Link to the recorded talk (in panopto) 

  A d-dimensional framework is a pair (G,p) consisting of a graph G and an embedding p of its vertex set into R^d. The framework is said to be rigid if there is no continuous motion of the vertices of G, starting from the positions prescribed by p, that preserves the lengths of all edges (other than the trivial motions -- rotations and translations of the whole graph). Recently, Jordán and Tanigawa introduced the d-dimensional algebraic connectivity of a graph G. This is a quantitative measure of the rigidity of G, which generalizes the classical notion of spectral expansion of graphs (corresponding to the d=1 case). In this talk I will present several results on the rigidity and d-dimensional algebraic connectivity of graphs. In particular, I will discuss the existence of "rigidity expander graphs" (infinite families of sparse graphs whose d-dimensional algebraic connectivity is bounded away from zero), and show several new results on the rigidity of random graphs. Based on joint work with Eran Nevo, Yuval Peled and Orit Raz, and with Michael Krivelevich and Peleg Michaeli.