Tutte Colloquium - Bill Jackson

Title: Rigidity of Simplicial Complexes

Abstract: A simplicial k-cycle is an abstract simplicial k-complex in which every (k-1)-face belongs to an even number of k-faces. A simplicial k-circuit is a minimal simplicial k-cycle (in the sense that none of its proper subcomplexes are simplicial k-cycles). The set of all simplicial k-circuits with vertex set V is the set of circuits of the simplicial k-matroid whose groundset is the set of all (k+1)-subsets of V. Examples of simplicial k-circuits occur naturally as boundary complexes of (k+1)-dimensional simplicial polytopes or, more generally, connected simplicial k-dimensional manifolds. In particular, every connected triangulated surface is a simplicial 2-circuit.

A celebrated result of Cauchy implies that the 1-skeleton of every simplicial convex polyhedron is rigid. This was extended to simplicial convex d-dimensional polytopes by Whiteley in 1984. These results become false if we remove the convexity constraint but Fogelsanger showed in his PhD thesis in 1988 that they continue to hold in the generic case. More precisely, he showed that the 1-skeleton of every simplicial (d-1)-circuit is generically rigid in \(\mathbb{R}^d\) whenever d is greater or equal than 3. Together with James Cruickshank and Shin-ichi Tanigawa, we recently adapted Fogelsanger's proof technique to obtain results on the global rigidity of simplicial (d-1)-circuits and the rigidity of symmetric simplicial (d-1)-circuits in \(\mathbb{R}^d\).

I will give a brief introduction to the theory of rigid frameworks, then describe Fogelsanger’s ingenious proof technique and outline how it can be adapted to work for global and symmetric rigidity.