# Programme

All lectures take place in lecture hall H2,
*Volksbankhörsaal*,
in
building 22A
(part of building 22 in the upper middle part of the map),
Universitätsplatz.

## Friday, 23 March

I will give an overview on cyclotomic conditions guaranteeing the existence of certain combinatorial designs. Time will probably not permit me to present all the conditions that I have in mind so that I will try to select some of the nicer ones.

I will start from the old conditions by R.C. Bose that I improved in the 1990s about the existence of an elementary abelian (q, 4, 1)- or (q, 5, 1)-BIBD with a multiplier of order 3 or 5, respectively, fixing all the base blocks.

Then I will touch on the famous asymptotic theorem by R.M. Wilson about the existence of elementary abelian BIBDs that has been recently revisited by Anita Pasotti and myself, and I will finish with some unpublished conditions of mine about the existence of optimal optical orthogonal codes of weight 4 or 5.

The talk reviews some results and open problems related to joint work of the speaker with Dieter Jungnickel.

The dinner takes place at Landhaus Hadrys; registration is required.

## Saturday, 24 March

The definition of a generalised polygon axiomises the basic property of the classical polygons in an attempt to find objects which often have a high degree of regularity and symmetry. When viewed as an incidence structure the classical polygons have lines which are incident with two points, points which are incident with two lines. These are examples of a partial linear space which we define in the following way.

A partial linear space is an incidence structure which consists of a set of points P and a set of lines L where the elements of L are subsets of P and two points are incident with (contained in) at most one line.

The incidence graph of a partial linear space is the bipartite graph whose vertices are the elements of P and L and where an edge joins an element x of P with an element m in L if the point x is incident with the line m.

The incidence graph of the classical n-gon is the cyclic graph on 2n vertices. Note that two vertices in a cyclic graph on 2n vertices are at distance at most n. Moreover the shortest cycle is clearly of length 2n. These are the properties that are generalised in the definition of a generalised polygon.

A generalised n-gon is a partial linear space whose incidence graph has diameter n and girth 2n. Or in other words, two vertices are at distance at most n and the shortest cycle has length 2n.

Assuming some non-degeneracy condition one can prove that a finite generalised polygon has an order (k,r), where every line is incident with exactly k points and every point is incident with exactly r lines. Note that some authors define such a finite generalised polygon to have order (k-1,r-1).

In 1964 Feit and Higman proved that, for r and k at least 3, there exist finite generalised n-gons if and only if n=3, 4, 6 or 8.

In this talk I will suggest an alternative sets of axioms which allow us to construct generalised pentagons, heptagons, etc. The talk will consist of basic properties of these pentagons, some sporadic constructions, some possible applications and some non-existence results for supposedly feasible parameters.

Tight sets in finite classical polar spaces have been recently investigated by different authors.

In this talk, we present recent results on i-tight sets in the finite classical polar
space
H(2n+1,q^{2}), arising from the non-singular Hermitian variety in PG(2n+1,q^{2}).

A generator of H(2n+1,q^{2}) is a 1-tight set of H(2n+1,q^{2}). A Baer subgeometry
PG(2n+1,q), contained in H(2n+1,q^{2}) and where the Hermitian polarity of H(2n+1,q^{2})
defines a symplectic polarity of this Baer subgeometry PG(2n+1,q), defines a (q+1)-tight
set of H(2n+1,q^{2}).

We prove that for i small, every i-tight set on H(2n+1,q^{2}) is the union of pairwise
disjoint
generators of H(2n+1,q^{2}) and of such Baer subgeometries PG(2n+1,q) contained in
H(2n+1,q^{2}).

The techniques used to prove these results extend techniques by Metsch and Storme on
maximal partial spreads in PG(3,q^{2}).