Flexibility in symplectic and Weinstein geometry

A Stein manifold is a complex manifold admitting a proper holomorphic embedding into C^n. A number of theorems due to Oka, Cartan, and Grauert (among others) show that Stein manifolds have geometric flexibility properties, compared to their compact/projective complex counterparts. A culmination of this is the work of Cieliebak-Eliashberg, which states that Stein manifolds can be completely classified (up to deformation) via their underlying symplectic geometry (called a Weinstein structure). While this Stein=Weinstein equivalence holds for all Stein manifolds, the relations between Weinstein manifolds and contact geometry gives rise to a "deeper level" of h-principle flexibility, which only holds in complex dimension >2 and furthermore only holds for a subclass of Weinstein manifolds. These techniques allow us to apply techniques from high dimensional smooth topology to prove theorems about symplectic and Stein manifolds, which have been used for applications such as classification theorems and constructions of exotic submanifolds.