Generalized complex geometry (introduced by Hitchin and Gualtieri in the early 2000's) interpolates between ordinary complex and symplectic geometry. Stable generalized complex manifolds (first introduced by Cavalcanti and Gualtieri in 2015) provide examples of generalized complex manifolds which admit neither a symplectic nor a complex structure. Their generalized complex structure is, up to gauge equivalence, fully determined by a Poisson structure which is symplectic everywhere except on a real codimension 2 submanifold, a so-called elliptic symplectic form. Like log symplectic structures, these are examples of Poisson structures which can be described in terms of symplectic forms for a Lie algebroid not isomorphic to the tangent bundle; Poisson structures of this type have been widely studied in recent years. I will describe how to generalise a range of symplectic techniques to elliptic symplectic geometry, and describe their natural generically Lagrangian submanifolds, in particular a new type called Lagrangian brane with boundary. I will discuss small deformations of select classes of submanifolds.

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