#
Monday Lecture and Colloquium

**Monday, February 8, 2010**

Freie Universität Berlin

Institut für Informatik

Takustrasse 9

14195 Berlin

room 005

** Lecture - 14:15**

###
Martin Aigner - Berlin

### From Irrational Numbers to Matchings: Markov´s Uniqueness Problem

*Abstract:*

A celebrated result in number theory is the Theorem of Markov which relates two
seemingly totally different subjects: approximations of irrational numbers and the
solutions of a certain equation. A proof, which Markov only sketched, was
eventually provided in detail by Frobenius precisely 100 years ago. In his paper
Frobenius mentioned a problem, now known as the uniqueness conjecture, which
has remained unsolved to this day. I will tell you about the theorem and the
conjecture, and discuss the connections to trees, groups, combinatorics of words,
lattice paths, and matchings of plane graphs.

**Colloquium - 16:00**

###
Bernd Schulze - Berlin

###
The orbit rigidity matrix of a symmetric framework

*Abstract:*

We present a new method for detecting non-trivial – non-congruent – finite
motions in symmetric bar and joint frameworks, i.e., motions that
continuously displace the joints of the framework while holding the
lengths of all bars fixed and changing the distance between two
unconnected joints. Basic to this method is the construction of an `orbit
rigidity matrix' whose columns and rows correspond to a set of
representatives for the vertex orbits and edge orbits of the underlying
graph of the framework, respectively. For frameworks that are generic
modulo the prescribed symmetry, a sufficiently large kernel of the `orbit
rigidity matrix' guarantees the existence of a non-trivial finite motion
in the framework. Thus, symmetric frameworks can often be shown to be
flexible by simply counting variables (vertex orbits) and equations (edge
orbits).
All the motions we detect with this new method have the nice property that
they preserve the symmetry of the framework throughout the path.

Letzte Aktualisierung:
03.02.2010