4/7/2004
Tom Mrowka
MIT and the Institute for Advanced Study
From foliations of three manifolds to representations of their fundamental groups
Abstract
Recent progress in understanding symplectic four-manifolds (due to Eliashberg building on a long line of work in particular that of Gabai , Thurston-Eliashberg and Giroux) combined with results in Gauge theory (in particular results of Feehan and Leness relating the Donaldson and Seiberg-Witten invariants) has led to the resolution of some long-standing questions about three manifolds. For example one can now show (independently of Perelman's work) that non trivial surgery on a non-trivial knot can never yield a counter example to the Poincar'e conjecture (the property P question). In general one can show taut foliations of three manifolds imply the existence non-trivial representations of their fundamental groups in to SU(2).