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Abstract: Let Pi be a finite projective plane of order n, and let G be a large (to be specific, |G| > (n^2+n+1)/2) abelian collineation group of Pi. Such planes have been classified into eight cases by Dembowski and Piper in 1967; the best-known case is that of a regular group (or Singer group). We survey the present state of knowledge about the existence and structure of such planes, making use of the close connection to difference sets and similar objects and the study of equivalent equations in certain group algebras. We shall explain the fundamental importance of this approach if one wants to obtain any deeper non-existence results. Several striking recent examples of unexpectedly short proofs for strong restrictions will be discussed.
Colloquium - 16:00
Abstract: J. Bokowski recently presented a family of self-dual 3-spheres based on a construction of G. Gevay. This family contains in particular the hypersimplex and the 24-cell.
In my talk I will show that this family of spheres can be seen as a very special case of the E-construction, which was invented by D. Eppstein, G. Kuperberg and G. M. Ziegler for the construction of 2-simple and 2-simplicial 4-polytopes and subsequently extended to arbitrary polytopes and, more general, to lattices by G. M. Ziegler and myself.
We will see that the spheres are E(C_m x C_n) for the product of two polygons C_m and C_n with m resp. n vertices and prove that for 1/n+1/m <= 1/2 we can realise these spheres as polytopes, while in general polytopality is not known. In particular we will see that this way we get a very flexible method to find new polytopal realisations of the hypersimplex and the 24-cell.