March 30, 2006. Sergei Anisov (Utrecht University, the Netherlands). From Voronoi diagramms to the topology of geometric 3-dimensional manifolds

The notion of a simple polyhedron (recall that neighborhoods of bertice of simple polyhedra are homeomorphic to the standard "6-wing batterfly" and all edges of it are triple lines) appeared in B.Casler's papers on three-dimensional topology. A simple polyhedron P embedded into a 3-manifold M is said to be a simple spine of M if M (punctured at a point if M is a closed manifold) can be collapsed onto P. Spines are a useful tool in algorithmic topology.

Cut locus of a Riemannian manifold M with respect to its point x (that is, the closure of the set of points y such that the shortest geodesic between x and y in M is not unique) is a standard object to study in differential geometry.

Voronoi diagrams are a classical tool and a classical object to study in computational geometry.

Simple polyhedra, "typical" cut loci in 3-manifolds, and "typical" Voronoi diagrasm have the same local structure. This simple observation enables us to apply ideas and methods of geometry and singularity theory to topological questions about spines of 3-manifolds. As a byproduct, one gets unexpected results from combinatorics.

List of talks at previous sessions of the seminar.