Mathematical Sciences: Topology

存在唯一性的可计算障碍理论 一般齐性空间的Seifert构造 典型的纤维，最近由K.B.教授开发。李 和F.雷蒙德 雷蒙德教授将完善这一理论， 几何应用 特别是，变形理论的 保持纤维结构的几何结构将是 研究了各种塞弗特纤维和拓扑结构的 将研究相应的模空间。 斯科特教授将进一步研究几个问题 围绕着三个重要的概念， 3-流形及其基本群的拓扑。 考虑 具有无穷基本的闭定向不可约三维流形 组 如果M和N是同伦等价的，那么它们是否 同胚的？ 如果基本群包含一个 同构于整数的子群，M是经典的Seifert 三歧管？ 泛覆盖同胚于欧几里德吗 空间？ 这些拓扑研究揭示了 哪些几何物体可以从简单的几何物体 对象 代数或微分方程的解空间 产生了这些相同的物体，这使得它们的结构 特性普遍存在，了解它们很重要。

A computable obstruction theory for existence and uniqueness of Seifert constructions with general homogeneous spaces as typical fiber, has been recently developed by Professors K.B. Lee and F. Raymond. Professor Raymond will refine this theory and develop geometric applications. In particular, the deformation theory of geometric structures that preserve the fiber structure will be studied for various Seifert fiberings and the topology of the corresponding moduli spaces will be investigated. Professor Scott will investigate further several problems centered about three important conjectures that relate the topology of 3-manifolds with their fundamental groups. Consider closed oriented irreducible 3-manifolds with infinite fundamental groups. If M and N are homotopically equivalent, are they homeomorphic? If the fundamental group contains a central subgroup isomorphic to the integers, is M a classical Seifert 3-manifold? Is the universal covering homeomorphic to Euclidean space? These topological investigations shed light on the ways in which geometric objects can be assembled from simpler geometric objects. Solution spaces to algebraic or differential equations give rise to these same objects, which makes their structural properties pervasive and understanding them important.