Pieter Vandecasteele

Valuations and hyperplanes of dual polar spaces

Valuations were introduced in De Bruyn and Vandecasteele (Valuations of near polygons, preprint, 2004) as a very important tool for classifying near polygons. In the present paper we study valuations of dual polar spaces. We will introduce the class of the SDPS-valuations and characterize these valuations. We will show that a valuation of a finite thick dual polar space is the extension of an SDPS-valuation if and only if no induced hex valuation is ovoidal or semi-classical. Each SDPS-valuation will also give rise to a geometric hyperplane of the dual polar space.

The classification of the slim dense near octagons

We classify all dense near octagons with three points on each line.

Near polygons with a nice chain of sub-near polygons

In (Eur. J. Combin. 24 (2003) 631) we defined a class of near polygons and conjectured that the near 2n-gons from this class are precisely those near polygons which satisfy the following properties: (i) every line is incident with exactly three points, (ii) every two points at distance 2 have at least two common neighbours, (iii) there exists a chain F0 ⊂ F1 ⊂ ..... ⊂ Fn of geodetically closed sub-near polygons with the property that the sub-near 2i-gon Fi, i ∈ {0,...,n - 1}, is big in the sub-near 2(i + 1)-gon Fi+1. In the present paper we present a proof of this conjecture.

Two conjectures regarding dense near polygons with three points on each line

We introduce two conjectures concerning the structure of dense near polygons with three points on each line. The first conjecture deals with the whole class of such near polygons. The second conjecture deals with only those near polygons which have a nice chain of subgeometries. Although the second conjecture is implied by the first one, we introduce it because this conjecture is more likely to be proved in the near future. We prove some results which are special cases of the second conjecture and say what case is still open.