2012 Fiscal Year Final Research Report

Formal KZ equation on moduli spaces and multiple polylogarithms

Research Project

Project/Area Number	22540035
Research Category	Grant-in-Aid for Scientific Research (C)
Allocation Type	Single-year Grants
Section	一般
Research Field	Algebra
Research Institution	Waseda University
Principal Investigator	UENO Kimio 早稲田大学, 理工学術院, 教授 (70160190)
Project Period (FY)	2010 – 2012
Keywords	KZ 方程式 / モジュライ空間 / 多重対数関数 / 多重ゼータ値 / リーマン : ヒルベルト問題
Research Abstract	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.

(4 results)