Thas, Joseph A.; Matematični kolokvij maj 2007

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Finite Geometries : Classical Problems and Recent Developments

Joseph A. Thas

Univerza v Gentu, Belgija

17. maj 2007

In recent years there has been an increasing interest in finite projective spaces, and important applications to practical topics such as coding theory, cryptography and design of experiments have made the field even more attractive. In my talk some classical problems and recent developments will be discussed. First I will mention a purely combinatorial characterization of Hermitian curves in PG(2,q2); here, from the beginning, the considered pointset is contained in PG(2,q2). Next, a recent elegant result on semiovals in PG(2,q), due to A. Gács, will be given. A second approach is where the object is described as an incidence structure satisfying certain properties; here the geometry is not a priori embedded in a projective space. This will be illustrated by a characterization of the classical inversive plane in the odd case. Another recent beautiful result in Galois geometry is the discovery of an infinite class of hemisystems of the Hermitian variety in PG(3,q2), leading to new interesting classes of incidence structures, graphs and codes; before this result, just one example for GF(9), due to Segre, was known.

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