Mathematical Sciences: The Topology of Varieties

Hain打算利用Hodge理论的技术研究紧黎曼曲面空间的拓扑结构和n × m矩阵在复数上的拓扑结构，这些复数的子矩阵都是非零的。可能的应用包括构造欧拉二对数的推广，它满足阿贝尔五项泛函方程对二对数的自然推广。研究生将使用Hain编写的计算机程序来帮助研究二重算术。该程序分解对称群的表示，这些表示产生为巨大的张量积。一台高性能的个人计算机将专门用于这项任务。

Hain intends to use the techniques of Hodge theory to study the topology of the space of compact Riemann surfaces and the topology of the space of n x m matrices over the complex numbers, all of whose minors are non-zero. Possible applications include the construction of generalizations of Euler's dilogarithm which satisfy natural generalizations of Abel's 5-term functional equation for the dilogarithm. A graduate student will employ a computer program Hain has written to aid in investigating the dilogarithm. The program decomposes representations of symmetric groups, which arise as huge tensor products. A high-powered personal computer will be devoted to this task.