Pure Maths Colloquium: Tony Nixon

A framework is a geometric realisation into Euclidean \(d\)-space of a graph with edges represented by stiff bars and vertices by universal joints. The framework is globally rigid if every other realisation, in the same dimension, with the same edge lengths arises from an isometry of the space. I will try to give a gentle introduction to the theory of global rigidity in the case that the realisation is generic. Specifically, I will describe that generic global rigidity depends only on the underlying graph and the characterisation of such graphs when \(d=2\). The corresponding situation when \(d>2\) is open. However, in \(3\)-dimensions, if we restrict to triangulations then a recent result of Cruickshank, Jackson and Tanigawa gives a combinatorial characterisation. In current joint work with Sean Dewar and Eleftherios Kastis, we prove a similar characterisation for all projective planar graphs. I will describe these results and then conclude with an application of our work to maximum likelihood estimation in Gaussian graphical models.