The rough path signature variety

Signature tensors are a useful tool in the study of paths X. When X runs among a given class of path (e.g. polynomial, piecewise linear, etc), the signature of X parametrizes an algebraic variety. The geometry of this variety reflects some of the properties of the chosen class of path. The most important example is the class of rough paths, that are widely studied in stochastic analysis. Their signature variety presents many similarities to the Veronese variety, and we'll illustrate the first nice results, as well as some open questions.