Pure Maths Colloquium: Zeb Rocklin

Systems such as vertebrates, origami sheets, umbrellas and robots are flexible, meaning that they can deform at their joints despite the rigidity of their constitutive elements. Since the time of JC Maxwell, rigidity theory has described such phenomena by identifying the spatial embeddings of graphs that preserve the lengths of their edges. I will focus on the rigidity of metamaterials, which repeat periodically in their bulk but also include boundaries. Building on the work of Kane and Lubensky (2013), I will show that the bulk structure generates an abstract space of modes and that periodic bulk modes lie at the boundary between modes on disjoint surfaces. These modes then generate a topological invariant that generates and places boundary modes. Mechanical criticality, generalized from Calladine for multiply periodic structures, is a crucial control parameter.