Curves and 3-Manifolds

我们在这个提议中描述了三个问题，都在一般的结理论和3流形领域。3-manifold的heegard拆分是将3-manifold分解成简单的部分，称为handlebody。这种分解是研究许多关于3流形的有趣的开放性问题的有效方法，但是在我们关于Heegaard分裂的基本信息中有一些令人吃惊的空白。我们知道给定同一个3-流形的两个heegard分裂，一个可以通过一系列的移动从一个移动到另一个。这个提议中的第一个问题是，在这些移动中，3歧管的两个不同的heegard分裂可以“相距多远”。第二个问题是探索庞加莱猜想的一个特殊方面，这个方面是由邓伍迪最近的尝试和弗里德曼和丘的早期工作提出的。我们将仔细研究类似于邓伍迪提出的计划是否可以在真正的三球上工作，从一个更简单的三角二球问题开始。最后一个问题是将本文作者最近关于平面中浸入曲线的工作推广到三维空间中的浸入2球。我们将研究奇点的集合（包括，例如，自交曲线和零曲率点），并尝试推导奇点的数量和类型之间的关系，类似于平面或投影二维空间中浸入曲线的已知结果。我们还描述了这些和其他相关的问题是如何被提议者用来激发对该领域和数学的兴趣的，从高中学生，包括本科生和研究生，到博士后。低维拓扑研究的是二维、三维和四维空间的性质。这是一个曾经完全属于“纯粹”数学领域的领域，在这个领域的研究主要是为了它的复杂性和美丽。随着我们继续了解宇宙，从空间本身的形状到DNA链的结，这个抽象领域与现实世界之间的深刻联系越来越明显。本提案旨在探索三维空间和结理论中的一些基本问题，包括理解三维空间不同分解之间的关系以及在标准三维宇宙中浸没的二维球体的复杂性。

We describe three problems in this proposal, all in the general area of knot theory and 3-manifolds. A Heegaard splitting of a 3-manifold is a decomposition of the 3-manifold into simple pieces, called handlebodies. This decomposition is an effective way to study many of the interesting open questions about 3-manifolds, but there are some startling gaps in our basic information about Heegaard splittings. We know that given two Heegaard splittings of the same 3-manifold, one can move from one to the other by a series of moves. The first problem in this proposal asks how "far apart" two different Heegaard splittings of a 3-manifold can be under these moves. The second problem is to explore a particular aspect of the Poincare Conjecture prompted by Dunwoody's recent attempt and suggested by earlier work of Freedman and Yau. We will look carefully at whether a plan analogous to what Dunwoody proposed can be made to work on a real 3-ball, beginning with an even simpler version of the question on a triangulated 2-sphere. The final problem is a generalization of recent work of the proposer on immersed curves in the plane to immersed 2-spheres in 3-space. We will look at the set of singularities (including, for example, curves of self-intersection and points of zero curvature) and try to derive relationship between numbers and types of singularities, similar to the kinds of results known for immersed curves in the plane or in projective 2-space. We also describe how these and other related problems have been used by the proposer to stimulate interest in the field and in mathematics in general at levels ranging from high school students, including undergraduates and graduate students, through postdocs.Low-dimensional topology is the study of properties of spaces in dimensions two, three, and four. It is a field that used to lie squarely in the realm of "pure" mathematics, and research in the field was pursued largely for its intricacy and beauty. As we continue to understand the universe, from the shape of space itself to the knotting of strands of DNA, the deep connections between this abstract area and the real world are increasingly apparent. This proposal aims to explore some of the fundamental questions in 3-dimensional spaces and knot theory, including understanding the relationships between different decompositions of 3-dimensional spaces and quantifying the complexity of 2-dimensional spheres immersed in a standard 3-dimensional universe.