A codimension-1 foliation F of a 3-manifold M is called taut if there exists a closed curve in M that intersects each leaf of F transversely.Existence of taut foliations imply useful properties for a 3-manifold.
I will give a motivation for the study of taut foliations of 3-manifolds and then focus on taut foliations of punctured surface bundles. In particular, I shall give an outline of the proof of the result that Dehn-filling the boundary along slopes sufficiently close to the slope of the fiber of the surface-bundle produces closed manifolds with taut-foliations. This is joint work with Rachel Roberts.
I shall give all pre-requisites specific to the theory of 3-manifolds