Curves and 3-Manifolds

曲线和 3 流形

• 批准号: 0306599
• 负责人: Abigail Thompson
• 金额: $ 11.75万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306599&HistoricalAwards=false
• 关键词: Curves; Manifolds

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.

我们描述了三个问题，在这个建议中，所有在一般领域的纽结理论和3-流形。 三维流形的Heegaard分裂是将三维流形分解成简单的块，称为单块体。 这种分解是研究许多关于三维流形的有趣的开放问题的有效方法，但是在我们关于Heegaard分裂的基本信息中存在一些惊人的空白。 我们知道，给定同一个三维流形的两个Heegaard分裂，一个可以通过一系列移动从一个移动到另一个。在这个建议中的第一个问题是问如何“远离”两个不同的Heegaard分裂的3-流形可以在这些举动。 第二个问题是探索邓伍迪最近的尝试所提出的庞加莱猜想的一个特定方面，并由弗里德曼和丘的早期工作提出。 我们将仔细研究一个类似于邓伍迪所提出的方案是否可以在真实的3球上工作，从一个更简单的关于三角形2球的问题开始。 最后一个问题是一个推广最近的工作的提议者沉浸曲线在平面上沉浸2球在3空间。 我们将研究奇点的集合（例如，包括自相交曲线和零曲率点），并试图推导奇点的数量和类型之间的关系，类似于平面或射影2-空间中浸入曲线的已知结果。 我们还描述了如何使用这些和其他相关的问题已经被提议者，以激发兴趣，在该领域和数学一般的水平，从高中生，包括本科生和研究生，通过postdocs.Low-dimensional拓扑是研究的性质的空间在二维，三维和四维。 这是一个曾经完全属于“纯”数学领域的领域，该领域的研究主要是为了它的复杂性和美丽。 随着我们不断了解宇宙，从空间本身的形状到DNA链的打结，这个抽象领域与真实的世界之间的深层联系越来越明显。该计划旨在探索三维空间和纽结理论中的一些基本问题，包括理解三维空间的不同分解之间的关系，以及量化沉浸在标准三维宇宙中的二维球体的复杂性。