The quantitative nature of symplectic cohomology

Symplectic cohomology has been extensively studied on exact symplectic manifolds. How far does our intuition stretch when we move to non-exact cases? We present a tool – completed symplectic cohomology – that has interesting properties on non-exact manifolds. We exhibit its behavior on semi-monotone negative line bundles, where mirror symmetry coarsely guides our computations. We end with a discussion on a conjectural dynamical application.