Robert Connelly - Cornell University

The Kneser-Poulsen Conjecture in the plane

Abstract: We give a proof of the planar case of a longstanding conjecture of Kneser (1955) and Poulsen (1954). We prove that if a finite set of disks in the plane is rearranged so that the distance between each pair of centers does not decrease, then the area of the union does not decrease, and the area of the intersection does not increase. The idea is to use a previously known version of the conjecture in higher dimensions, where the centers are allowed to move continuously and expansively only, and relate the higher dimensional result to the planar case.

Colloquium - 16:00

David Bremner - New Brunswick, Canada

Matroid Polytope Completion

Abstract: Matroid polytopes are the oriented matroid analogue of point sets in convex position, or equivalently of convex polytopes. Since matroid polytopes can be encoded as e.g. a vector of signs, they provide a nice discrete search space.

Similar to the question of polytope realizability (but hopefully easier!), one can consider a class of problems of matroid polytope realizability. In this talk I will discuss computational experiments (and software) with the the following related questions. Given a (possibly empty) simplicial complex Delta and a (possibly empty) partial chirotope (i.e. some of the orientations of affine bases),

1. Find all of the matroid polytopes M consistent with \chi and containing Delta on their boundary complex.
2. Add the additional restriction that two marked facets s and t are no closer in the ridge graph of M than in the ridge graph of M.