Localization and flexibilization in symplectic geometry

Localization is an important construction in algebra and topology that allows one to study global phenomena a single prime at a time. Flexibilization is an operation in symplectic topology introduced by Cieliebak and Eliashberg that makes any two symplectic manifolds that are diffeomorphic (plus a bit of tangent bundle data) become symplectomorphic. In this talk, I will explain that it is fruitful to view flexibilization as a localization (at the 'prime' zero). I will also give examples of new localization functors of symplectic manifolds (up to stabilization and subcriticals), which interpolate between flexible and rigid symplectic geometry and can be viewed as symplectic analogs of topological localization of Sullivan, Quillen, and Bousfield.