Iterated Integrals on the Digraphs
组织者
演讲者
Yunpeng Zi
时间
2026年05月28日 15:00 至 16:15
地点
A3-4-301
线上
Zoom 518 868 7656
(BIMSA)
摘要
Iterated integral is a classical geometric structure on the real line. It came from a classical question of solving differential equations namely find the solution to the equation
$\frac{dy}{dt}=A(t)y(t)$ with $y(0)=Id$. It was extended to the differentiable space by K.T.Chen in 1950s. And after 30 years of exploration Chen and his student Richard Hain find the iterated integral is the key to the Homotopy period problem, which asking whether one can detect all generators of homotopy group by taking integrals with arbitrary seclecting the differential forms on the smooth manifolds. R.Hain further extended this geometric structure to algebraic varieties which assigning a pure Hodge structure to the fundamental group of algebraic varieties.
I will talk about a recent project cooperated with Zhang and Yau where we extend the iterated path integrals from smooth manifolds to digraphs and develop the associated algebraic and geometric structures. Iterated path integrals on a digraph naturally give rise to the iterated path algebra and the iterated loop algebra, both defined as quotient algebras of a shuffle algebra, with the latter carrying a canonical Hopf algebra structure.
We construct a non-degenerate pairing between elementarily equivalent classes of loops on a digraph and the iterated loop algebra. By restricting to iterated path integrals that are invariant under $C_\partial$-homotopy, a distinguished subalgebra is obtained which, under this pairing, corresponds to the group algebra of the fundamental group. We further show that this subalgebra is a homotopy invariant and forms a Hopf algebra with involutive antipode.
$\frac{dy}{dt}=A(t)y(t)$ with $y(0)=Id$. It was extended to the differentiable space by K.T.Chen in 1950s. And after 30 years of exploration Chen and his student Richard Hain find the iterated integral is the key to the Homotopy period problem, which asking whether one can detect all generators of homotopy group by taking integrals with arbitrary seclecting the differential forms on the smooth manifolds. R.Hain further extended this geometric structure to algebraic varieties which assigning a pure Hodge structure to the fundamental group of algebraic varieties.
I will talk about a recent project cooperated with Zhang and Yau where we extend the iterated path integrals from smooth manifolds to digraphs and develop the associated algebraic and geometric structures. Iterated path integrals on a digraph naturally give rise to the iterated path algebra and the iterated loop algebra, both defined as quotient algebras of a shuffle algebra, with the latter carrying a canonical Hopf algebra structure.
We construct a non-degenerate pairing between elementarily equivalent classes of loops on a digraph and the iterated loop algebra. By restricting to iterated path integrals that are invariant under $C_\partial$-homotopy, a distinguished subalgebra is obtained which, under this pairing, corresponds to the group algebra of the fundamental group. We further show that this subalgebra is a homotopy invariant and forms a Hopf algebra with involutive antipode.