# An algebraic geometry of paths via the iterated-integral signature

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.

Abstract:

*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.