Mathematical Sciences: Representations of Three-Manifold Groups

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Andrew Casson will continue to study the geometrization problem for 3-manifolds. The recent solution of the Seifert Fiber Space Conjecture has reduced this to the case of irreducible 3- manifolds whose fundamental groups contain no free Abelian subgroups of rank two. Casson will concentrate on such manifolds which have infinite fundamental group; according to the "Geometrization Conjecture," these manifolds should all admit hyperbolic structures. He will seek simplified proofs of Thurston's now classical results establishing this conjecture for manifolds with non-trivial boundary, in the hope of developing methods which apply to the still unsolved case of manifolds without boundary. Robion Kirby will continue to search for topological interpretations of Witten's 3-manifold invariants for Sl(2,Z), and particularly for the case of finite coverings. It is a surprising fact that although we live in a three dimensional space, a so-called 3-manifold, and so are blessed with a natural intuition about such geometric objects, in the end this does not carry us as far as we might have expected, for questions which have been settled by algebraic calculations for higher dimensional manifolds still remain baffling in the 3-dimensional case. The most famous of these is the celebrated conjecture of Poincare from around the turn of the century concerning 3- dimensional spheres, where precisely the original 3-dimensional case is the only one still open. The investigators are pursuing a variety of questions about 3-dimensional manifolds with slightly strange notions of distance on them, so-called hyperbolic metrics, but time and time again these questions have been shown to have clear relevance to the case of manifolds with a more familiar notion of distance.