This week I am in Brighton, at a summer school on Finite Geometry aimed at PhD students, organised by Fatma Karaoglu.
Despite the aim of the summer school, there are a few people here who are certainly not PhD students, including old friends Jim Davis and David Wales.
There are four short courses, given by Simeon Ball, Anton Betten, Jef Thas, and me. Jef and I lectured twice a day for the first two days, and then bowed out to be replaced by the other two for the last two-and-a half days. There were also many contributed talks, and GAP sessions run by my colleagues Alexander Konovalov and Chris Jefferson.
My talks were loosely based on the third series of my Advanced Combinatorics lectures in St Andrews. They were centred around the classification of graphs with the strong triangle property: each edge is contained in a triangle such that any further vertex is joined to exactly one vertex of the triangle. These graphs fall into two infinite families and three sporadic examples. If this suggests links to the ADE affair, that is correct: it leads fairly directly to a classification of the ADE root systems. But it leads in other directions too. It is the classification of generalised quadrangles with three points on a line, and the proof slightly tweaked shows the non-existence of infinite quadrangles with this property. It generalises to the classification of graphs with the triangle property: each edge is contained in a triangle such that any further vertex is joined to one or all vertices of the triangle. This is the theorem of Shult and Seidel, an immediate predecessor of the celebrated Buekenhout–Shult theorem classifying all the finite-rank polar spaces.
The notes are on the course website, here.
Jef spoke about arcs and caps in finite projective spaces, from Segre’s classification of conics in projective planes over fields of odd order (including Segre’s important “lemma of tangents”) to results on generalisations where instead of points one considers subspaces of arbitrary direction, and the connections with linear MDS codes. A very full and detailed survey, and even more valuable when the notes become available (as they should be soon).
Anton Betten’s topic is computational results in finite geometry. He began with the observation that, just because a problem is hard, that doesn’t mean we shouldn’t solve it as best we can. He quoted me on this – it is a sentiment I agree with! He remarked that he has some difficulty with NSF grants precisely because computer scientists who referee his proposals think that the problems are too hard. After this he plunged into four particular problems. The first was the classification of things called BLT-sets, which give rise to (often new) projective planes and generalised quadrangles, among other things. He has pushed the classification up to q = 67; the numbers of examples do not grow too rapidly. The others are parallelisms in 3-dimensional projective space, optimal linear codes, and cubic surfaces. He showed us his app for showing features of the 27 lines on a cubic surface, running on his iPhone on the document camera – you can find this app, called “Clebsch”, on the app store if you have an iPhone. (Does it sound as if I know what I am talking about??)
Simeon is telling us more about arcs and caps, and applying the polynomial method to these questions. He and Michel Lavrauw have a theorem which is a considerable improvement of Segre’s in odd characteristic, and (with a lot of effort) also gives results in higher dimensions. He concluded the series by proving the prime case of the celebrated MDS conjectures. Simeon’s notes are also available, here.
In his last talk, Anton showed us some details of what the computer actually does in these classifications. Typically there is a huge bump which takes a lot of the time. In a nice piece of synergy, in Simeon’s last talk, he showed us a method which allows one to tunnel through the bump: a condition on a (rather large) matrix built from a small arc allows one to show that it has no extension to a large arc of some size. New developments should arise from this!
Among the contributed talks, I am only going to mention two. Michael Reynolds is developing a prgram, along the lines of GeoGebra, for doing metric geometry over finite fields. I am not terribly impressed with the app, which represents the affine plane over a finite field as a square grid: lines don’t look much like lines! But I like the philosophy, which is to avoid all use of real arithmetic, transcendental functions, etc. Thus, the squared distance is given by a quadratic form; orthogonality by the associated bilinear form; and instead of angle θ, he uses cos2θ, which can be found “rationally” using the cosine rule.
Sandra Kingan did her best to persuade finite geometers that they should be interested in matroids. She talked about the representability problem over GF(5). The difficulty of the problem is greatly increased by the fact that the same matroid can have different representations. But what are inequivalent arcs of the same size in a projective plane but different representations of a uniform matroid?
Thursday night brought a very enjoyable dinner in a restaurant in Hove. No pictures, I’m afraid; I forgot my camera! There will be a conference photo eventually …