Subflexible symplectic manifolds

After recalling some recent developments in symplectic flexibility, I will introduce a class of open symplectic manifolds, called "subflexible", which are not flexible but become so after attaching some Weinstein handles. For example, the standard symplectic ball has a Weinstein subdomain with nontrivial symplectic topology. These are exotic symplectic manifolds with vanishing symplectic cohomology. I will explain how to study them using a deformed version of symplectic cohomology, and how this invariant can computed using the machinery of Fukaya categories and Lefschetz fibrations. This is partly based on joint work with Emmy Murphy.