Mathematical Sciences: Representations of Braid and Mapping Class Groups

点击翻译按钮获取中文摘要

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. ***