|
|
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
|
