Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink

Algebraic cycles and motivic generic iterated integrals

Following the work of Gangl, Goncharov and Levin in [GGL], we will give a combinatorial framework for motivic study of iterated integrals on the affine line. We will show that under a certain genericity condition these combinatorial objects yield to elements in the motivic Hopf algebra constructed in Bloch-Kriz [BK]. It will be shown that the Hodge realization of these elements coincides with the Hodge structure induced from the fundamental torsor of path of punctured affine line.

Algebraic cycles and motivic iterated integrals II

This is a sequel to our previous paper (joint with Furusho). It will give a more natural framework for constructing elements in the Hopf algebra of framed mixed Tate motives according to Bloch and Kriz. This framework allows us to extend our previous results to interpret all multiple zeta values (including the divergent ones) and the multiple polylogarithms in one variable as elements of this Hopf algebra. It implies that the pro-unipotent completion of the torsor of paths on projective line minus three points, is a mixed Tate motive in the sense of Bloch-Kriz. Also It allows us to interpret t…

Galois symmetries of fundamental groupoids and noncommutative geometry

We define motivic iterated integrals on the affine line, and give a simple proof of the formula for the coproduct in the Hopf algebra of they make. We show that it encodes the group law in the automorphism group of certain non-commutative variety. We relate the coproduct with the coproduct in the Hopf algebra of decorated rooted planar trivalent trees - a planar decorated version of the Hopf algebra defined by Connes and Kreimer. As an application we derive explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf algebra. We give a criteria for a motivic iterated integral…

平面曲线的多变量强单值群猜想

The Multivariable Strong Monodromy Conjecture for Plane Curves

该研究针对全纯芽组建立迭代残差障碍，证明局部多变量拓扑ζ函数的极大阶极超平面及约化平面曲线芽组的实际极超平面均含于Bernstein–Sato零轨迹，从而证明平面曲线的拓扑多变量强单值群猜想。

Extensions of Motives and the Fundamental Group

In this paper we construct extensions of the Mixed Hodge structure on the fundamental group of a pointed algebraic curve. These extensions correspond to the regulator of certain explicit motivic cohomology cycles in the self product of the curve which were first constructed by Bloch and Beilinson. This leads to a new iterated integral expression for the regulator. Our result is a generalization of a result of Colombo's where she constructs the extension corresponding to a motivic cycle class in the Jacobian of a hyperelliptic curve constructed by Collino. This is to appear in the Mathematical …

The polylog quotient and the Goncharov quotient in computational Chabauty-Kim theory II

Building on work by Dan-Cohen--Wewers, Dan-Cohen [DC], and Brown, we push the computational boundary of our explicit motivic version of Kim's method in the case of the thrice punctured line over an open subscheme of Spec ZZ. To do so, we develop a refined version of the algorithm of [DC] tailored specifically to this case. We also commit ourselves fully to working with the polylogarithmic quotient. This allows us to restrict our calculus with motivic iterated integrals to the so-called depth-one part of the mixed Tate Galois group studied extensively by Goncharov. An application was given in p…