Mathematical Sciences: "Knot Theory and 3-Manifolds"

这项研究主要集中在纽结理论和三维流形领域。汤普森将要研究的问题之一是Property-P猜想。称3-球面上的纽结具有性质P，如果没有对该纽结的非平凡运算产生同伦的3-球面。长期以来，人们一直猜想3-球面中的所有非平凡纽结都具有性质，P·Gordon和Luecke最近证明了对非平凡纽结进行非平凡运算不能得到3-球面本身。人们仍然有可能在三维球上做一个结的手术，并得到一个与庞加莱猜想相反的例子。这个问题的解决方法涉及到Gabai关于缝合流形的工作的推广，以及考虑是否可以对伪3-球面上的一个纽结进行手术并获得S3的等价问题。此外，最近与Scharlemann的工作提出了关于3-流形中嵌入有限图的几个有趣的问题。Thompson将尝试将他们刻画S3中无结平面图的工作推广到3-球面中的任意有限图。这些都是关于三流形的基本问题。它们是开放的这一事实表明，我们对三维流形仍然知之甚少。尽管我们生活在其中，但事实是这样的，因此这样的拓扑问题甚至可能具有宇宙学意义。

The research is in the areas of knot theory and 3- dimensional manifolds. One of the problems Thompson will work on is the property-P conjecture. A knot in the 3-sphere is said to have property P if no non-trivial surgery on the knot yields a homotopy 3-sphere. It has long been conjectured that all non- trivial knots in the 3-sphere have property P. Gordon and Luecke have recently shown that non-trivial surgery on a non-trivial knot cannot yield the 3-sphere itself. There remains the possibility that one could do surgery on a knot in the 3-sphere and obtain a counter-example to the Poincare conjecture. The suggested approach to this problem involves a generalization of work of Gabai on sutured manifolds, as well as considering the equivalent problem of whether one can do surgery on a knot in a fake 3-sphere and obtain S3. In addition, recent work with Scharlemann has suggested several interesting questions about imbedded finite graphs in 3-manifolds. Thompson will try to generalize their work characterizing unknotted planar graphs in S3 to arbitrary finite graphs in the 3-sphere. These are fundamental questions about three-manifolds. The fact that they are open points up how much we still do not know about three-dimensional manifolds. This is so despite the fact that we live in one, and so such topological questions might even have cosmological significance.