Monday, January 12, 2015
Freie Universität Berlin
Institut für Informatik
Takustr. 9
14195 Berlin
Room: 005
Lecture - 14:15
Abstract:
The talk studies analogues to matroids and oriented matroids for complex projective point configurations.
Such generalizations can be obtained by introducing a convex phase function that plays the analogous the role of
the sign function for oriented matroids. Commonalities and differences to the theory of oriented matroids are studied.
In particular criteria for realizability in the generic case are given and will be related to certain highly symmetric syzygies
in invariant theory and a rigidity property.
Colloquium - 16:00
Abstract:
A classic theorem in combinatorial design theory is Fisher's inequality, which states that a family F of subsets of $\{1,2,...,n\}$ with all pairwise intersections of size λ can have at most n non-empty sets. One may weaken the condition by requiring that for every set in F,all but at most k of its pairwise intersections have size λ. We call such families $(k,\lambda)$-Fisher. Vu was the first to study the maximum size of such families, proving that for k = 1 the largest family has 2n-2 sets, and characterizing when equality is attained. In this talk we present a refined version of these results, showing how the size of the maximum family depends on the parameterλ. In particular we show that for small λ one essentially recovers Fisher's bound. We also solve the next open case of k = 2 and obtain the first non-trivial upper bound for general k.