The last day of the conference was over. Here are a couple of closing remarks. We have certainly heard an amazing collection of talks over the last week!
I didn’t yet mention the other short course, given by Simeon Ball and Aart Blokhuis, on the polynomial method in finite geometry. Among other things, they showed us four different ways to get bounds for a set of points in a projective or affine space meeting every hyperplane in at least (or at most) a certain number of points: one based on polynomials over the finite field of the geometry, one over the complex numbers, one over the p-adic numbers, and one using the resultant. But there was a lot of detail, and I didn’t even attempt to take detailed notes. I hope the slides will be made available at some point!
The highlight of my advanced combinatorics course in St Andrews this year was the construction and characterisation of the strongly regular graphs associated with non-singular quadratic forms over the field GF(2). These featured in the talks by Alice Hui (who is using Godsil–McKay switching to find other graphs cospectral with them, and Sebi Cioabă (together with lots more in a packed programme). Peter Sin also talked about cospectral graphs, more precisely, an infinite number of pairs of graphs which (unlike Alice’s) have the same spectrum and the same Smith normal form. These are the Paley and Peisert graphs on q vertices, where q is the square of a prime congruent to 3 (mod 4). These have a common description. The subgroup of index 4 in the multiplicative group has 4 cosets with a cyclic structure. Taking the first and third cosets in cyclic order gives the Paley graph, the first and second give the Peisert graph.
Dirk Hachenberger was counting elements of a finite field which are both primitive (i.e. generate the multiplicative group) and normal (an element is normal if it and its conjugates under the Frobenius map span the field as vector space over the base field). Remarkably, it was only proved in 1992 by Lenstra and Schoof that primitive normal elements exist, but Dirk has lower bounds for the number, and it seems that, at least for extensions of large degree of a fixed base field, almost all primitive elements are normal.