Scharlemann will apply new 3-manifold techniques arising from the theory of sutured manifolds and graph-theoretic combinatorics to a variety of open problems in knot and link theory, including the additivity of unknotting number and thecabling conjecture. His students will work on related problems. The topological classification of surfaces is a classical matter, depending basically on how many holes they have. The corresponding problem in one higher dimension is vastly more complicated and an active subject of current research by numerous geometric topologists, including Scharlemann and his students.
