Mathematical Sciences: Representations of Braid and Mapping Class Groups

数学科学：辫子和映射类群的表示

• 批准号: 9503069
• 负责人: Richard Hain
• 金额: $ 8.09万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-08-01 至 1998-10-01
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9503069&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Representations; Braid; Mapping

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. ***

9503069 Hain Hain打算继续利用霍奇理论研究曲线的模空间的拓扑。特别是，他对计算Torelli群的第二同调以及理解Torelli群的子群非常感兴趣，Torelli群是Johnson同态的核心。与杨军一起，利用经典的第四重对数构造了GL_4(C)上的一个局部可积上循环，它代表了第四个Borel调节子。最后，他打算将Vassiliev不变量理论中的某些构造从亏格0推广到正亏格。对称性是二十世纪数学的一个重要和中心主题。最自然的对称集合之一是具有g个孔的甜甜圈表面的对称集合。虽然这类对称群的研究已有近100年的历史，但仍有大量的中心问题尚未解决。海恩打算在未来三年内致力于其中的几个项目。他的结果对现代粒子物理学、数论和代数几何的某些分支、多项式方程的解的研究都有潜在的应用。海恩还有第二个重大项目。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定义的，它们是众所周知的对数函数的推广，在科学和经济学中非常重要。多对数最近开始出现在物理学和数论中。重要的是要确定它们的基本性质。***