I gave this talk on March 31, 2022 (which also happens to be Trans Day of Visibility) at Technische Universität Berlin for the Rough Algebra Day organised by Yannic Vargas, Sylvie Paycha, Bernd Sturmfels and Peter Friz.
In this presentation we discuss research opportunities starting from the observation that the iterated-integral signature constructs a correspondence between finite or infinite dimensional varieties of paths and finite or infinitely generated ideals in the shuffle algebra. This leads to a notion of Zariski topology on path space. We in particular explain how the splitting of the shuffle product into two half-shuffles yields a very promising concept of half-shuffle ideals which are conjectured to correspond to varieties that contain subpaths. In general, we give an outlook on how the algebraic classification of iterated-integral based path variaties can be developed. Further aspects of this research proposal are the study of singularities in path varieties and rough paths on classical affine varieties.
Find the slides here.