Mathematical Sciences: Dehn Surgery and 3-Manifold Theory

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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.