Our Faculty of Engineering, Computing and Mathematics ran, what I think, was an excellent idea; to support “seed” projects of each of the “critical mass research groups” of the faculty. So in December, we put forward a plan to do a few interesting things with the funds given to us by the faculty, that would give returns. One of these was our retreat in February, which we already knew would give back more than was put in, as we had been doing this for a few years already. The other more significant expense was to set up a meeting between us and the Monash group of combinatorialists. This meeting has also coincided with the visits of Peter Cameron and Rosemary Bailey, who are the “professors-at-large” supported by the seed funding. So Ian Wanless, Graham Farr, Kerri Morgan, Daniel Horsely, and Darcy Best flew over from Melbourne and have been in Perth this week, and we have had talks and discussions over the last three days. The title of the meeting has now officially been coined “MUSIC” after a suggestion by Peter Cameron: Monash UWA Symposium In Combinatorics.

Some slides of the talks are provided below:

More will be posted here soon once the meeting has concluded …

Congratulations to my postdoc Irene Pivotto and husband Robin Christian on the birth of their first child, Martin, born last week at St John of God hospital in Subiaco (which is where my daughters were both born).

This is actually the second CMSC baby in a year, as Alice Devillers and Sam Norton had baby Emilia late last year – at the time I wasn’t keeping up with SymOmega at all due to pressure of work, so missed announcing it.

Better late then never though, so congratulations to both sets of parents!

It’s been a long time since posts, mainly due to the fact that logistical issues caused all my year’s teaching to be compressed into first semester (that’s late-Feb to early-June for any readers not used to Southern Hemisphere habits). It was pretty hard, especially as one of my units is a 550-student first-year Engineering Maths that I had not taken before.

But after many weeks of weekends, evenings or nights spent desperately trying to finish lecture notes, tutorials and solutions for the next day’s lectures, workshops and tutes, the semester eventually ended.

So rather than stay home to attend to the vast number of overdue non-teaching tasks (admin, refereeing, bureaucracy) that I’d had to resolutely ignore during the semsester, instead I flew straight to Germany for a week-long meeting on Graph Polynomials at Schloss Dagstuhl (in Saarland, southern Germany).

It’s been a while since the last post, and much has happened. Last week, I attended “Combinatorics 2016” in Maratea Italy; a beautiful spot for a conference. I gave two talks: a short talk that was originally planned, plus I filled in for Tim Penttila’s plenary lecture since he was unable to make it at the last minute. Tim’s talk was about three instances where algebra and geometry are intimately linked:

1. A proof of Wedderburn’s little theorem using the Dandelin-Gallucci theorem;
2. A proof of the Artin-Zorn theorem by using the Glauberman-Heimbeck theorem;
3. An alternative approach to proving the Burn-Hanson-Johnson-Kallaher-Ostrom theorem.

The algebraic statements of these results are fundamental in algebra, and they are accordingly:

1. A finite division ring is a field;
2. A finite alternative division ring is a field;
3. A finite Bol quasifield is a nearfield.

And the beautiful geometric counterparts are:

1. A finite Desarguesian projective plane is Pappian;
2. A finite Moufang projective plane is Pappian;
3. A finite Bol projective plane is coordinatised by a nearfield.

The short talk I gave was on joint work with Tim on the foundations of hyperbolic plane geometry, but more about that in a later post.

There were many talks, and I didn’t attend all of them, but the highlights for me were:

• Ferdinand Ihringer’s talk “New bounds on the Ramsey number $r(I_n,L_3)$
• Jan De Beule’s plenary lecture “Arcs in vector spaces over finite fields”
• Geertrui Van de Voorde’s talk “Point sets in $PG(2,q)$ such that every line meets in 0, 2, or t points”
• Daniel Horsely’s plenary lecture “Extending Fisher’s inequality to coverings and packings”
• Zsuzsa Weiner’s talk “A characterisation of Hermitian varieties”

This week, I attended another conference, but shorter. It was in the La Rioja wine region of north-west Spain, the second joint meeting of the Royal Spanish, Belgian, and Luxembourg mathematical societies. We had a special session on combinatorial and computational geometry which was perhaps the most international of the special sessions. From finite geometry, we had talks by myself, Aida Abaid, Maarten De Boeck, Nicola Durante, and Ferdinand Ihringer (pictured below).

