University of St Andrews University of St Andrews

MT5821: Advanced Combinatorics

PhD studentships at University of East Anglia
The School of Mathematics at UEA is advertising
five fully-funded studentships starting October 2019,
in the areas of Representation of symmetric groups;
2-representation theory and categorification;
generic representations of p-adic groups;
stochastic wave impact; and wave attenuation in ice.
Email Vanessa Miemietz (v.miemietz@uea.ac.uk)
for more information.

The module

The module will run in Semester 2 (28 January to 25 April), with Spring break from 18 March to 29 March.

NEW: The examination will be on Thursday 23 May 2019 at 09:30 in the Stewart Room.

The course will this year cover the third of the approved syllabuses: Finite Geometry. The module descriptor says,

Projective and polar spaces: geometry of vector spaces, combinatorics of projective planes, sesquilinear and quadratic forms and their classification, diagram geometry, classical groups.
You will also be introduced to root systems (one of the most pervasive ideas in mathematics, related to the classification of Lie algebras, singularity theory, cluster algebras, etc. – though we will not cover these topics!).

Course information sheet (including some graph theory definitions).

Timetable

Lectures will be in Room 1A on Mondays (odd weeks), Wednesdays and Fridays 12:00-13:00.

Two tutorial hours have been scheduled. These are Mondays 15:00-16:00 and Tuesdays 11:00-12:00 (not week 1), both in Room 3B. Both tutorials will run, but this may be reviewed after we see how things work out. It is not necessary to attend both.

The Petersen graph
The Petersen graph

Lecture notes

The Fano plane
The Fano plane

Problems and solutions

Weekly problems, and occasional revision problems, will also be posted here. The assessment is 100% exam, but it is recommended that you try your hand at some of the problems. Work handed in will be returned to you with comments as quickly as possible.

A challenge

You know three strongly regular graphs on 16 points: L2(4) (the 4×4 lattice, or the line graph of K4,4); the Clebsch graph (problem 1.2 on Revision Sheet 1), and the Shrikhande graph (problem 1.5 on Revision Sheet 1). For each of these graphs, take the adjacency matrix, and replace 0 by −1; the result is a Hadamard matrix. Are these Hadamard matrices equivalent?

Exam resources

There is a sample exam paper and the 2016 exam paper available. I do not have solutions to these papers, but I will be happy to go over your attempts at either or both with you before the exam.

Other resources

There are some comments on algebraic background and on diagonalisation of real symmetric matrices.

In the theorem about the Strong Triangle Property, we found possible values of m (the number of triangles through a vertex) to be 2, 3, 5 or 11. This document contains a construction for m = 5, and a nonexistence proof for m = 11.

You can find here the slides of a talk on The ADE affair, one of the most pervasive themes in mathematics, which will be touched on in the lectures.

Peter J. Cameron
Room 317, Mathematical Institute
pjc20@st-andrews.ac.uk
6 March 2019