Discrete Geometry and Graph Theory
(An informal summer seminar organized with Károly Bezdek, Bob Connelly and Kostya Rybnikov)
Place: Malott 224
Time: Tuesday, Thursday, Friday 3:00 to 4:00 PM, from June 27, 2000 to August 4, 2000.
Topics: The Colin de Verdiére graph invariant, Stress matrices, global rigidity, symmetric polyhedra.
Abstracts and speakers (subject to change and extension):
It might be interesting to look at some of the papers by Lovasz on his home page at Yale. Or you can use his Microsoft homepage.
Bob Erdahl visited on August 3 at 3 PM in Malott 224. Here is an abstract of his talk:
Space filling zonotopes, dicings and regular matroids
A space filling zonotope is one that can be used to tile space, in a facet-to-facet way, by translates. A dicing, on the other hand, is a way to cut space up - it is similar to dicing a carrot. I will explain how space filling zonotopes and dicings are dual notions in the theory of geometric lattices, and how the study of this duality has led to a partial answer to a question that was asked by George Voronoi over 90 years ago. Another interesting fact is that affine classes of space filling zonotopes are in one-to-one correspondence with regular matroids. I will explain how these notions fit into Voronoi's therory of lattice types. I show that the Voronoi polytope for a lattice can always be written as a Minkowski sum of simpler Voronoi polytopes. These summands are the building blocks, and correspond to Voronoi polytopes for "edge forms" in Voronoi's theory of lattice types. In 2- and 3-dimensions there is only one type of building block, a line segment. All Voronoi polytopes in these dimensions are Minkowski sums of line segments, so are zonotopes. In four dimensions a single new type of building block appears, the 24-cell, and in higher dimensions there are more. A non-trivial edge form is characterized by the property that the affine type of its Voronoi cell changes under any perturbation different from scaling.
We hope to continue the seminar this Fall.
Last updated: August 17, 2000