Department Colloquium

Location: CL 435

Speaker: Francis Bischoff, University of Regina

Title:  The signature of paths and surfaces

Abstract:

There are two prominent sources of groups in mathematics: symmetries and topological spaces, and they are related through the theory of differential equations, or more generally, gauge theory. Namely, by solving a differential equation, a path in a topological space can be represented as a symmetry of a vector space. In this way, the concatenation structure on paths is reflected in the composition of symmetries. The universal example of this correspondence is the path signature, a construction that assigns a non-commutative power series to a path in Euclidean space, generalizing the canonical homomorphism from the free group on \(n\)-generators into the free \(R\)-algebra on \(n\)-generators. Remarkably, the signature completely characterizes a path up to reparametrization and cancellation and can be used to construct the solutions to a family of differential equations. For this reason, the path signature has found applications in various fields, such as stochastic analysis and machine learning, where it can be used to encode time series data. 

I plan to discuss ongoing work to generalize the signature so that it can be used to encode two-dimensional surfaces. To formulate this construction, we must first generalize the notion of a group to 2 dimensions, so that we can encode the concatenation structure of surfaces. In fact, this 2-dimensional group structure has existed in the literature since the 1940s under the name of crossed modules, and the higher dimensional generalization of the signature was suggested in work of Kapranov. I will present the construction of a crossed module of piecewise linear surfaces, and a theorem characterizing the kernel of the surface signature map. This is joint work with Darrick Lee and Camilo Arias Abad.