Mathematical Sciences: Representations of Three-Manifold Groups

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Andrew Casson will study aspects of the geometrization problem for 3-manifolds. He will attempt to improve the "Strong torus theorem" to conclude that irreducible 3-manifolds which contain essential singular tori either contain essential embedded surfaces of genus one or are Seifert fibered spaces. His method is related to one used by Tukia. He will also attempt to find good usable conditions on a 3-manifold guaranteeing the existence of representations of the fundamental group in PSL2(C). In particular, he will seek methods of finding hyperbolic or other geometrically interesting representations. Robion Kirby will continue studying Witten's program for defining an invariant of framed 3-manifolds via an "average" of the Chern-Simons invariants for SU(n) valued connections. He will also work on the diffeomorphism classification of smooth 4-manifolds obtained by "Gluck twists" on imbedded 2-spheres, or a generalization to immersed 2-spheres with one double point. Yakov Eliashberg will study invariants of Legendrian knots. All three projects concern different aspects of 3-dimensional manifolds, which are natural geometric objects describable locally by 3 independent coordinates. The structure of 3-manifolds has relevance to other parts of mathematics inasmuch as such manifolds occur as solution sets of ordinary or differential equations. Relevance to physical theories is also evident.