# Talk: Orthogonal and special linear invariants of paths via the iterated-integral signature: Overview and new developments

I gave this blackboard talk on May 13, 2022 at the *15th Annual ERC Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis*.

The talk was largely based on joint work with Joscha Diehl, Micheal Ruddy, Jeremy Reizenstein and Nikolas Tapia.

Abstract:

*Looking at the action of the orthogonal group, we apply Fels-Olver’s moving frame method paired with the log-signature transform to construct a set of integral invariants for curves in R^d from the iterated-integrals signature. In particular we show that one can algorithmically construct a set of invariants that characterize the equivalence class of the truncated iterated-integrals signature under orthogonal transformations, which yields a characterization of a curve in R^d under rigid motions and an explicit method to compare curves up to these transformations. In this talk, we furthermore present a so far unpublished, more explicit new description of the moving frame via the QR-decomposition of a certain matrix build from the level two signature data of the curve.*

*Furthermore, this talk discusses the new result that such a full characterization of a path up to group action (and tree-like equivalence) via iterated-integral invariants does not exist for the special linear group. We instead hint a so far only conjectured kind of “Determinantensatz” for the iterated-integral signature which would describe the equivalence relation on paths given by only looking at special linear iterated-integral invariants.*