m-systems of polar spaces and SPG reguli

It will be shown that every m-system of W2n+1(q), Q(-)(2n+1,q) or H(2n, q(2)) is an SPG regulus and hence gives rise to a semipartial geometry. We also briefly investigate the semipartial geometries, associated with the known m-systems of these polar spaces

Partial ovoids and partial spreads in finite classical polar spaces

We survey the main results on ovoids and spreads, large maximal partial ovoids and large maximal partial spreads, and on small maximal partial ovoids and small maximal partial spreads in classical finite polar spaces. We also discuss the main results on the spectrum problem on maximal partial ovoids and maximal partial spreads in classical finite polar spaces

Hyperplanes of symplectic dual polar spaces : a survey

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Partial Ovoids and Partial Spreads of Classical Finite Polar Spaces

2000 Mathematics Subject Classification: 05B25, 51E20.We survey the main results on ovoids and spreads, large maximal partial ovoids and large maximal partial spreads, and on small maximal partial ovoids and small maximal partial spreads in classical finite polar spaces. We also discuss the main results on the spectrum problem on maximal partial ovoids and maximal partial spreads in classical finite polar spaces.The research of the fourth author was also supported by the Project Combined algorithmic and the oretical study of combinatorial structur es between the Fund for Scientific Research Flanders-Belgium (FWO-Flanders) and the Bulgarian Academy of Science

Incidence geometry from an algebraic graph theory point of view

The goal of this thesis is to apply techniques from algebraic graph theory to finite incidence geometry. The incidence geometries under consideration include projective spaces, polar spaces and near polygons. These geometries give rise to one or more graphs. By use of eigenvalue techniques, we obtain results on these graphs and on their substructures that are regular or extremal in some sense. The first chapter introduces the basic notions of geometries, such as projective and polar spaces. In the second chapter, we introduce the necessary concepts from algebraic graph theory, such as association schemes and distance-regular graphs, and the main techniques, including the fundamental contributions by Delsarte. Chapter 3 deals with the Grassmann association schemes, or more geometrically: with the projective geometries. Several examples of interesting subsets are given, and we can easily derive completely combinatorial properties of them. Chapter 4 discusses the association schemes from classical finite polar spaces. One of the main applications is obtaining bounds for the size of substructures known as partial m- systems. In one specific case, where the partial m-systems are partial spreads in the polar space H(2d − 1, q^2) with d odd, the bound is new and even tight. A variant of the famous Erdős-Ko-Rado problem is considered in Chapter 5, where we study sets of pairwise non-trivially intersecting maximal totally isotropic subspaces in polar spaces. A combination of geometric and algebraic techniques is used to obtain a classification of such sets of maximum size, except for one specific polar space, namely H(2d − 1, q^2) for odd rank d ≥ 5. Near polygons, including generalized polygons and dual polar spaces, are studied in the last chapter. Several results on substructures in these geometries are given. An inequality of Higman on the parameters of generalized quadrangles is generalized. Finally, it is proved that in a specific dual polar space, a highly regular substructure would yield a distance- regular graph, generalizing a result on hemisystems. The appendix consists of an alternative proof for one of the main results in the thesis, a list of open problems and a summary in Dutch