional Modules There are a range of additional optional modules that you will be able to take during your fourth year that will continue building your knowledge and skills from previous modules to a more  advanced level of study. You can find more information about this course in the programme specification. Optional modules are usually available at levels 5 and 6, although optional modules are not offered on every course. Where optional modules are available, you will be asked to make your choice during the previous academic year. If we have insufficient numbers of students interested in an optional module, or there are staffing changes which affect the teaching, it may not be offered. If an optional module will not run, we will advise you after the module selection period when numbers are confirmed, or at the earliest time that the programme team make the decision not to run the module, and help you choose an alternative module.