Multiple polylogarithms, polygons, trees and algebraic cycles

Functional equations of polygonal type for multiple polylogarithms in weights 5, 6 and 7

We present new functional equations in weights 5, 6 and 7 and use them for explicit depth reduction of multiple polylogarithms. These identities generalize the crucial identity $\mathbf{Q}_4$ from the recent work of Goncharov and Rudenko that was used in their proof of the weight 4 case of Zagier's Polylogarithm Conjecture.

Cycle complex over the projective line minus three points : toward multiple zeta values cycles

In this paper, the author constructs a family of algebraic cycles in Bloch's cubical cycle complex over the projective line minus three points which are expected to correspond to multiple polylogarithms in one variable. Elements in this family are in particular equidimensional over the projective line minus three points. In weight greater or equal to $2$, they are naturaly extended as equidimensional cycle over the affine line. This allows to consider their fibers at the point 1 and this is one of the main differences with Gangl, Goncharov and Levin work where generic arguments are imposed for…

On the Goncharov depth conjecture and a formula for volumes of orthoschemes

We prove a conjecture of Goncharov, which says that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. We give an explicit formula for this presentation, involving a summation over trees that correspond to decompositions of a polygon into quadrangles. Our second result is a formula for volume of hyperbolic orthoschemes, generalizing the formula of Lobachevsky in dimension $3$ to an arbitrary dimension. We show a surprising relation between two results, which comes from the fact that hyperbolic orthoschemes are parametrized by configurations of p…

Multiple logarithms, algebraic cycles and trees

This is a short exposition--mostly by way of the toy models ``double logarithm'' and ``triple logarithm''--which should serve as an introduction to a forthcoming article in which we establish a connection between multiple polylogarithms, polygons, trees and algebraic cycles.

通过同调循环上的卷积得到亚纯高亏格积分核

Meromorphic higher-genus integration kernels via convolution over homology cycles

研究如何通过同调循环上的卷积，利用全纯阿贝尔微分和素形式直接构造恩里克斯核，证明其在卷积和模变化下的封闭性，强化了与非全纯模张量的相似性。

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…