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Lectures and Colloquia during the semester



May 3, 2004

Technische Universität Berlin
Straße des 17. Juni 136
10623 Berlin
Math building - Room MA 042           - map -
Lecture - 14:15

Aart Blokhuis -Eindhoven University of Technology

Special Point Sets in Finite Projective Planes

Abstract: The motivating theorem for the study of special point sets in finite projective planes is Segre's famous result that every (q+1)-arc in a desarguesian projective plane of odd order q is a conic. Here a k-arc is a set of k points, no 3 collinear. The result says that simple combinatorial properties of the set guarantee an algebraic structure. Another well-known result of this type is due to Bruen and says that a (non-trivial) blocking set of size q+\sqrt q+1 in a plane of order q is a Baer subplane. Here a blocking set is a collection of points with at least 1 point on every line, and a Baer subplane is a substructure isomorphic to a plane of order \sqrt q.

In the last 15 years algebraic techniques, in part based on work by Rédei on lacunary polynomials, have led to a much better understanding of the structure of special point sets, and in the talk an exposition of these developments will be given.


Colloquium - 16:00

Stephan Hell -Technische Universität Berlin

On the number of Tverberg partitions in the prime power case

Abstract: In 1966, Helge Tverberg showed that any set of (d+1)(q-1)+1 points in d-dimensional Euclidean space admits a partition into q subsets such that the intersection of their convex hulls is non-empty. Such partitions are called Tverberg partitions; the result is best possible: For less than (d+1)(q-1)+1 points the statement does not hold.

Another natural question is to ask for a lower bound for the number of Tverberg partitions. How many Tverberg partitions are there for a given set of points? Gerard Sierksma conjectured that there are at least ((q-1)!)^d many for (d+1)(q-1)+1 points in d-dimensional Euclidean space. The conjecture is still not proved. In this talk we will show how to extend the currently best known lower bound for q being prime of Vucic and Zivaljevic to the prime power case.


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