Ruth and Irving Adler Expository Lecture in Mathematics

Birkhoff sections have been invented... by Poincaré in his work on celestial mechanics. Birkhoff made an extensive use of this concept in dynamical systems. Sometimes, one can find surfaces transverse to the trajectories of a vector field in a 3-manifold. If this happens, the dynamics of the vector field is somehow reduced to the dynamics of the first return map on the surface. This technique is especially useful to describe the behavior of geodesics on surfaces. In this expository talk, I would like to describe old and new examples and discuss some open problems related to the existence of these sections.