Fibred 3-manifolds and beyond

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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.