Mathematical Sciences: Dehn Surgery and 3-Manifold Theory

吴计划继续他的德恩手术和3-流形理论的研究。其主要任务是寻求对产生可约或边界可约3-流形的“特殊”Dehn手术的一些满意的理解。对于边界可约性问题，他将专注于手柄上结点手术的1-桥猜想。对于可约性问题，他将从寻找更多例外Dehn手术的例子开始，因为这将有望导致对手术流形的可约性的一些内在(甚至可能是完整的)理解。此外，他还将研究三维流形中包含非分离约化球的纽结上外科手术的布线猜想。三维流形是非常基本的几何对象。它们是具有欧几里得三维空间的局部特征的空间，但可能全局复杂得多。二维的例子更容易描绘--许多表面不是平面，但看起来像是蚂蚁在上面爬行。由于我们生活在一个3维的流形中，忽略了时间维度，它们也具有宇宙学上的兴趣，而且有令人惊讶的数量是未知的。

Wu plans to continue his study on Dehn surgery and 3- manifold theory. The principal task is to search for some satisfactory understanding of those "exceptional" Dehn surgeries which yield reducible or boundary reducible 3-manifolds. For the boundary reducibility problems, he will concentrate on the 1- bridge conjecture for surgeries on knots in handlebodies. For the reducibility problems, he will start by searching for more examples of exceptional Dehn surgeries, as this will hopefully lead to some intrinsic (probably even a complete) understanding of the reducibility of surgered manifolds. Also, he will study the cabling conjecture for surgeries on knots in 3-manifolds which contain a non-separating reducing sphere. Three-dimensional manifolds are very basic objects of geometric interest. They are spaces which have the local character of Euclidean 3-space but may be globally much more complicated. Examples in two-dimensions are easier to picture -- many surfaces are not the plane but would appear to be to ants crawling on them. Since we live in a 3-dimensional manifold, ignoring the time dimension, they are of cosmological interest as well, and there is a surprising amount not known about them.