Darrick Lee (University of Edinburgh) – Thin Homotopy and the Signature of Piecewise Linear Surfaces

Thin homotopy is an equivalence relation on paths which consist of two basic equivalences: reparametrizations and cancellation of retracings. This naturally arises in geometry when studying the invariances inherent in parallel transport on principal bundles. The path signature is the parallel transport map with respect to a “universal” translation-invariant connection on $R^n$, which has recently been used to develop the theory of rough paths. In fact, the path signature characterizes thin homotopy classes of paths.

In this talk, we discuss the generalization of this story to piecewise linear surfaces. We develop an algebraic model of piecewise linear paths and surfaces akin to the free group construction, and use this to study the relationship between thin homotopy of surfaces, and the notion of a “surface signature.” Based on joint work with Francis Bischoff.