Fibred 3-manifolds and beyond

DMS-0312442 Rachel Roberts Roberts教授将研究有关3-流形拓扑的几个问题。特别是，她建议调查一系列问题所提出的她早期的工作（联合约翰Shareshian和梅兰妮斯坦）证明存在无限多双曲流形不包含Reebless叶状。她还提出了一项研究，特别是Reebless叶理和基本层压作为她提出的方法的一部分，以证明真理的财产P猜想的结，这说，没有反例庞加莱猜想可以产生Dehn手术结。 三维流形是由三维空间块按照一定的规则粘合在一起而形成的空间。在全局上，所获得的空间通常是相当复杂的。这些空间在物理和自然科学的许多背景下自然出现，并模拟了许多有趣的现象。作为一个例子，我们注意到最近的联合研究拓扑学家和宇宙学家提出了一个可能令人惊讶的模型，我们的空间宇宙。作为另一个例子，三维流形的研究在三维空间中弦的打结和连接的研究中很重要，这反过来又在DNA结构的研究中很重要。 三维流形研究的主要目标是发现所有这些空间的有用描述，并开发有用的工具来使用它们。在这项研究中，罗伯茨教授调查了与这一目标有关的问题。

DMS-0312442Rachel Roberts Professor Roberts will investigate several problems concerning the topology of 3-manifolds. In particular, she proposes investigating a series of questions suggested by her earlier work (joint with John Shareshian and Melanie Stein) demonstrating the existence of infinitely many hyperbolic manifolds containing no Reebless foliation. She also proposes a study of particular Reebless foliations and essential laminations as part of her proposed approach to proving the truth of the property P conjecture for knots, which says that no counterexample to the Poincar\'e Conjecture can be generated by Dehn surgery on knots. Three-manifolds are spaces formed by gluing together blocks of three-dimensional space according to certain prescribed rules. Globally, the spaces obtained are usually quite complex. These spaces arise naturally in many contexts in the physical and natural sciences and model many interesting phenomena. As an example of this, we note the recent joint research of topologists and cosmologists suggesting a perhaps surprising model for our spatial universe. As another example, the study of 3-manifolds is important in the study of the knotting and linking of strings in three-dimensional space, which in turn is important in the study of the structure of DNA. A primary goal of the study of 3-manifolds is to discover a useful description of all such spaces and to develop useful tools for working with them. In this research, Professor Roberts investigates questions related to this goal.