Mathematical Sciences: Representations of Three-Manifold Groups

Andrew Casson将继续研究3流形的几何化问题。Seifert纤维空间猜想的最新解将其简化为不可约的3-流形的情况，其基本群不包含秩2的自由阿贝尔子群。卡森将集中研究具有无限基本群的流形；根据“几何化猜想”，这些流形都应该承认双曲结构。他将寻求瑟斯顿在非平凡边界流形中建立这一猜想的经典结果的简化证明，希望开发适用于无边界流形的尚未解决的情况的方法。robon Kirby将继续为Sl（2，Z）寻找Witten的3流形不变量的拓扑解释，特别是对于有限覆盖的情况。令人惊讶的是，尽管我们生活在一个三维空间，一个所谓的三维流形中，因此我们对这些几何物体有一种天生的直觉，但最终这并没有把我们带得像我们所期望的那样远，因为那些在高维流形中通过代数计算已经解决的问题，在三维情况下仍然令人困惑。其中最著名的是庞加莱在世纪之交提出的关于三维球体的著名猜想，正是在这个猜想中，最初的三维情况是唯一仍然开放的。研究人员正在研究各种关于三维流形的问题，这些流形上有一些奇怪的距离概念，即所谓的双曲度量，但这些问题一次又一次地被证明与具有更熟悉的距离概念的流形的情况有明确的关联。

Andrew Casson will continue to study the geometrization problem for 3-manifolds. The recent solution of the Seifert Fiber Space Conjecture has reduced this to the case of irreducible 3- manifolds whose fundamental groups contain no free Abelian subgroups of rank two. Casson will concentrate on such manifolds which have infinite fundamental group; according to the "Geometrization Conjecture," these manifolds should all admit hyperbolic structures. He will seek simplified proofs of Thurston's now classical results establishing this conjecture for manifolds with non-trivial boundary, in the hope of developing methods which apply to the still unsolved case of manifolds without boundary. Robion Kirby will continue to search for topological interpretations of Witten's 3-manifold invariants for Sl(2,Z), and particularly for the case of finite coverings. It is a surprising fact that although we live in a three dimensional space, a so-called 3-manifold, and so are blessed with a natural intuition about such geometric objects, in the end this does not carry us as far as we might have expected, for questions which have been settled by algebraic calculations for higher dimensional manifolds still remain baffling in the 3-dimensional case. The most famous of these is the celebrated conjecture of Poincare from around the turn of the century concerning 3- dimensional spheres, where precisely the original 3-dimensional case is the only one still open. The investigators are pursuing a variety of questions about 3-dimensional manifolds with slightly strange notions of distance on them, so-called hyperbolic metrics, but time and time again these questions have been shown to have clear relevance to the case of manifolds with a more familiar notion of distance.