For a set S of n points in R^d, their pairwise distances determine their locations up to symmetry.

Rigidity, in a nutshell, says that when points are in general positions, a small fraction of (carefully chosen) distances determines the rest.

In the talk, we will discuss what can be achieved without assuming general position. Let P denote the set of pairs in S for which the distance is known, we consider two contrasting scenarios when d=1: (1) P is arbitrary, and (2) P is random.

I’ll describe recent works and directions, as well as the case d>1.

Based on joint work with Itai Benjamini.