Mathematical Sciences: Low Dimensional Manifolds and Knot Theory

9626550戈登·卡梅伦戈登将继续调查Dehn手术。主要目标是进一步限制各种特殊（非双曲）手术对双曲结。在许多情况下，这种手术的数量和性质的已知界限现在相当尖锐，似乎最终可能（至少对瑟斯顿的几何化猜想取模）给出所有特殊手术的基本完整描述。所使用的方法将主要基于Gordon和John Luecke在他们之前关于Dehn手术的工作中开发的组合拓扑技术，例如结补猜想的证明。该项目涉及结理论和三维拓扑，后者的总体目标是了解三维流形的结构。这些物体在局部就像普通的三维欧几里得空间，但其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。三维拓扑学的一个重要方面是结理论——结是一个以某种方式嵌入普通三维空间的闭环。一方面，关于3-流形的结果通常作为特殊情况给出关于节的信息，另一方面，各种各样的数学方法可以应用于节的研究，从而导致关于3-流形的新信息。（最近，通过这种方式发现了三维拓扑和量子场论之间的深层联系。）将结理论与3-流形的一般理论联系起来的主要载体是Dehn手术，这是当前项目的重点。在Dehn手术中，将结周围的固体管移除并以不同的方式“重新缝合”，从而获得新的3歧管。如果一个人允许链接（即几个环连接在一起）和结，那么每个3-流形都可以用这种方式得到，因此对这种结构的充分理解将对3-流形的一般理论具有重要意义。＊＊＊

9626550 Gordon Cameron Gordon will continue to investigate Dehn surgery on knots. The main goal is further to circumscribe the various exceptional (non-hyperbolic) surgeries on hyperbolic knots. The known bounds on the number and nature of such surgeries are now quite sharp in many cases, and it seems that it may eventually be possible (at least modulo the Geometrization Conjecture of Thurston) to give an essentially complete description of all exceptional surgeries. The methods used will be based largely on the combinatorial- topological techniques developed by Gordon and John Luecke in their previous work on Dehn surgery, such as the proof of the Knot Complement Conjecture. This project deals with knot theory and 3-dimensional topology, the general goal of the latter being to understand the structure of 3-dimensional manifolds. These are objects that are locally like ordinary 3-dimensional Euclidean 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 important aspect of 3-dimensional topology is the theory of knots---a knot being a closed loop embedded somehow in ordinary 3-dimensional space. On the one hand, results about 3-manifolds often give information about knots as special cases, while on the other hand, a wide variety of mathematical methods can be applied to the study of knots, leading in turn to new information about 3-manifolds. (Recently, deep connections between 3-dimensional topology and quantum field theory were discovered in this way.) The main vehicle by which knot theory relates to the general theory of 3-manifolds is Dehn surgery, and this is the focus of the current project. In Dehn surgery, a solid tube around a knot is removed and "s ewn back" differently, giving a new 3-manifold. If one allows links (i.e., several loops linked together) as well as knots, then every 3-manifold can be obtained in this way, so a sufficiently good understanding of this construction would have important implications for the general theory of 3-manifolds. ***