Lectures and Colloquia during the semester

**Monday, June 7, 2004**

Freie Universität Berlin - Institut für
Informatik

Takustraße 9

14195 Berlin

Room 005
- map -

**Lecture - 14:15**

### Richard Weiss -Tufts University

### Generalized Polygons and Spherical Buildings

*Abstract:*
A generalized polygon is a bipartite graph whose diameter
equals half the length of a minimal circuit. Generalized
polygons can be viewed as spherical buildings of rank two.
Spherical buildings, in their turn, can be viewed as certain
edge-colored graphs whose rank is defined to be the number
of colors. Spherical buildings were introduced by Jacques
Tits. He showed in 1974 that the only spherical buildings
of rank at least three are those arising from simple algebraic
groups (and certain variations of these groups).
This is a beautiful example of a theorem in which a few
purely geometrical hypotheses end up implying that the only examples display
a high degree of symmetry. Since the class of generalized
polygons include the incidence graphs of arbitrary projective
planes and these are too numerous to classify, it is not possible to extend
Tits' result to arbitrary spherical
buildings of rank two. In recent work, however, Tits and the
speaker have succeeded in extending Tits' original theorem
under the so-called Moufang condition. We will give a brief overview
of these results for non-group theorists using only
the language of graph theory.

**Colloquium - 16:00**

### Mark de Longueville -Freie Universität Berlin

### Cake, Groups, Topology and all that

*Abstract:*
For a certain class of cake division problems the existence of a
solution was shown by topological methods. To be more precise: by
cousins of the Borsuk-Ulam theorem for more general group actions.

In a very special case a constructive existence proof is known. Here
the relation between the topological theorem and its combinatorial
counterpart which yields the alternative proof is well understood.

The talk will concentrate on the same relation in more general cases,
i.e., the relation between the cousins of the Borsuk-Ulam theorem and
its combinatorial counterparts.

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