# New directions in additive combinatorics: day 1

I know I’m not going to be able to keep this up, but I felt I needed to report on the first day of what is turning out to be an excellent conference at the National University of Singapore. This will leave me with the issue that my reporting of subsequent talks in this conference will not be at the same degree as I will present here, even though they might be of the same standard. So I apologise in advance.

First of all, we kicked off with Ben Green’s first lecture in his series on Finite Field Models (and in particular, to problems simulating outstanding problems in additive combinatorics). What was quite novel about this talk was that the news about the Sunflower Conjecture was placed in the centre of his talk: it was a news breaking talk. Add to this Tao’s reformulation of the Croot-Lev-Pach principle, and it really felt like everything we were being told was hot off the press. The next talk was by Peter Cameron, and he spoke about some problems to do with synchronisation (for automata and permutation groups). I was reminded that we still not have an idea what is going on with acting on *k*-subsets for . Something perhaps worth thinking about when I have time. Then Kai-Uwa Schmidt spoke about his recent results on *o-polynomials*; which are the functions you get when you coordinatise a hyperoval in a finite Desarguesian projective plane. I remember seeing his papers on the arxiv some time ago, so it was nice to see it all placed in context: some very nice recent results there on a problem that hasn’t seen many advances for a while. There were two citations of SymOmega in his talk: one of them was a comment I made about hyperovals, the second was a comment from Tim Penttila.

In the afternoon, Peter Keevash set the scene for his series on designs. He gave his big result of yesteryear, and all of its many consequences: Wilson’s Conjecture and the Existence Conjecture. He has only just outlined the strategy behind the proof, so the best is yet to come. I then gave the first talk after afternoon tea: it was a tad rushed, but I managed to get to the end. After my talk was a summary of the EKR problem for finite polar spaces by Klaus Metsch. Klaus has produced many interesting results in this area of late, and often by applying the Hoffmann bound to linear combinations of adjacency matrices of an association scheme! He showed that good old finite geometry techniques pair up well with the algebraic combinatorial techniques, when both cannot do the job alone.

There might be more reporting tomorrow … if I have time.

Thanks for the nice summary.

The link for the (weak) sunflower conjecture and the Croot-Lev-Pach bound do not work. At least not without a WordPress account.

Hi John,

In the same vein as Ferdinand, thanks for the reporting!

The links that Ferdinand mentioned aren’t working for me either. The sunflower conjecture has an easy to process Wikipedia statement, however there’s none for Croot-Lev-Pach.

Best wishes,

Jon

I’ve fixed the links. Thanks for that!

The Wikipedia article only mentions the sunflower conjecture, not the solved weak sunflower conjecture/Erdos-Szemeredi sunflower conjecture (now theorem, I guess), so linking to that would not be too helpful.

Weak sunflower conjecture, again, is a well-established name, but a bit confusing because there are also weak Delta systems aka weak sunflowers, which also have their own conjecture by Erdos and Rado, but are not involved in the statement of the Erdos-Szemeredi sunflower conjecture.

Oh, “non-uniform 3-sunflower conjecture” would be good descritptive name, but I am a few decades too late to have any influence on naming conventions.