Low-dimensional Manifolds and Knot Theory

项目编号：dms -9971718题目：低维流形和结理论摘要：Gordon教授的项目重点是研究3-流形上的Dehn填充，特别是双曲流形上各种非双曲填充之间距离的最优界的求解程序。这些边界中的许多现在是已知的，唯一剩下的情况是(1)在3球（双曲）结的补上的可约填充；(2)透镜空间结上的环面填充；(3)其中一种填充物为小塞菲特纤维空间的情况。戈登教授将继续研究这些问题，以及与Dehn填充物有关的其他问题。一个特别的努力将被用来攻击布线猜想，它断言不存在上述类型(1)的填充。构成Gordon教授项目背景的数学领域是三维拓扑学，其总体目标是理解三维流形的结构。这些物体在局部就像普通的三维空间，但它们的整体结构可能相当复杂。由于我们生活在三维流形中，有人可能会说三维拓扑旨在描述我们空间宇宙的数学可能性。这一目标仍未实现，尽管有诱人的证据表明，这样的描述最终可能成为可能。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，在两年的时间里，将通过卫星进行某些观测，在此基础上，将尝试用数学方法确定宇宙的几何结构。三维拓扑的一个重要方面是结理论——结是一个以某种纠缠方式嵌入三维空间的闭环——而结理论与三维流形的一般理论相关联的主要方式之一是通过Dehn手术，在这个过程中，围绕着结的固体管从空间中移除，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种结构。虽然这项工作的推动力是希望进一步了解三维拓扑结构，但有趣的是，关于Dehn手术的一个定理，即循环手术定理，最近被用于确定某些酶在DNA链上的拓扑作用。

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.