Low-dimensional Manifolds and Knot Theory

提案编号：DMS-9971718标题：低维流形和纽结理论首席研究员：Cameron MCA。Gordon摘要：Gordon教授的项目重点是研究三维流形上的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优界的程序。这些界限中的许多现在都是已知的，剩下的唯一情况是(1)3-球面上(双曲)纽结的补集上的可约填充；(2)透镜空间中纽结上的环形填充；以及(3)其中一个填充是小Seifert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Dehn填充物的问题。将作出特别的努力来反驳布线猜想，该猜想断言不存在上述类型(1)的填充物。形成戈登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。这些物体像普通的三维空间一样是局部性的，但其全局结构可能相当复杂。由于我们生活在一个三维流形中，人们可能会说，三维拓扑学的目的是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，这样的描述最终可能是可能的。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，两年后将从卫星上进行某些观测，并在此基础上尝试从数学上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--纽结是在三维空间中以某种纠缠方式嵌入的闭合环--纽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德恩手术，这是一种将纽结周围的实心管从空间中移除并以不同方式缝合回来的过程，从而产生了新的三维流形。在过去的几年里，人们投入了大量的精力来研究这种建筑。尽管这项工作的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，循环手术定理，最近被用来确定某些酶在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.