Unit name | Combinatorics |
---|---|

Unit code | MATH20002 |

Credit points | 20 |

Level of study | I/5 |

Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |

Unit director | Dr. Wolf |

Open unit status | Not open |

Pre-requisites |
Linear algebra and geometry (Math 11005), Probability 1 (Math 11300), Analysis 1 (MATH11006), Number Theory and Group Theory (MATH11511) (From 2015/16: Analysis 1A, Foundations and proof, Introduction to group theory will replace Analysis 1 and NTGT as prerequisites) |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

Aims

This new unit serves as an introduction to combinatorics, developing fundamental aspects of a diverse range of topics in discrete mathematics such as counting, generating functions, extremal graph theory, Ramsey theory and random walks. The unit aims to develop and improve students’ problem-solving and theorem-proving skills, building on those acquired in first-year courses. Moreover, it seeks to enhance students’ appreciation of the interconnectedness of different areas of mathematics by introducing probabilistic, analytic and algebraic techniques.

Description

Combinatorics is the study of discrete structures, which are ubiquitous in our everyday lives. While combinatorics has important practical applications (for example to networking, optimisation, and statistical physics), problems of a combinatorial nature also arise in many areas of pure mathematics such as algebra, probability, topology and geometry.

The course will start with a revision of various counting techniques, and take a close look at generating functions. The unit will then proceed to introduce the basic notions and fundamental results of graph theory, including Turán’s theorem on independent sets, Hall’s marriage theorem, Euler’s formula for planar graphs and Kuratowski’s theorem. In the last part of the unit probabilistic and algebraic methods will be used to study more advanced topics in graph theory, including Ramsey’s theorem and random walks.

Students who successfully complete the unit should: be proficient at counting rearrangements of finite sets; have acquired facility with the computation and application of generating functions; be familiar with the basic definitions and concepts in graph theory, including trees, cycles, connectivity, matchings, planarity; understand, be able to prove and apply the fundamental results derived in the course, and solve unseen problems of a similar kind; understand and be able to apply methods from elementary probability, analysis and linear algebra to a range of problems in discrete mathematics, including Ramsey theory, isoperimetry and random walks.

In addition, students should have learnt how to give a mathematical formulation to word problems of a discrete nature; improved their problem-solving and theorem-proving skills; gained an appreciation of how methods from probability, analysis and algebra can be used to solve problems in discrete mathematics.

Lectures, including examples and revision classes, supported by lecture notes with problem sets and model solutions.

80% examination & 20% coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period. Your unit mark after reassessment will be based entirely on the resit exam.

Lecture notes and handouts will be provided covering all the material presented in lectures.

The following supplementary texts provide additional background reading: Combinatorics: Topics, Techniques, Algorithms by Peter Cameron A course in combinatorics by J.H. van Lint and R.M. Wilson Notes on introductory combinatorics by George Pólya, Robert Tarjan and Donald Woods