Low-dimensional Manifolds and Knot Theory

低维流形和纽结理论

• 批准号: 9971718
• 负责人: Cameron Gordon
• 金额: $ 18.29万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-15 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971718&HistoricalAwards=false
• 关键词: Low; dimensional; Manifolds; Knot; Theory

Proposal Number: DMS-9971718Title: Low-dimensional manifolds and knot theoryPrincipal Investigator: Cameron McA. Gordon Abstract: The focus of Professor Gordon's project is the study of Dehn filling on 3-manifolds, and in particular the program to obtain the optimal bounds on the distances between the various kinds of non-hyperbolic fillings on hyperbolic manifolds. Many of these bounds are now known, the only remaining cases being (1) reducible fillings on complements of (hyperbolic) knots in the 3-sphere; (2) toroidal fillings on knots in lens spaces; and (3) the case where one of the fillings is a small Seifert fiber space. Professor Gordon will continue to investigate these questions, as well as other problems concerning Dehn fillings. A special effort will be made to attack the Cabling Conjecture, which asserts that no fillings of type (1) above exist. The area of mathematics which forms the context of Professor Gordon's project is 3-dimensional topology, the general goal of which is to understand the structure of 3-manifolds. These are objects that are locally like ordinary 3-dimensional space, but whose global structure may be quite complicated. Since we live in a 3-manifold, one might say that 3-dimensional topology aims to describe what the mathematical possibilities are for our spatial universe. This aim is still not realized, although there is tantalizing evidence that such a description might ultimately be possible. One very interesting application of the mathematical theory of 3-dimensional manifolds is the "circles in the sky" project, in which, in two years' time, certain observations will be made from a satellite, on the basis of which an attempt will be made to mathematically determine the geometric structure of the universe. An important aspect of 3-dimensional topology is the theory of knots - a knot being a closed loop embedded in some tangled fashion in 3-dimensional space - and one of the main ways in which knot theory relates to the general theory of 3-manifolds is through Dehn surgery, a process in which a solid tube around a knot is removed from space and sewn back in differently, giving a new 3-manifold. Much effort has been devoted over the last few years to investigating this construction. Although the impetus for this effort is the desire to further our understanding of 3-dimensional topology, it is interesting to note that a theorem about Dehn surgery, the Cyclic Surgery Theorem, has recently been used to determine the topological action of certain enzymes on strands of DNA.

提案编号：DMS-9971718标题：低维流形和纽结理论主要研究者：卡梅隆麦卡。戈登摘要：戈登教授的项目的重点是研究3-流形上的Dehn填充，特别是获得双曲流形上各种非双曲填充之间距离的最佳界限的程序。 许多这样的界限现在都是已知的， 仅有的剩余情况是（1）在3-球面中的（双曲）纽结的补上的可约填充;（2）在透镜空间中的纽结上的环形填充;以及（3）其中一个填充是小的塞弗特纤维空间的情况。 戈登教授将继续调查这些问题，以及其他有关德恩填料的问题。 我们将特别努力攻击布线猜想，它断言不存在上述类型（1）的填充。 该领域的数学形成的背景下，戈登教授的项目是三维拓扑，其总体目标是了解结构的3流形。这些物体在局部类似于普通的三维空间，但其整体结构可能相当复杂。 由于我们生活在一个三维流形中，人们可能会说三维拓扑旨在描述我们的空间宇宙的数学可能性。 这一目标仍然没有实现，尽管有诱人的证据表明这种描述最终是可能的。 三维流形数学理论的一个非常有趣的应用是“天空中的圆圈”项目，在该项目中，将在两年内从一颗卫星上进行某些观测，并在此基础上试图用数学方法确定宇宙的几何结构。 三维拓扑学的一个重要方面是纽结理论--纽结是以某种缠结的方式嵌入三维空间中的闭环--纽结理论与三维流形的一般理论联系的主要方式之一是通过德恩手术，在这个过程中，围绕纽结的实心管从空间中移除，并以不同的方式缝回，得到一个新的三维流形。 在过去的几年里，人们投入了大量的精力来研究这一结构。 虽然这一努力的动力是希望进一步了解我们的三维拓扑结构，有趣的是，注意到一个定理德恩手术，循环手术定理，最近已被用来确定某些酶的拓扑作用链的DNA。