Using foliations to understand manifolds

One approach to understanding a complicated object involves breaking the object into simpler pieces that fit back together in constrained ways. In this talk, the complicated object is an n-manifold, and the simpler pieces are obtained from foliations, in various ways, including foliation charts, leaves of a foliation, branched surfaces, and sutured manifolds. Work of Thurston in the 1970s highlighted the power of codimension one. More recently, the Floer homology theories have led to new insights. After introducing the definition of foliation and describing some basic foliation results, I will specialize to the special case of codimension one foliations in 3-manifolds. Included in the discussion will be results of Gabai, Thurston, Eliashberg-Thurston, Bowden, Kazez-R, and Delman-R.