Mathematical Sciences: Rigidity

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R. Connelly intends to study some fundamental problems in the theory of rigid structures. Typical subjects include minimal cabling, self-stresses and Gale diagrams, generic global rigidity, and the pre-stress stability of certain symmetric "tensegrids". A basic concept that has proved surprisingly useful is the energy form associated to a self-stress for a tensegrity framework. Its rank, index, and positive definiteness are all related to important rigidity properties and should prove useful in the future. Connelly also plans to study certain packing and covering problems, particularly those related to rigidity. The mathematical theory underlying the rigidity of frameworks composed of rigid rods or of such frameworks reinforced by bracing cables is surprisingly amenable to analysis by general principles as well as of obvious practical relevance. Packing and covering problems also support a rich theory and have been found relevant to the design of error-correcting codes among other things.