Tony Nixon

On the d-dimensional rigidity problem

A bar-joint framework (G,p) in Euclidean d-space is rigid if the only edge-length preserving continuous motions of the vertices arise from isometric transformations. It is known that, when (G,p) is generic, its rigidity depends only on the underlying graph G, and is determined by the rank of the edge set of G in the generic d-dimensional rigidity matroid. Complete combinatorial descriptions are known when d=1,2 but obtaining a corresponding characterisation when d>2 is a long-standing open problem. In this talk I will give a gentle introduction to combinatorial rigidity. I will also report on recent advances for nearly planar graphs and for graphs with sufficiently large degree vertices.