MT5861: Advanced Combinatorics

That's it! |
---|

That was the last presentation of the module in its current form. Thank you all for your attendance and comments, and best wishes for whatever you do next. |

The next presentation is scheduled for 2021-22, Semester 2 (commencing January 2022).

There are three syllabuses for this module, available here. In 2020-21, the third syllabus, on topics such as strongly regular graphs, root systems, designs, finite geometries, and Hadamard matrices, will be covered.

Course material from previous presentations of the module can be found here.

There will be two pieces of assessed coursework, each worth 10% of the marks for the module. The examination will be worth 80%.

The first assessed coursework is here, and the solutions are here. Marks and marked scripts are on MMS.

The second assessed coursework is here, and the solutions are here. Marks and marked scripts are on MMS.

The lectures are given in-person; video recordings will be available as technology permits.

This table now includes short summaries of the lectures which may help those who cannot attend in person.

The sections in the table below correspond roughly to weeks, but with some slippage.

**Added 12 March:** Lecture 15 was not recorded due to a problem with Windows. Please see the summary below for the content of this lecture.

Section | Topic | Notes | Problems | Summaries |
---|---|---|---|---|

1 | Strongly regular graphs | Part 1 | Sheet 1, solutions | Lect1, Lect2, Lect3 |

2 | Designs and projective planes | Part 2 | Sheet 2, solutions | Lect4, Lect5, Lect6 |

3 | Root systems | Part 3 | Sheet 3, solutions | Lect7, Lect8, Lect9 |

4 | Graphs with least eigenvalue −2 | Part 4 | Sheet 4, solutions | Lect10, Lect11 |

5 | Hadamard matrices | Part 5 | Sheet 5, solutions | Lect12, Lect13 |

6 | Projective spaces | Part 6 | Revision1, solutions | Lect14, Lect15 |

7 | Quadratic forms | Part 7 | Sheet 7, solutions | Lect16, Lect17 |

8 | The triangle property | Part 8 | Sheet 8, solutions | Lect18, Lect19 |

9 | Partitions into strongly regular graphs | Part 9 | Revision 2, solutions | Lect20, Lect21, Lect22 |

I may post extra material about interesting topics related to the module, or PhD studentships that come to my attention, here.

- Diagonalising real symmetric matrices
- Finite fields
- Another approach to root systems
- Strong triangle property: last steps

- Parameters of strongly regular graphs (by Andries Brouwer)
- Sagemath program to construct strongly regular graphs (by Nathann Cohen and Dima Pasechnik)
- List of strongly regular graphs on up to 64 vertices (by Ted Spence)
- Slides of lecture and problem session on algebraic graph theory; problems and solutions, video of talk, thanks to Dr Sonwabile Mafunda (University of Johannesburg, South Africa)
- Proof of the Friendship Theorem by Joshua Paik, a former MT5861 student
- Slides for a SUMS talk on the "ADE affair"
- A picture of the Fano plane in a paper on ecology
- 2-designs from a statistical viewpoint, from the Sage Encyclopedia of Research Design

Coursework sheets can be downloadad from the table above. You are welcome to hand in written work to be marked. I suggest on each sheet a maximum number of questions you should do. My solutions will be posted after the hand-in date.

Revision problems on the first half of the module are available here, and on the second half here; solutions are here and here.

Past exam papers on Syllabus 3 are available here. Note that the module was previously called MT5821: it was a 20-credit module and the exam was 2 hours 30 minutes.

I will make
lecture notes available in advance of the lectures, so as to give you maximum
flexibility in planning your time. But as usual I strongly advise you to
attend the lectures, and then make use of the notes or slides to clarify
things that were not clear from the lectures.
**Warning**: The notes or slides may change, subject to correction of
mistakes and comments from the audience!

Assessment will be 20% coursework and 80% examination.There will be a coursework sheet provided every week. These are optional except for two which will be clearly indicated in advance, which will be each worth 10% of the overall mark for the module.

Please do not hesitate to contact me by email if you have any questions.

Peter J. Cameron

17 April 2022