An algebraic geometry of paths via the iterated-integrals signature

Contrary to previous approaches bringing together algebraic geometry and signatures of paths, 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. 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. While halfshuffle varieties are stable under stopping paths at an earlier time, we furthermore study varieties that are stable under 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, we see that studying halfshuffle varieties naturally leads to a generalization of classical algebraic curves, surfaces and affine varieties in finite dimensional space, where these generalized algebraic sets are now described through iterated-integral equations.

As a highlight for this talk, I will illustrate how we can use this machinery to translate every system of ODEs with polynomial coefficients into a purely algebraic description of the variety of solutions to the ODE system.