Origins and applications of signatures

In the early twentieth century, Élie Cartan solved the equivalence problem for submanifolds under the action of a Lie group. In essence, two (suitably regular) submanifolds can be locally mapped to one another by a group transformation if and only if their differential invariants have identical functional relationships. Cartan's result was subsequently reformulated by the author by introducing the notion of a signature, which is the submanifold parametrized by the fundamental differential invariants. The subsequent equivariant method of moving frames made this result completely algorithmic, and applicable to arbitrary Lie group (and even Lie pseudo-group) actions. In this talk, I will discuss some of the history, survey basic ideas and algorithms, and present a few of the many applications, including the automatic reassembly of objects: jigsaw puzzles, egg shells, and broken bones. I will endeavor to keep the talk accessible to a general audience.