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.