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:
- A proof of Wedderburn’s little theorem using the Dandelin-Gallucci theorem;
- A proof of the Artin-Zorn theorem by using the Glauberman-Heimbeck theorem;
- 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:
- A finite division ring is a field;
- A finite alternative division ring is a field;
A finite Bol quasifield is a nearfield.
And the beautiful geometric counterparts are:
- A finite Desarguesian projective plane is Pappian;
- A finite Moufang projective plane is Pappian;
- 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 “
- Jan De Beule’s plenary lecture “Arcs in vector spaces over finite fields”
- Geertrui Van de Voorde’s talk “Point sets in 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).