Varieties of Signature Tensors

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals, and they are central to the theory of rough paths in stochastic analysis.  For some special families of curves, such as polynomial paths and piecewise-linear paths, their parametrized signature tensors trace out algebraic varieties in the space of all tensors. We introduce these varieties and examine their fundamental properties, while highlighting their intimate connection to the problem of recovering a path from its signature. This is joint work with Peter Friz and Bernd Sturmfels.