Abstract

Generalised complex geometry (introduced by Hitchin and Gualtieri in the early 2000’s) interpolates between ordinary complex and symplectic geometry. Stable generalised complex manifolds (first introduced by Cavalcanti and Gualtieri in 2015) provide a class of examples of generalised complex manifolds that admits neither a symplectic nor a complex structure. Their generalised complex structure is, up to gauge equivalence, fully determined by a Poisson structure which is symplectic everywhere except on a real codimension 2 submanifold. I will give an introduction on how to apply symplectic techniques to this class of manifolds, and their natural submanifolds, generalised complex branes, in particular a new class of Lagrangian branes with boundary, and outline how we hope to use these to define a Fukaya category for certain types of stable generalised complex manifold.