## Learning finite geometry

I have just finished my first year as a PhD student in the Incidence Geometry research group at the Ghent University. In the past one year I have learned, or at least tried to learn, some finite geometry and related areas. I wouldn’t say that I know a lot about what is classically considered to be finite geometry. For example, I doubt if I can prove the classification of bilinear forms over finite fields, or say anything interesting about unitals, blocking sets,  generators of polar spaces, projective embeddings of point-line geometries, diagram geometries, buildings etc. beyond the basic definitions. But luckily I was still able to do some interesting research in this area since it required only a good combinatorial understanding of generalized polygons and near polygons. I did have to get used to a relatively new concept called valuations of near polygons, but it wasn’t very challenging (at least in hindsight). A preprint of my first research paper is available here.

Some of my recent work, while still being related to valuations, uses more group theory, graph theory, and computer programming than finite geometry. It is quite exciting and I would love to build up on it but I would also like to learn more finite geometry. Especially since I can’t possibly get a better environment than Ghent for learning that. So, here are brief descriptions of some classical topics that I would like to get more comfortable with:

1. Blocking sets
This is a very active area of finite geometry with historical roots in game theory. The name blocking coalition was used by Moses Richardson in his 1955 paper “On Finite Projective Games” where lines of a finite projective plane (representing a subset of some finite set of people) were called minimal winning coalitions while a set of points that doesn’t contain any line but intersects all lines was called a blocking coalition, what we now call a blocking set. It is an interesting exercise to show that every projective plane except the Fano plane has a blocking set (hint: size 2n where n is the order of the plane). When talking about hypergraphs, blocking sets are usually called transversals or hitting sets.

Lecture notes by Aart Blokhuis titled “Blocking sets in projective and affine planes” is  probably a good place to start learning about this topic. These objects have also been studied for polar spaces (another classical thing that I want to learn properly), see the PhD thesis of Jan De Beule . Relationship of blocking sets with coding theory and other recent problems in the theory of blocking sets are discussed here. Some algebraic techniques which are beyond my current understanding can be seen in this paper by Sziklai and Szőnyi.

Here’s a small list of people who work on these objects: Aart Blokhuis, Tamás Szőnyi, Peter Sziklai, Simeon Ball, Michel Lavrauw, Leo Storme, Zs Weiner, Klaus Metsch, Jan De Beule, Geertrui Van de Voorde.

2. Unitals
A unital is a $2$$(n^3 + 1, n + 1, 1)$ ($n \geq 3$block design. If the blocks are certain sets of collinear points in some projective plane of order $n^2$ then the unital is called embeddable. Historically unitals came from Hermitian curves in the projective planes $PG(2,q^2)$ where $q$ is a prime power. There is a full book devoted to this topic and it seems like a good place to learn about unitals.

It can be easily shown that if a unital is embeddable in a projective plane $PG(2,q^2)$ then every line of the plane intersects the unital in $1$ or $q+1$ points. Therefore, every such unital is a blocking set of the projective plane. In fact, it was proved by Bruen and Thas in 1977 that if $B$ is a minimal blocking set in $PG(2,q)$ then $|B| \leq q \sqrt{q} + 1$ with equality holding if and only if $q$ is a square and $B$ is a unital.

An important open problem regarding unitals is discussed on this blog post.

3. Polar Spaces
This is one of the crucial topics in finite geometry that I am really uncomfortable with.
I started learning about this topic from Peter Cameron’s notes but soon I discovered a gentler introduction by Simeon Ball, Introduction to finite geometry. Polar spaces are not that hard to define axiomatically, it’s a point line geometry where given a point p and a line L either there is a unique point on L that is collinear with p or all points of L are collinear with the point p (and some other non-degeneracy conditions).  From this a notion of rank can be defined on the polar spaces, much like the projective spaces, and the smallest ranked polar spaces are the same thing as generalized quadrangles, something I am quite comfortable with.  I guess it’s the classical part of bilinear forms and polarities on finite vector spaces, from which the notion of polar spaces arose, that irks me. Perhaps I’ll get more comfortable with it once I prove some classical results myself.

Some recent work that considers intersection problems of the Erdős Ko Rado type in polar spaces is present in the PhD thesis of my colleague Maarten De Boeck. This seems quite interesting to me and I would definitely like to read more about that in near future.

In the last year I have also been quite active on Quora and answered some questions related to finite geometry  which can be found here, here and here. I even made my first ever wikipedia page, near polygons. I wish to continue with these things though I should probably reduce the amount of time I spend on them.

Finally I would like to mention some books and surveys on finite geometry:

Incidence Geometry by Eric Moorhouse
Foundations of Incidence Geometries by Johannes Ueberberg
Projective Geometry by Albrecht Beutelspacher
Projective Geometry: an Introduction by Rey casse

Books that I would like to read:
Diagram Geometry by Cohen and Buekenhout
Generalized Polygons by Hendrik Van Maldeghem
Points and Lines by Ernest E. Shult
Classical Groups by Larry C. Grove
Topics in Finite Geometry: Ovals, Ovoids and
Generalized Quadrangles by S. E. Payne
A geometrical picture book by Polster

Edit: here’s a new survey paper by Jeff Thas and James Hirschfeld on open problems in finite projective spaces.

Some helpful online resources for learning finite geometry:
http://finitegeometry.org/
http://cage.ugent.be/geometry/fining.php