Mathematical Sciences: Presidential Young Investigator Award

这一总统青年研究员奖将加强对约翰·E·卢克在链接上的德恩手术的研究的支持，这是三维流形拓扑中的一个主题。三维流形M中的环是嵌入在M中的圆的集合。纽结是一个分量的环。今年的工作将有两个方面：(1)理解具有双曲补的纽结上的Dehn运算何时产生非双曲流形，(2)将关于纽结上的Dehn运算的结果推广到关于具有多个分量的环上的Dehn运算的结果。流形是自然的研究对象，是像欧几里德空间那样局部但一般不是全局的空间。三维情形是闭流形尚未完全分类的最低维。结和链接也是自然物体，令人惊讶的是，关于它们的一些问题仍然没有答案。

This Presidential Young Investigator Award will enhance support for John E. Luecke's research in Dehn Surgery on Links, a topic in the topology of three-dimensional manifolds. A link in a 3-dimensional manifold, M, is a collection of circles embedded in M. A knot is a link of one component. The work this year will have two aspects: (1) understanding when a non-hyperbolic manifold results from Dehn surgery on a knot with hyperbolic complement, (2) generalizing results about Dehn surgery on knots to results about Dehn surgery on links with more than one component. Manifolds are natural objects of study, being spaces which are locally but in general not globally like Euclidean space. The case of three dimensions is the lowest dimension for which the closed manifolds have not been fully classified. Knots and links are also natural objects, about which it is surprising how basic are some of the questions still unanswered.