What I have learned in finite geometry

On September 2nd, 2014 I wrote a blog post titled learning finite geometry, in which I described how much I have learned in my first year of PhD and more importantly, the topics that I wish to learn while I am in Ghent. Since I’ll be leaving Ghent in a few days, I feel like this is the right time reflect on that old post. I am quite happy to see that beyond just learning those topics, I have also been able to do some research on them. Here are the topics that I described in my previous blog post:

1. Blocking sets
Two of my papers have results on blocking sets. In the first one, I used the Alon-Furedi theorem to prove old and new results on partial covers and blocking sets with respect to hyperplanes in finite Desarguesian affine/projective spaces. I have given a new common framework for treating some of the problems on blocking sets in an easy way using the polynomial method. See section 6 of my paper for the details.

In my second paper, I have proved a generalisation of a classical result in finite geometry from the 1970s due to Aiden A. Bruen and Jeff Thas. Moreover, I have done so with a new method involving the expander mixing lemma which has lead to a unified (and simple) approach to problems related to blocking sets, Nikodym sets and tangency sets. This method is quite interesting in its own right as it has also lead to some advancements in the cage problem.

2. Unitals
These objects appear in my second paper on blocking sets, as they were used by Sam to give a construction of large minimal multiple blocking sets in finite projective planes. In fact, as Sam has described in his blog post, an open problem on unitals was one of my motivations to look at these blocking sets. Sadly, we haven’t been able to solve that open problem yet.

3. Polar Spaces
After the end of my first year, I had seen polar spaces appear in almost every other research paper and talk in finite geometry. Even though I had read and understood the definitions of these objects in my first year, I wasn’t really comfortable with them. I am not sure when exactly that changed, but it was certainly quite gradual. In my paper with John and Gordon, on regular induced subgraphs of generalized polygons, I have used some of the basic theory of polar spaces to give new constructions of small $k$-regular graphs of girth $g$ with $g = 8, 12$. Before that I also refereed a paper which helped me revisit some of the basics of the axiomatic theory of polar spaces. Overall, I can say that I have made pretty good progress towards understanding polar spaces. I now wish that more mathematicians would learn the basic theory of these beautiful geometrical objects, especially because people sometimes obtain results on specific classes of finite polar spaces without realising it (see this for example). The new book “Finite Geometry and Combinatorial Applications” by Simeon Ball can be a useful resource for anyone interested in doing so.

Besides these topics, I have also learned about several combinatorial problems in which finite geometry plays an important part. For example, the cage problem, forbidden subgraphs, and ramsey numbers. During my postdoc years, I will work on finding more such combinatorial applications of finite geometry, and using various tools from combinatorics to solve problems in finite geometry.