The Gavin Brown Prize is a “best paper” prize awarded annually by the Australian Mathematical Society, for work in any field of mathematics published no more than 10 years before the award.
To our great surprise but obviously immense delight, our 2010 paper “Every flock generalized quadrangle has a hemisystem” was awarded the 2020 prize.
This was the first paper that arose from our first joint ARC Discovery grant back in 2009, just after I had moved from the CS department to the Maths department at UWA, and started working with John and Michael.
I won’t discuss the actual details of the mathematics too much here, but just enough to describe what the problem is (was). A finite generalized quadrangle (GQ) is a particular type of finite point-line geometry that has no triangles, and flock generalized quadrangles are a large family of GQs. A hemisystem H is a subset of the lines of the GQ that contains exactly half of the lines on each point; from this it is essentially obvious that H must contain exactly half of the lines.
Hemisystems give rise to other interesting combinatorial and geometric objects, and so over several decades various researchers had tackled the question of when a GQ contains a hemisystem. At the time we started the work, the only recent progress that had been made was the discovery of an infinite family of hemisystems by Cossidente and Penttila in 2005.
Our paper answered the question in almost the strongest possible fashion – a construction for hemisystems in the very large family of flock GQs. The construction was elegant, the family of GQs to which it applied was large and it constructed exponentially many hemisystems. As it had previously been conjectured that hemisystems were very scarce, this was all totally unexpected.
In many areas of finite geometry, the usual case is that new interesting geometric configurations are found by intricate constructions that work in particular small families of geometries. In our ARC Discovery grant application we had “promised” to find some new small hemisystems, or possibly an infinite family for a particular class of GQs. As smashed through this goal in our first year, we were super happy at the time, and wrote a few SymOmega posts about it, such as the following one from John:
But to have this recognised by our colleagues and to be placed in the prestigious company of the former (and future) winners of the Gavin Brown Prize is something we could never have anticipated, but greatly appreciate.