Mathematical Sciences: Topology

以一般均匀空间为典型纤维的Seifert结构的存在唯一性，由K.B. Lee和F. Raymond教授最近发展了一个可计算的阻塞理论。雷蒙德教授将完善这一理论并开发几何应用。特别是，几何结构的变形理论，保持纤维的结构将研究各种Seifert纤维和相应的模空间的拓扑结构。斯科特教授将进一步研究几个问题，这些问题集中在三个重要的猜想上，这些猜想与3-流形的拓扑结构及其基本群有关。考虑具有无限基群的闭取向不可约3流形。如果M和N是同伦等价的，它们是同胚的吗？如果基本群包含一个与整数同构的中心子群，M是一个经典的塞弗特3流形吗？全称覆盖是否同胚于欧几里德空间？这些拓扑学研究揭示了几何物体可以由更简单的几何物体组装起来的方式。代数或微分方程的解空间会产生这些相同的对象，这使得它们的结构特性变得普遍，理解它们很重要。

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.