### Abstract File Upoad

### Abstract or Additional Information

Given a graph that is realized in some Euclidean space, with edges of fixed lengths joining the vertices, when does there exist other configurations with the same edge lengths? There are three avors of this question:

(a) Local rigidity - infinitesimal rigidity: There is no continuous motion of the vertices other than the "trivial ones" that are restrictions of rigid motions of the whole space.

(b) Global rigidity: There are no other non-congruent configurations with the same corresponding edge lengths in the same Euclidean space.

(c) Universal rigidity: There are no other non-congruent configurations with the same corresponding edge lengths in any higher dimensional Euclidean space.

Each of the rigidity flavors above have their own techniques, refinements, and examples to be explained later.