An algebraic geometry of paths via the iterated-integrals signature
A talk in the Bielefeld Stochastic Afternoon series by
Rosa Preiss
| Abstract: | 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 are defined by relations holding all along their trajectory. In particular, this allows us to define rough paths on affine varieties in a purely algebraic and convenient way. While halfshuffle varieties are stable under stopping paths at an earlier time, we furthermore study varieties that are stable under concatenation of paths. These correspond to Hopf ideals of the tensor algebra, and the associated coordinate rings are once again Hopf algebras. https://arxiv.org/abs/2311.17886 Within the CRC this talk is associated to the project(s): B8 |