Robert Hardt

Professor
Department of Mathematics
Rice University

Varifolds were originally introduced to describe various 2 dimensional minimal surfaces and soap film models. A varifold is stationary in a region  U  if its first variation of its mass is zero under deformations supported in U. A stationary one-dimensional varifold may model a  spider-web (possibly of variable thickness) where the region  U  is the complement of the attaching points for the web. F.Almgren and W.Allard studied their regularity in 1976. After reviewing some previous applications of one dimensional varifolds, we discuss a new one involving Michell trusses. These are cost-optimal balanced structures consisting of bars and cables. Introduced in a 1904 paper of an economist A.G. Michell, they have been treated in the Mechanical Engineering  literature and in interesting papers by R.Kohn and G. Strang (1983) and by G.Bouchitte, W.Gangbo, and  P.Sepulcher (2008). There are many basic open questions about the location and structure of Michel trusses.