
Reinhard Laue University of Bayreuth, Germany
Feb. 16  Mar. 9, 2005 School of Mathematics, IPM

Group Actions and Their Homomorphisms 
Feb. 16, 2005, 13:3014:30 
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

tDesigns from KramerMesner Martices 
Feb. 21, 2005, 13:3014:30 
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, 13:3014:30 
The construction requires prescribed groups of automorphisms. Some groups are discussed here that yielded interesting designs.
 Symmetric Groups for Graphical Designs
 KramerMesner Matrices with Polynomial Entries
 Projective Groups for 7Designs and Steiner Systems
 Open Problems

Direct Constructions of Large Sets 
Feb. 26, 2005, 13:3014:30 
The famous recursive constructions of $t$designs by Teirlink
and AjoodaniNamini, 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, 13:3014:30 
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, 13:3014:30 
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
