Mathematical Sciences: Representations of Braid and Mapping Class Groups

9503069 Hain Hain打算用Hodge理论继续研究曲线模空间的拓扑结构。特别是他对计算Torelli群的第二同态，以及了解Torelli群的子群，即Johnson同态的核非常感兴趣。与杨军一起，他打算尝试用代表第四Borel调节器的第四经典多对数在GL_4(C)上构造一个局部可积的循环。最后，他打算将瓦西里耶夫不变量理论中的某些构造从0格推广到正格。对称是20世纪数学的一个重要的中心主题。最自然的对称集合之一是有g个洞的甜甜圈表面的对称集合。尽管这类对称群已经被研究了近100年，但仍有许多核心问题尚未解决。海恩打算在未来三年内完成其中几项工作。他的研究结果对现代粒子物理学、数论和代数几何的某些分支以及多项式方程解的研究具有潜在的应用价值。海恩还有第二个大项目。这是对经典多对数的研究。这些函数是200年前由英国数学家斯宾塞（Spence）定义的，是对著名的对数函数的推广，对数函数在科学和经济学中非常重要。最近，多对数开始出现在物理学和数论中。确定它们的基本性质是很重要的。＊＊＊

9503069 Hain Hain intends to continue his study of the topology of moduli spaces of curves using Hodge theory. In particular, he is very interested in computing the second homology of the Torelli group, and understanding the subgroup of the Torelli group, which is the kernel of the Johnson homomorphism. With Jun Yang, he intends to try to construct a locally integrable cocycle on GL_4(C), using the fourth classical polylogarithm, which represents the fourth Borel regulator. Finally, he intends to extend certain constructions in the theory of Vassiliev invariants from genus 0 to positive genus. Symmetry is an important and central theme in twentieth century mathematics. One of the most natural sets of symmetries is the set of symmeteries of the surface of a donut with g holes. Although such groups of symmeteries have been studied for almost 100 years, there are a large number of central unresolved problems. Hain intends to work on several of these during the next three years. His results have potential applications to modern particle physics and to certain branches of number theory and algebraic geometry, the study of solutions of polynomial equations. Hain has a second major project. This is the study of classical polylogarithms. These functions were defined 200 years ago by Spence, an English mathematician, and are generalizations of the well-known logarithm function so important throughout the sciences and economics. Polylogarithms have started to appear recently in physics and number theory. It is important to establish their basic properties. ***