Most closed 4-manifolds don't admit symplectic forms, but they admit "near-symplectic" 2-forms that are symplectic away from some circles. I will introduce them to probe homotopy 4-spheres, by counting certain Riemann surfaces in the complement of these circles. The dream is that these will be sensitive enough to distinguish potentially exotic 4-spheres... but maybe it's a nightmare. Speaker(s): Chris Gerig (SCGP, Stony Brook)