Mathematical Sciences: Geometric Topology

比尔曼预计，算法的发展将从L和L的两个定向链条开始，分别由闭合的n和r-辫子表示(或者由分别具有n和r个塞弗特圆的任意链接图表示)，并将决定这些链条是否为同位素。对于L是r分量解链的标准代表的特殊情况，她预计算法将是多项式时间算法。摩根计划研究双曲n流形的族的紧致性问题，更广泛地研究紧致族的问题。他打算使用Donaldson提出的涉及规范理论的新概念来研究代数曲面的C-无穷分类。霍奇森的研究主要涉及使用几何方法研究三维流形。他的主要项目将涉及瑟斯顿引入的双曲德恩手术空间的研究。通过使用包括霍奇理论和几何极限分析在内的技术，他希望在这些Dehn手术空间的局部和全局结构上获得新的结果。这将导致在某些三维流形上构造几何结构的新方法，例如圆上的曲面丛。他还打算继续研究双曲结构变形时几何的变化，特别是可能发生的退化类型：包括退化到更低的维度极限和沿不可压缩表面分裂。这将包括一些涉及在3-流形上构造几何结构的计算研究。他还将研究最近由Casson和Floer定义的3-流形的新不变量。

Birman anticipates the development of an algorithm which will begin with two oriented links L and L', represented by closed n and r-braids respectively (or alternatively by arbitrary link diagrams with n and r Seifert circles respectively), and which will decide whether these links are isotopic. For the special case where L' is the standard representative of the r- component unlink, she expects that the algorithm will be a polynomial-time algorithm. Morgan plans to study compactness questions for families of hyperbolic n-manifolds, and more generally the problem of compactifying such families. He intends to study the C-infinity classification of algebraic surfaces using new ideas introduced by Donaldson involving gauge theory. Hodgson's research deals mostly with the investigation of 3- dimensional manifolds using geometric methods. His main projects will involve the study of the hyperbolic Dehn surgery spaces introduced by Thurston. By using techniques including Hodge theory and analysis of geometric limits he hopes to obtain new results on the local and global structure of these Dehn surgery spaces. This would lead to new methods for constructing geometric structures on certain 3-manifolds, for example surface bundles over the circle. He also intends to continue his investigations into the variation of geometry as hyperbolic structures are deformed, especially the kinds of degeneration that can occur: including degeneration to lower dimensional limits and splitting along incompressible surfaces. This will include some computational studies involving the construction of geometric structures on 3-manifolds. He will also study the new invariants of 3-manifolds defined recently by Casson and Floer.