Creating Periodic Orbits of Reeb Vector Fields in Three Dimensions

We will give an introduction to Reeb vector fields on odd-dimensional manifolds; these are a special case of more general vector fields whose dynamics will be studied in the Special Year. Much more is known about Reeb vector fields in three dimensions than in higher dimensions, thanks to Seiberg-Witten theory. In particular, a remarkable "closing lemma" due to Irie shows that in three dimensions one can create a periodic orbit of a Reeb vector field through a given region by a C-infinity small perturbation. We will discuss a quantitative refinement of this result, asserting roughly that one can create a periodic orbit of period at most L by a perturbation of size on the order of 1/L.