Obstructions to Dehn surgeries (Please note the date!)

Lickorish and Wallace proved that all closed orientable 3-manifolds can be obtained by Dehn surgery along a link in S^{^3}. Thurston observed that this proof gives rise to a connected graph that contains all closed orientable 3-manifolds, where every vertex of the graph corresponds to a 3-manifold and an edge corresponds to a Dehn surgery. From this perspective, one can organize all 3-manifolds and ask the question how close is a manifold M to S^{^3}. The most natural question is what manifolds are distance 1 from S^{^3}, i.e. Dehn surgery along a knot. Surprisingly, the answer is unknown. In my talk,  I will provide some background and discuss recent progress on this question.