Geometry Seminar

A major theme in symplectic geometry is understanding rigidity phenomena of symplecitc objects. In this talk I will explain a newly discovered rigidity phenomena concerning (codimension 2) symplectic submanifolds loosely summarized as "symplectic submanifolds don't like to bend too quickly." This is made precise by the notion of anchored symplectic embeddings: given two four-dimensional symplectic manifolds, together with knots in their boundaries, we define an “anchored symplectic embedding” to be a symplectic embedding, together with a two-dimensional symplectic cobordism between the knots (in the four-dimensional cobordism determined by the embedding). Using pseudoholmorphic curves, we find many examples where ordinarily symplectic embeddings exist, but they cannot be upgraded to anchored symplectic embeddings unless one enlarges the target domain. This is joint work with Michael Hutchings, Agniva Roy and Morgan Weiler.