# Topics in Discrete Mathematics 34

## Unit aims

This is a topics course aimed at deepening and broadening the students' knowledge of various aspects of discrete mathematics, as well as illustrating connections between discrete mathematics and other areas such as algebra, probability, number theory, analysis and computer science.

## Unit description

Discrete mathematics refers to the study of mathematical structures that are discrete in nature rather than continuous, for example graphs, lattices, partially ordered sets, designs and codes. It is a classical subject that has become very important in real-world applications, and consequently it is a very active research topic.

This topics course exposes the students to a selection of advanced topics in discrete mathematics. These may include (but are not restricted to) advanced topics in graph and hypergraph theory, design and coding theory, combinatorial topics in group theory, as well as probabilistic, algebraic and Fourier-analytic methods throughout discrete mathematics.

While results and problems of recent origin may be included in the syllabus, the instructors aim to make the material accessible to all students fulfilling the prerequisites by providing complete lectures notes and including all necessary background material.

The unit is suitable for students with a firm grasp of the basic concepts in Combinatorics, and likely of interest to those with an interest in number theory, algebra, probability and/or theoretical computer science.

## Relation to other units

The course follows on from Combinatorics. It complements Complex Networks and the Data Structures and Algorithms unit in Computer Science.

## Learning objectives

In accordance with the specific syllabus taught in any particular year, students who successfully complete the unit should:

- have developed a solid understanding of the advanced concepts covered in the course;
- be able to use techniques from algebra, analysis and probability to solve problems in discrete mathematics;
- have a good grasp of the applications of combinatorial techniques in other areas of mathematics and to real-world problems.

By pursuing an individual project on a more advanced topic students should have:

- developed an awareness of a broader literature;
- gained an appreciation of how the basic ideas may be further developed;
- learned how to assimilate material from several sources into a coherent document.

## Transferable skills

The ability to think clearly about discrete structures and the ability to analyse complex real-world problems using combinatorial abstractions.

## Syllabus

Topics covered might include (but are not restricted to) any of the following:

- random graphs
- permutation groups
- introduction to the theory of hypergraphs
- advanced topics in design and coding theory
- random walks on groups
- probabilistic methods in combinatorics
- discrete Fourier analysis and applications
- connections with information theory and algorithms

## Reading and References

Lecture notes and handouts will be provided covering all the main material.

The following supplementary texts provide background reading prior to the start of the course:

- Peter J. Cameron,
*Combinatorics: Topics, Techniques, Algorithms*. CUP, 1995. - Dieter Jungnickel,
*Graphs, networks, and algorithms*. Springer, 2005. - Ian Anderson,
*A First Course in Discrete Mathematics*. Springer, 2001.

**Unit code:** MATHM0009

**Level of study:** M/7

**Credit points:** 10

**Teaching block (weeks):** 2 (13-18) **Lecturers:** Dr Alex Malcolm and Dr Dan Martin

## Pre-requisites

Students must have taken MATH20002 Combinatorics and either MATH21800 Algebra 2 or MATH21100 Linear Algebra 2. For joint Mathematics and Computer Science students, it would be desirable to have taken COMS21103 Data Structures and Algorithms. Students may not take this unit if they have taken the corresponding Level H/6 unit MATH30002 Topics in Discrete Mathematics 3.

## Co-requisites

None

## Methods of teaching

Lectures, including examples and revision classes, supported by lecture notes with problem sets and model solutions. Self-study with directed reading based on recommended material.

## Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

- 100% Exam

NOTE: Calculators are not allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.