This research proposal focuses on four types of problems related to subsets in finite projective and
polar spaces, for which the research is currently making a lot of progress. Projective and polar
spaces are geometries that arise from vector spaces.
The four types of subsets I will study are Erdos-Ko-Rado sets, Cameron-Liebler classes, tight sets
and Kakeya sets. There are important links between these subsets.
An Erdos-Ko-Rado set is a set of subspaces pairwise meeting in at least a point, e.g. the set of all
subspaces through a fixed point. The Erdos-Ko-Rado problem asks to classify the (large) Erdos-Ko-
Rado sets.
A Cameron-Liebler line class in a projective space is a set of lines such that each line in the set
meets a fixed number of lines in the set and such that each line not in the set meets an other fixed
number of lines in the set, e.g the set of all lines in a plane. This definition has recently been
generalised to subspaces. It is asked to classify all small Cameron-Liebler classes.
A tight set in a polar space is a set of points combinatorially behaving as a set of pairwise disjoint
subspaces of maximal dimension. The main question is to classify the smallest tight sets. These
results depend heavily on the type of the polar space.
A Kakeya set in a projective plane is a set of lines in a projective plane, one through each point on
a fixed 'line at infinity'. Its size is the number of points it covers. It is asked to classify the small
Kakeya sets.