Volume rigidity of simplicial complexes

Abstract

This thesis develops a theory of rigidity of frameworks of simplicial complexes subject to maximal-simplex-volume constraints inspired by the well-studied theory of rigidity of frameworks of graphs subject to edge-length constraints. We take three main approaches: (simplicial) combinatorial, proving combinatorial conditions for generic local and global rigidity in all dimensions; algebro-combinatorial, exploring techniques of Bulavka et al. [2022] and conjecturing a lower bound on the rank of a simplicial complex in the volume rigidity matroid; and geometric, giving bounds on the number of embeddings of generic frameworks of bipyramids and showing that global rigidity is not a generic property of simplicial complexes in general. We additionally provide notes on ongoing and potential future research areas in volume rigidity theory that are, as of yet, underdeveloped, such as forbidden sign patterns and the rigidity with respect to volume constraints on non-maximal simplices.