An algebraic geometry of paths via the iterated-integrals signature
"Contrary to previous approaches bringing together algebraic geometry and signatures of paths [AFS19,Gal19,CGM20], we introduce a Zariski topology on the space of paths itself, and study path varieties consisting of all paths whose signature satisfies certain polynomial equations [Pre23]. Particular emphasis lies on the role of the non-associative halfshuffle, which makes it possible to describe varieties of paths that satisfy certain relations all along their trajectory. Specifically, we may understand the set of paths on a given classical algebraic variety in R^d starting from a fixed point as a path variety, e.g. all paths on a sphere. While the characteristic geometric property of halfshuffle varieties is that they are stable under stopping paths at an earlier time, we furthermore study varieties corresponding to Hopf ideals that are stable under the natural operation of concantenation of paths. We point out how the notion of dimension for path varieties crucially depends on the fact that they may be reducible into countably infinitely many subvarieties. Finally, studying halfshuffle varieties naturally leads to a generalization of affine algebraic curves, e.g. including the graph of the exponential function.
As an application within algebraic geometry, we present some recent joint work with Felix Lotter that uses the new halfshuffle approach to significantly improve the understanding of the classical finite dimensional projective ""Varieties of Signature Tensors"" introduced in [AFS19].
[Pre23] Rosa Preiß. An algebraic geometry of paths via the iterated-integral signature. Preprint, November 2023. arXiv:231.17886 [math.RA].
[AFS19] Carlos Améndola, Peter Friz, and Bernd Sturmfels. Varieties of Signature Tensors. Forum of Mathematics, Sigma, 7:e10, 2019.
[Gal19] Francesco Galuppi. The rough Veronese variety. Linear Algebra and Its Applications, 583:282–299, December 2019.
[CGM20] Laura Colmenarejo, Francesco Galuppi, and Mateusz Michałek. Toric geometry of path signature varieties. Advances in Applied Mathematics, 121:102102, August 2020"
