Speaker: Robert Connelly, Cornell University
Title: Universal Rigidity
Time: 2:30 PM, Monday, November 16, 2015
Place: Malott 206
Abstract: A framework is a finite graph G whose vertices are in ℝd and whose edges are fixed length bars. It is universally rigid if it is rigid not just in ℝd but in any ℝD, D>d, containing ℝd. It turns out that if any framework is universally rigid, there is sequence of matrix stress certificates that guarantee universal rigidity, and for the special case of complete bipartite graphs, the certificate simplifies to the condition that the partitions are not separable by a quadric surface. Furthermore for any abstract graph, a theorem of L. Lovasz (interpreted by A. Alfakih) implies that if it is appropriately connected, it always has a universally rigid realization in ℝd. If the graph is generically globally rigid when restricted to ℝd, then it has a universally rigid and infinitesimally rigid configuration. This provides a certificate for generic global rigidity that gives an explicit graph that is globally rigid in ℝd, which had not been known in dimension three or higher. This is joint work with Steven Gortler.