多元微积分（一）

Elliptic multizetas and the elliptic double shuffle relations

We define an elliptic generating series whose coefficients, the elliptic multizetas, are related to the elliptic analogues of multiple zeta values introduced by Enriquez as the coefficients of his elliptic associator; both sets of coefficients lie in $\mathcal{O}(\mathfrak{H})$, the ring of functions on the Poincaré upper half-plane $\mathfrak H$. The elliptic multizetas generate a $\mathbb Q$-algebra $\mathcal{E}$ which is an elliptic analogue of the algebra of multiple zeta values. Working modulo $2πi$, we show that the algebra $\mathcal{E}$ decomposes into a geometric and an arithmetic part…

A $t$-motivic interpretation of shuffle relations for multizeta values

Thakur (2010) showed that, for $r,$ $s\in \mathbb{N}$, a product of two Carlitz zeta values $ζ_A(r)$ and $ζ_A(s)$ can be expressed as an $\mathbb{F}_p$-linear combination of $ζ_A(r+s)$ and double zeta values of weight $r+s$. Such an expression is called shuffle relation by Thakur. Fixing $r,$ $s\in \mathbb{N}$, we construct a $t$-module $E'$. To determine whether an $(r+s)$-tuple $\mathfrak{C}$ in $\mathbb{F}_q(θ)^{r+s}$ gives a shuffle relation, we relate it to the $\mathbb{F}_q[t]$-torsion property of the point $\mathbf{v}_\mathfrak{C}\in E'(\mathbb{F}_q[θ])$ constructed with respect to the …

多重zeta值与奇偶性结果之间的特殊关系

Special relations between multizeta values and parity results

研究了D.塔库尔Thakur的multizeta值的F_p-跨度是一个代数（Thakur.函数字段多zeta值的混洗关系）。特别地，乘积ε（a）ε（b）是多ζ值的线性组合。本文讨论了作者和D.证明了对于小值或特殊的a族，如何将a（a）a（b）写成多zeta值的F_p-线性组合.同时证明了Thakur提出的宇称猜想。

Double shuffle relations for arborified zeta values

Arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees. This approach allows to study their convergence domain and to relate them to multizeta values. Generalisations to rooted trees of the stuffle and shuffle products are defined and studied. It is further shown that arborifed zeta values are algebra morphisms for these new products on trees.

正特征下的着色多重 zeta 值

Colored multizeta values in positive characteristic

该研究考察正特征下 Shen-Shi 着色多重 zeta 值，建立非零性、和混洗关系、$t$-motivic 解释，并证明不同权重的值在 $\overline{k}$ 上线性无关。

多重值：$\mathfrak{M}_{0,n}$上的李代数和周期

Multizeta values: Lie algebras and periods on $\mathfrak{M}_{0,n}$

本文主要研究多重zeta值之间的代数和几何关系。在第二章中，我们证明了一个结果，它给出了双混洗李代数在深度1和深度2上的深度分次片的维数。在第三章和第四章中，我们研究了在$\mathfrak{M}_{0,n}$上表示为周期的多ζ值之间的几何关系。在这项研究中的关键成分是特殊的部分紧化的模空间与multizeta值的顶维de Rham上同调。在第三章中，我们给出了这种上同调的多边形基的显式表达式。在第四章中，我们将这种方法推广到其它部分紧化模空间的上同调基的显式刻画。本文最后给出了一个新的$Pic(\overline{\mathfrak{M}}_{0,n})$表示。