An overarching goal in the study of 3-manifolds, post-Perelman, is to put them in some satisfactory order. One model for an order can be found in 2-dimensions; surfaces, orientable or not, with or without boundary, come in a neat package, which is well-understood. Thanks to Thurston and Perelman one now knows inexpressibly more than before about the geometry of all 3-manifolds, yet one still seeks an elegant framework in which to place them all. This quest leads to some long-standing problems, as well as introducing some new ones. This proposal describes problems in this general area which can be considered long-standing, although from a new point of view. For example, the PI will build on the fundamental work on the stabilization problem for Heegaard splittings which was completed during the previous grant, working with J. Hass and W. Thurston, and proposes to complete the second half of this conjecture. The PI introduces a new knot invariant, related to earlier joint research with Scharlemann, and suggests methods to answer some very basic questions regarding it. In addition the PI includes a problem closely related to the long-standing slice-ribbon conjecture. The PI also discuss her extensive outreach activities, which are intimately entwined with her research.
A 3-dimensional manifold is an object that looks, to a local observer, like 3-dimensional Euclidean space. That is, it looks like the room in which you are likely to be sitting. But just as several thousand years ago mankind knew only the local, not the global, shape of the earth, at present we know only about the local, not the global, shape of the 3-dimensional universe in which we live. So 3-dimensional spaces are in many respects important objects of study. Tremendous progress in this field has been made in recent years, and there is now a hope of understanding the full sweep of 3-dimensional spaces. The problems in this proposal address some of the longstanding structural questions in the field, with the aim of uncovering at least part of what is sure to be, when it is ultimately discovered, the elegant framework which underlies it.
