Invariant Sets and the Topology of ECH Curves

The ECH "U-map" produces many pseudo-holomorphic curves, but in applications it is sometimes important to have some control on the topology of these curves.  I want to explain how we know that, at least in the case of rational homology spheres, "most" of these curves have negative Euler characteristic at most 2 .  Then I'll briefly explain a story that leverages this fact to detect interesting invariant sets.   The results I'll state are all joint work with Rohil Prasad, though the circle of ideas has a long history which I'll also survey.