“Bulletin Board”

 School of Mathematics - February 2, 2005

A Short Course on
Constructive Combinatorics via Group Actions, t-Designs as example Structures

Reinhard Laue
University of Bayreuth, Germany

Feb. 16 - Mar. 9, 2005
School of Mathematics, IPM

Reinhard Laue
University of Bayreuth, Germany

Feb. 16 - Mar. 9, 2005
School of Mathematics, IPM

Group Actions and Their Homomorphisms
Feb. 16, 2005, 13:30-14:30
Isomorphism classes of combinatorial objects are described as group orbits. Construction up to isomorphism is regarded from the point of group actions. Methods from this theory are presented that form a basis of the approach.
  • Group Orbits as Isomorphism Classes
  • Transfer into Groups
  • Homomorphisms
  • From Counting to Constructing
  • Open Problems

t-Designs from Kramer-Mesner Martices
Feb. 21, 2005, 13:30-14:30
Since constructing and classifying $t$-designs usually is considered as extremely difficult, these objects are chosen as a challenge. The construction generally follows the formalized approach of Kramer and Mesner. We introduce our realization in the DISCRETA project of Bayreuth University.
  • Patterns on Graphs
  • Orbit Incidences
  • Snakes and Ladders
  • Examples

Families of Automorphism Groups
Feb. 23, 2005, 13:30-14:30
The construction requires prescribed groups of automorphisms. Some groups are discussed here that yielded interesting designs.
  • Symmetric Groups for Graphical Designs
  • Kramer-Mesner Matrices with Polynomial Entries
  • Projective Groups for 7-Designs and Steiner Systems
  • Open Problems

Direct Constructions of Large Sets
Feb. 26, 2005, 13:30-14:30
The famous recursive constructions of $t$-designs by Teirlink and Ajoodani-Namini, Khos­rovshahi require starting points, some of which are constructed by DISCRETA. We present the specialization of the approach to this problem.
  • Projective Groups with one or two Orbit Types on $k$-Element Subsets
  • Randomized Selection of Disjoint $t$-Designs
  • Open Problems

Selected Orbits from Prescribed Stabilizers
Mar. 5, 2005, 13:30-14:30
Constructing selected columns of the Kramer Mesner matrix that correspond to orbits on subsets with special properties reduces the search space and may result in designs with practical importance. We firstly prescribe stabilizers of orbits of the prescribed group on blocks and secondly select from these orbits those that are partitioned into classes of disjoint blocks by overgroups of the stabilizers.
  • Steiner Systems
  • Resolvable Designs
  • Open Problems

Solving Isomorphism Problems
Mar. 9, 2005, 13:30-14:30
The designs constructed via a prescribed group of automorphisms often can be classified up to isomorphism by the same group theoretic methods that are used in their construction. We stepwise refine the approach in order to reduce the theoretical requirements needed for an algorithmic solution of the problem. Since the designs constructed by this approach have a large automorphism group, we use another construction method by Van Leijenhorst and Tran van Trung to obtain astronomical numbers of isomorphism types of $t$-designs, mostly with only very small automorphism groups.
  • Moebius Inversion
  • Local Methods
  • Glueing $t$-designs
  • Open Problems

back to top
scroll left or right