Lectures and Colloquia during the semester

**June 25, 2001**

Technische Universität Berlin

Straße des 17. Juni 136, 10623 Berlin

Math building - Room MA 042

** Lecture - 14:15**

### Jürgen Richter-Gebert - ETH Zürich

### Pappos Theorem --- ten proofs and three variations

*Abstract:*
The main topic of this talk is to show how inspection of
algebraic equations can be used to generalize proofs of
geometric theorems. We take the well known Theorem of
Pappos about nine points and nine lines in the projective plane
as a starting point for our investigation. Starting from there
we will look at several different ways of proving this theorem.
Each way of proving leads to different kinds of generalization.
By this we will obtain a large collection of geometric incidence
theorems including many well known theorems in euclidean and
projective geometry. In particular we will see, that the structure
of cycles in geometric simplicial complexes is responsible
for the algebraic cancelation patterns that arise in many proofs
of geometric theorems.

** Colloquium - 16:00**

### Simon King - Strasbourg

### Polytopality of triangulations

*Abstract:*
It is known that
the bridge number of a link formed by edges of a shellable triangulation
T of S^3 is majorized by a linear function of the number of tetrahedra of T. We construct a series of triangulations of S^3 together with two-component links formed by edges so that the bridge
number of the links grows exponentially in the number of tetrahedra.
Thus, there are triangulations that are ``arbitrary far'' from being
shellable. We prove an upper bound for the bridge number in terms
of the number of tetrahedra that does not depend on any geometric assumption
on T, by methods from normal surface theory.
Inspired by the bridge number, we define a numerical invariant
p(T), called \emph{polytopality}. It satisfies bounds similar to
the mentioned bounds for the bridge numbers.
We conjecture that T is polytopal if and only if p(T) equals the number of tetrahedra of T. We show that T can be transformed into the boundary complex of a 4--simplex by a sequence of local transformations whose length is majorized by a linear function of p(T). Thus we obtain a new recognition algorithm for S^3.

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