Deformation problems in Poisson and generalized complex geometry

In geometry, it is interesting to consider not just a specific geometric structure on a manifold, but the space of geometric structures of that type. We often consider this space modulo a natural equivalence relation. For example, we can look at symplectic structures modulo isotopies. Global statements are in many cases too much to ask for, which is why we consider small deformations of a structure, giving a description of nearby structures. In this proposal, the structures of interest are generalized complex (GC) and Poisson structures, which are used in physics in particular in mirror symmetry and string theory. We will use existing methods to study specific families of deformations (e.g. deformations subject to a constant rank constraint) and expand and build upon them to use them in this more general context. We will find algebraic structures that describe these deformations, study if infinitesimal transformations can be extended to a smooth 1-parameter family, find a geometric way to express the gauge equivalence and compare deformations of GC structures and their underlying Poisson structure. Furthermore, we will also consider deformations of isotropic involutive distributions in a fixed symplectic framework, and deformation of Poisson structures in context of a fixed involutive distribution.