Mathematical Sciences: Representations of Braid and Mapping Class Groups

小行星9503069海恩 海恩打算继续他的研究拓扑模空间的曲线使用霍奇理论。 特别是，他非常感兴趣的计算第二同源的Torelli组，并了解子群的Torelli组，这是内核的约翰逊同态。 与杨俊一起，他试图利用第四经典多重对数，即第四Borel调节子，构造GL_4（C）上的一个局部可积上圈。 最后，他打算将Vassiliev不变量理论中的某些构造从亏格0扩展到正亏格。 对称性是二十世纪世纪数学的一个重要的中心主题。 最自然的对称性集合之一是具有g孔的甜甜圈表面的对称性集合。 虽然这类对称群已经被研究了近100年，但仍有大量的核心问题尚未解决。 海恩打算在未来三年内研究其中的几个项目。 他的研究结果有潜在的应用，现代粒子物理学和某些分支的数论和代数几何，研究解决方案的多项式方程。 海恩还有第二个主要项目。 这是对经典多面体的研究。 这些函数是200年前由英国数学家斯宾塞定义的，并且是在科学和经济学中非常重要的著名对数函数的推广。 多面体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属性。 ***

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. ***