# Options in Mathematics and Computer Science

## Contents |

**In the third year, students studying the Mathematics and Computer Science degree can choose the balance between options in
Computer Science and in Mathematics, devoting between 25% and 75% of their time
to each of them. In the fourth year, students can devote between 0% to 100% of their time studyting either. **

# Third Year

**Third year students may choose up to two courses from the list of Computer Science options, together with further courses from the list of advanced options.
**

**On the Maths side, in their third year students can choose from options including: **

### Logic

Designed to give a rigorous mathematical treatment of the fundamental ideas and results of logic that is suitable for the non-specialist mathematicians and will provide a sound basis for more advanced study. Cohesion is achieved... Continued on the Maths Institute website.

### Set Theory

This course will introduce sets and their properties as a unified way of treating mathematical structures, including encoding of basic mathematical objects using set theoretic language. To emphasize the difference between intuitive collections and formal sets.... Continued on the Maths Institute website.

### Introduction to Representation Theory

This course gives an introduction to the representation theory of finite groups and finite dimensional algebras. Representation theory is a fundamental tool for studying symmetry by means of linear algebra: it is studied in a way in which a given group or algebra may act on vector spaces, giving rise to the notion of a representation. We start in a more general setting, studying modules over... Continued on the Maths Institute website.

### Group Theory and an Introduction to Character Theory

A finite group represents one of the simplest algebraic objects, having just one operation on a finite set, and historically groups arose from the study of permutations or, more generally, sets of bijective functions on a set closed under composition. Thus there is the scope for both a rich theory and a wide source of examples. Some of this has been seen in the course Group Theory, and this course will build on that. In particular... Continued on the Maths Institute website.

### Banach Spaces

Students will achieve a firm knowledge of real and complex normed vector spaces, with their geometric and topological properties. Continued on the Maths Institute website.

### Hilbert Spaces

On completing this coure students will appreciate the role of completeness through the Baire category theorem and its consequences for operators on Banach spaces. They will have a demonstrable knowledge of the properties of a Hilbert space, including orthogonal complements, orthonormal sets, complete orthonormal sets together with related identities and inequalities. They will be familiar with... Continued on the Maths Institute website.

### Techniques of Applied Mathematics

This course develops mathematical techniques which are useful in solving `real-world' problems involving differential equations, and is a development of ideas which arise in the second year differential equations course. The course aims to show in a practical way how equations `work', what kinds of solution behaviours can occur, and some techniques which are... Continued on the Maths Institute website.### Applied Partial Differential Equations

The Applied Partial Differential Equations course continues the Differential Equations course, and extends some of the techniques of B5a to partial differential equations. In particular, first-order conservation laws are solved and the idea of a shock is introduced; general nonlinear first-order partial differential equations are solved, the classification of second-order partial differential equations is extended to systems, with hyperbolic systems being solved by characteristic variables. Then Green's function, maximum principle and similarity variable methods are demonstrated for partial differential equations. Continued on the on the Maths Institute website.### Number Theory

Number theory is one of the oldest parts of mathematics. For well over two thousand years it has attracted professional and amateur mathematicians alike. Although notoriously `pure' it has turned out to have more and more applications as new subjects and new technologies have developed. Our aim in this course is to introduce students to some classical and important basic ideas of the subject...Continued on the Maths Institute website.

### Communication Theory

The aim of the course is to investigate methods for the communication of information from a sender, along a channel of some kind, to a receiver. If errors are not a concern we are interested in codes that yield fast communication; if the channel is noisy we are interested in achieving both speed and reliability. A key concept is that of information as reduction in uncertainty. Continued on the Maths Institute website.

### Applied Probability

This course is intended to show the power and range of probability by considering real examples in which probabilistic modelling is inescapable and useful. Theory will be developed as required to deal with the examples. Continued on the Maths Institute website.

# Fourth Year

Fourth year students choose all their Computer Science options from the list of advanced options.

Fourth year students can choose from any of the courses listed on the Maths Institute website under the section "Part C" on this page.