Formal KZ equation on moduli spaces and multiple polylogarithms

一篇题为《模空间M_(0，5)上的KZ方程与多重多对数的调和乘积》的论文发表在Proc.伦敦。数学课。SOC..。本文建立了模空间M_(0，5)的简杆代数等代数基础和模空间M_(0，5)的纤维空间结构等几何基础，证明了M_(0.5)上KZ方程基本解的分解定理。此外，我们还证明了分解定理等价于广义调和乘积关系，并且它们包含了一元多重对数的所有调和乘积。题为《多对数求逆公式与黎曼-希尔伯特问题》的论文也于去年发表。本文利用Riemann-Hilbert问题，在一定的渐近条件下，成功地刻画了多对数。

The paper entitled by ”KZ equation on the moduli space M_{0,5} and the harmonic product of multiple polylogarithms” appeared in Proc. London. Math. Soc.. In this article, we established the algebraic foundation such as the reduced bar algebra, and the geometric foundation such as the fiber space structure of the moduli space M_{0,5}, and showed the decomposition theorem of the fundamental solution of the KZ equation on M_{0.5}. Moreover we showed that the decomposition theorem is equivalent to the generalized harmonic product relations, and that they involve all the harmonic products of multiple polylogarithms of one variable. The paper entitled by ‘’The Inversion Formula of Polylogarithms abd the Riemann-Hilbert Problem” also was published last year. In this paper, we succeeded in characterzing polylogarithms by using the Riemann-Hilbert problem under a certain asymptotic condition.