My two favourite plenary lectures were the first and last of the conference: Sara Arias de Reyna  (A glimpse of the Langlands programme) and Jesús María Sanz Serna (Forests, Trees, Words, Letters).

The longer the conference goes, the more time I spend doing research with some of the participants, and I tend to day-dream more in the lecture, so the quality of reporting will inevitably be low. Ben Green completed his mini-course on finite field models in additive combinatorics, with many many applications of the Cauchy-Schwarz inequality. We then had a session of short talks by Peter Sin and Xiang-dong Hou. The former spoke on generalised adjacency matrices of graphs and when two such matrices of the same graph can be similar and be “Smith Normal Form” equivalent. The latter outlined a proof of a conjecture on “monomial” graphs, which has connections to generalised quadrangles (since the conjecture is about a girth 8 bipartite graph). After lunch, Peter Keevash finished off the proof of his fabulous theorem by wrapping up the strategy that he outlined in the first lecture. We then completed the day by a very nice session of short talks by Alice Hui and Sebastian Cioabă. The former gave a very nice result on switching strongly regular graphs arising from geometric configurations in symplectic spaces, and the latter gave a stimulating summary of the speaker’s work on different types of connectivity and expansion properties of distance-regular graphs and graphs coming from association schemes.

Yesterday, we had four 90 minute talks! We began with more of the details of Peter Keevash’s proof, that uses some interesting results on hypergraphs and counting paths. After morning tea, Aart Blokhuis delved into t-fold blocking sets, followed by Simeon’s introduction to the links with coding theory. In the afternoon, we moved to the Department of Mathematics (at NUS) to see a colloquium by Ben Green on “Permutations and Number Theory”; a fabulous talk, one of the best I’ve ever seen him give. Finally, the “young person’s talk” was given by Ameera Chowdhury who spoke on a cool way to view and prove the MDS conjecture for prime fields and in the De Beule – Ball bound case.

In the evening, I spent two hours with Qing Xiang, Tao Feng, and Koji Momihara chatting about some interesting directions and problems we could look at in finite geometry. These guys are very good with Gauss sums and cyclotomic constructions, and so we are looking for more problems of this kind. Watch this space … work is now underway to construct some interesting objects in finite geometry! What was most interesting in our two hour session was the bottomless amount of notes and random pieces of paper that Qing seems to have stashed in his bag. Often I would be at the whiteboard saying something like “and from some work I did ten years ago …” and then Qing would pull out the relevant pages of information from his mystery bag. It is conjectured that he could also find a rabbit in there with some extra effort.

I forgot to write the report last night as I got carried away with some mathematical discussions with a colleague; better late than never! First, I missed two talks today, due to forgetting the time mainly when I was talking with Simeon Ball about $(q+1)$-arcs of projective spaces. We’ve ended up doing something, and that’s what has occupied me in the last while. Anyway, Aart Blokhuis and Simeon Ball began their mini-course on “Polynomial methods in finite geometry” yesterday, beginning with blocking sets. What I took away was that Hasse derivatives do something that the standard derivatives do not, but I’m still at a bit of loss why they are so successful in capturing the information about directions determined by a function. After morning tea, Lev spoke on the state of the art on quadratic residues and difference sets. I found this harder to follow, but there were some very interesting tables on computer output where some strange things happen. Five primes come out as solutions on a test of trillions of integers. Luckily we will have the slides posted on the webpage so I can remember exactly what these computations were about. Then Stefaan De Winter gave a beautiful talk on partial difference sets, where he and his co-authors have knocked off most of Ma’s list of parameters on at most 100 vertices. This was very impressive. There’s more on this talk over at Peter Cameron’s blog.

In the afternoon, Ben Green gave the second part of his series; this time on Rusza’s results and various improvements and advances thereof. Today (the third day) he will be giving a colloquium in the mathematics department here. As I said above, I lost track after afternoon tea and missed Ken Smith and Jim Davis’s talks; but then regained my composure and attended Oktay Olmez’s talk on directed strongly regular graphs and partial geometric designs; a great finish to another fantastic day at NUS!

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 $S_n$ acting on k-subsets for $k>3$. 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.