## Pushing things around

- Date: 10/12/2007

Robert Connelly (Cornell University)

University of Calgary

Unsolved for over twenty-five years, a surprisingly difficult

conjecture stated that a non-crossing polygonal chain of fixed-length

edges in the plane can be continuously opened without crossing. Gunter

Rote, Erik Demaine and I proved this Carpenter's Rule conjecture in

2000.

Unsolved for almost fifty years, M. Kneser and E. M. Poulsen

conjectured that if a finite collection of circular disks in the plane

are picked up and re-positioned in the plane so that no pair of centers

gets closer together, then the area of the union never gets smaller.

Karoly Bezdek and I proved this Kneser-Poulsen conjecture in 2002.

These two seemingly disparate results, the proofs of the Carpenter's

Rule conjecture and the Kneser-Poulsen conjecture, can be combined to

extend the Carpenter's Rule result to chains of appropriately slender

sets, which can open without self-intersection. One of the critical

ingredients in the Kneser-Poulsen result is a formula by Balazs Csikos

for the change in the area of unions and intersections of disks as the

centers are moved continuously. This brings up a large class of

problems simultaneously generalizing the notions of packings and

coverings of disks. For example, it shows how to find the critical area

configurations of the union of one set of disks, with one larger

radius, intersected with the union of another set of disks, with

another smaller radius.

PIMS Distinguished Lecture 2007