Heegaard Splittings, Knots and 3-Manifolds

后佩雷尔曼时代，研究3流形的一个首要目标，就是把它们排列成某种令人满意的顺序。在二维空间中可以找到一个阶的模型；表面，可定向或不可定向，有或没有边界，都是一个整洁的包，这是很容易理解的。由于瑟斯顿和佩雷尔曼，我们现在对所有3-流形的几何知识比以前多了很多，但我们仍然在寻找一个优雅的框架来放置它们。这种探索导致了一些长期存在的问题，同时也引入了一些新的问题。本建议描述了这一普遍领域的问题，这些问题可以被认为是长期存在的，尽管是从一个新的角度来看。例如，PI将与J. Hass和W. Thurston合作，在之前的资助中完成Heegaard分裂的稳定问题的基础工作，并建议完成这一猜想的后半部分。PI引入了一个新的结不变量，与早先与Scharlemann的联合研究有关，并提出了一些方法来回答有关它的一些非常基本的问题。此外，PI还包括一个与长期存在的条状猜想密切相关的问题。PI还讨论了她广泛的外展活动，这些活动与她的研究密切相关。三维流形是一个物体，对于局部观察者来说，看起来像三维欧几里得空间。也就是说，它看起来像你可能坐的房间。但是，正如几千年前人类只知道地球的局部形状，而不知道地球的全局形状一样，目前我们只知道我们所生活的三维宇宙的局部形状，而不知道地球的全局形状。所以三维空间在很多方面都是重要的研究对象。近年来，这一领域取得了巨大的进展，现在有希望了解三维空间的全面扫描。本提案中的问题解决了该领域一些长期存在的结构性问题，其目的是揭示至少一部分，当它最终被发现时，它背后的优雅框架肯定是什么。

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.