Geometric Construction of Functional Equations and Bloch Elements

9704353杨这个项目是研究几何拓扑学和数论的交界点。更具体地说，研究了经典多对数的函数方程，并将其与Grassman多对数联系起来；研究了双曲3-流形与代数K-理论中Bloch群之间的密切关系。要解决的一个重要问题是用双曲流形实现Bloch群中的元素--这个问题的解决直接关系到K3刚性猜想。双曲3-流形是三维流形或一般曲线空间的重要组成部分。这些是非欧几里得平面几何的推广的负曲空间。许多双曲三维流形可以由某些离散对称群表示为标准的‘平坦’双曲三维空间的商，这一观察结果允许研究人员使用代数技巧来研究双曲流形。

9704353 Yang This project deals with research at the interface between geometric topology and number theory. More specifically, the investigator is to study the functional equations of the classical polylogarithms and to relate this to the grassman polylogarithms; and to investigate the close relationship between hyperbolic 3-manifolds and the Bloch group in algebraic K-theory. One important question to be addressed is the realization of elements in Bloch group by hyperbolic manifolds - a solution to this question has a direct bearing on the K3 rigidity conjecture. Hyperbolic 3-manifolds are important building blocks of three-dimensional manifolds or curved spaces in general. These are negatively curved spaces generalizing the non-Euclidean plane geometries. Many hyperbolic 3-manifolds can be given as quotients of the standard 'flat' hyperbolic three space by certain discrete symmetry groups, and this observation allows researchers to use algebraic techniques to study hyperbolic manifolds.