Monday, June 23, 2014
Freie Universität Berlin
Institut für Informatik
Lecture - 14:15
Universality of geometric realization spaces for classes of combinatorial objects is a quite common phenomenon. Universality means essentially that for each semi-algebraic set there exists a combinatorial object of the given class such that its realization space is in some sense equivalent to the given semi-algebraic set. The proofs always give some kind of encoding of semi-algebraic sets by combinatorial objects of the type under consideration.
After a brief overview of several known universality theorems I state a universality theorem for realization spaces of polyhedral maps (i.e. dissections of a closed 2-manifold into polygons) and give a fairly extensive sketch of the proof.
Colloquium - 16:00
In 1983 Brehm described a triangulation of the Möbius Strip that does not admit a geometric realization in R^3. His proof uses conditions on the linking numbers of pairs of polygonal cycles on the surface to construct a contradiction. We show how this approach can be systematically transferred to give necessary conditions for the realizability of triangulated orientable surfaces.
PhD defense lecture